{"http:\/\/dx.doi.org\/10.14288\/1.0354251":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Mathematics, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Hutchcroft, Thomas","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2017-08-14T17:07:53Z","type":"literal","lang":"en"},{"value":"2017","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Doctor of Philosophy - PhD","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"We prove several theorems concerning random walks, harmonic functions, percolation, uniform spanning forests, and circle packing, often in combination with each other. We study these models primarily on planar graphs, on transitive graphs, and on unimodular random rooted graphs, although some of our results hold for more general classes of graphs. Broadly speaking, we are interested in the interplay between the geometry of a graph and the behaviour of probabilistic processes on that graph. Material taken from a total of nine papers is included. We have also included an extended introduction explaining the background and context to these papers.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/62595?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"Discrete probability and the geometryof graphsbyThomas HutchcroftB.A., University of Cambridge, 2012M.Math., University of Cambridge, 2013M.A., University of Cambridge, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c\u00a9 Thomas Hutchcroft 2017AbstractWe prove several theorems concerning random walks, harmonic functions, percolation, uniformspanning forests, and circle packing, often in combination with each other. We study these mod-els primarily on planar graphs, on transitive graphs, and on unimodular random rooted graphs,although some of our results hold for more general classes of graphs. Broadly speaking, we areinterested in the interplay between the geometry of a graph and the behaviour of probabilisticprocesses on that graph. Material taken from a total of nine papers is included. We have alsoincluded an extended introduction explaining the background and context to these papers.iiLay SummaryThere is a deep and well-developed connection between the geometry of a space and the wayprobabilistic processes behave on that space. A classical example is given by a heavy particle (e.g.a mote of dust) that moves randomly through space due to its bumping into a large number oflight particles (e.g., air molecules): The particle will move much faster in negatively curved spacesthan in flat space. In this thesis we continue to develop this connection in a discrete setting, whereour spaces are modeled by graphs. We prove several new results connecting the geometry of agraph to the behaviour of random processes on it. We are particularly interested in random walk,a discrete version of the random motion mentioned above, percolation, which models a randomporous material, and the uniform spanning forest, a probability model that is closely connected tothe theory of electrical networks.iiiPreface\u2022 Chapter 2 is adapted from the paper Critical percolation on any quasi-transitive graph ofexponential growth has no infinite clusters [129], of which I am the only author. This paperwas published in Comptes Rendus Mathematique in 2016.\u2022 Chapter 3 is adapted from the paper Collisions of random walks in reversible random graphs[133], by Yuval Peres and myself. Research was conducted as an equal collaboration, while Idid most of the writing. This paper was published in Electronic Communications in Proba-bility in 2015.\u2022 Chapter 4 is adapted from the paper Wired cycle-breaking dynamics for uniform spanningforests [127], of which I am the only author. This paper was accepted for publication in TheAnnals of Probability in 2015.\u2022 Chapter 5 is adapted from the preprint Interlacements and the wired uniform spanning forest[128], of which I am the only author. This paper was accepted for publication in The Annalsof Probability in 2017.\u2022 Chapter 6 is adapted from the paper Indistinguishability of trees in uniform spanning forests[131], by Asaf Nachmias and myself. Research was conducted as an equal collaboration, whileI did most of the writing. The paper was accepted for publication in Probability Theory andRelated Fields in 2016.\u2022 Chapter 7 is adapted from the paper Unimodular hyperbolic triangulations: circle packing andrandom walk [21], by Omer Angel, Asaf Nachmias, Gourab Ray, and myself. Research wasconducted as an equal collaboration, while I did most of the writing. The paper appeared inInventiones Mathematicae in 2016.\u2022 Chapter 8 is adapted from the preprint Boundaries of planar graphs: a unified approach [132],by Yuval Peres and myself. Research was conducted as an equal collaboration, while I didmost of the writing. The paper was accepted for publication in The Electronic Journal ofProbability in 2017.\u2022 Chapter 9 is adapted from the preprint Hyperbolic and parabolic unimodular random maps[20], by Omer Angel, Asaf Nachmias, Gourab Ray, and myself. Research was conducted asan equal collaboration, while I did most of the writing. The preprint has been submitted forpublication.ivPreface\u2022 Chapter 10 is adapted from the preprint Uniform spanning forests of planar graphs [130], byAsaf Nachmias and myself. Research was conducted as an equal collaboration, while I didmost of the writing. The preprint has been submitted for publication.\u2022 Chapter 1, the introduction, contains small amounts of the material from all the other chap-ters.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 What is this thesis about? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Random walks on graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 The Nash-Williams Criterion and electrical networks . . . . . . . . . . . . . 21.2.2 The isoperimetric approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Bounded harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.4 The Poisson boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.5 Infinite electrical networks and harmonic Dirichlet functions . . . . . . . . . 121.3 Circle packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.1 Planar graphs and maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.2 Circle packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.3 Harmonic Dirichlet functions on planar graphs . . . . . . . . . . . . . . . . . 211.3.4 Double circle packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.5 Circle packing and the degree . . . . . . . . . . . . . . . . . . . . . . . . . . 251.4 Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.4.1 The number of infinite clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 281.4.2 The geometry of infinite clusters . . . . . . . . . . . . . . . . . . . . . . . . . 311.5 Uniform spanning forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.5.1 Uniform spanning trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.5.2 Sampling using random walks . . . . . . . . . . . . . . . . . . . . . . . . . . 34viTable of Contents1.5.3 Uniform spanning forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.5.4 The number of trees in the wired forest . . . . . . . . . . . . . . . . . . . . . 361.5.5 Geometry of trees in the wired forest . . . . . . . . . . . . . . . . . . . . . . 381.5.6 The interlacement Aldous-Broder algorithm . . . . . . . . . . . . . . . . . . 411.5.7 Indistinguishability of trees and the geometry of trees in the free forest . . . 431.5.8 Uniform spanning forests of planar graphs . . . . . . . . . . . . . . . . . . . 451.5.9 USFs of multiply-connected planar maps . . . . . . . . . . . . . . . . . . . . 461.6 Unimodular random graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.6.1 Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491.6.2 Invariant amenability and nonamenability . . . . . . . . . . . . . . . . . . . 501.6.3 Unimodular random planar maps . . . . . . . . . . . . . . . . . . . . . . . . 511.6.4 The dichotomy theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.6.5 Boundary theory of unimodular random triangulations . . . . . . . . . . . . 561.7 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57I Two Short Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 Critical percolation on any quasi-transitive graph of exponential growth has noinfinite clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Collisions of random walks in reversible random graphs . . . . . . . . . . . . . . 643.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 Proof of Theorem 3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.3.1 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.3.2 Continuous-time random walk . . . . . . . . . . . . . . . . . . . . . . . . . . 68II Uniform Spanning Forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 Wired cycle-breaking dynamics for uniform spanning forests . . . . . . . . . . . 714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.1.1 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 The wired uniform spanning forest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3 Wired cycle-breaking dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.1 Update-tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.4 Proof of Theorem 4.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79viiTable of Contents4.5 Reversible random networks and the proof of Theorem 4.1.1 . . . . . . . . . . . . . 815 Interlacements and the wired uniform spanning forest . . . . . . . . . . . . . . . 845.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.2.1 Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.2.2 Excessive ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2.3 Ends in unimodular random rooted graphs . . . . . . . . . . . . . . . . . . . 895.3 Background and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3.1 Uniform spanning forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3.2 The space of trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3.3 The interlacement process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.3.4 Hitting probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.4 Interlacement Aldous-Broder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5 Proof of Theorems 5.2.1, 5.2.2 and 5.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . 985.5.1 Unimodular random rooted graphs . . . . . . . . . . . . . . . . . . . . . . . 1015.5.2 Excessive ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.6 Ends and rough isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.7 Closing discussion and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.7.1 The FMSF of the interlacement ordering . . . . . . . . . . . . . . . . . . . . 1055.7.2 Exceptional times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.7.3 Excessive ends via update tolerance . . . . . . . . . . . . . . . . . . . . . . . 1095.7.4 Ends in uniformly transient networks . . . . . . . . . . . . . . . . . . . . . . 1096 Indistinguishability of trees in uniform spanning forests . . . . . . . . . . . . . . 1106.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.1.1 About the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.1.2 Background and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.1.3 Component properties and indistinguishability on unimodular random rootednetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.2 Indistinguishability of FUSF components . . . . . . . . . . . . . . . . . . . . . . . . 1226.2.1 Cycle breaking in the FUSF . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.2.2 All FUSF components are transient and infinitely-ended . . . . . . . . . . . 1246.2.3 Pivotal edges for the FUSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2.4 Proof of Theorem 6.1.9 for the FUSF . . . . . . . . . . . . . . . . . . . . . . 1286.3 The FUSF is either connected or has infinitely many components . . . . . . . . . . 1316.4 Indistinguishability of WUSF components . . . . . . . . . . . . . . . . . . . . . . . . 1346.4.1 Indistinguishability of WUSF components by tail properties. . . . . . . . . . 1346.4.2 Indistinguishability of WUSF components by non-tail properties. . . . . . . 139viiiTable of ContentsIII Circle Packing and Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 1497 Unimodular hyperbolic triangulations: circle packing and random walk . . . . 1507.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1547.3 Background and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.3.1 Unimodular random graphs and maps . . . . . . . . . . . . . . . . . . . . . . 1567.3.2 Random walk, reversibility and ergodicity . . . . . . . . . . . . . . . . . . . 1577.3.3 Invariant amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.3.4 Circle packings and vertex extremal length . . . . . . . . . . . . . . . . . . . 1627.4 Characterisation of the CP type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657.4.1 Completing the proof of the Benjamini-Schramm Theorem . . . . . . . . . . 1667.5 Boundary theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687.5.1 Convergence to the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . 1697.5.2 Full support and non-atomicity of the exit measure . . . . . . . . . . . . . . 1717.5.3 The unit circle is the Poisson boundary . . . . . . . . . . . . . . . . . . . . . 1747.6 Hyperbolic speed and decay of radii . . . . . . . . . . . . . . . . . . . . . . . . . . . 1787.7 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807.8 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1828 Boundaries of planar graphs: a unified approach . . . . . . . . . . . . . . . . . . . 1848.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1848.1.1 Circle packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.1.2 Robustness under rough isometries . . . . . . . . . . . . . . . . . . . . . . . 1878.1.3 The Martin boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1888.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1908.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1908.2.2 Embeddings of planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 1908.2.3 Square tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1908.3 The Poisson boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.3.1 Proof of Theorem 8.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1968.4 Identification of the Martin boundary . . . . . . . . . . . . . . . . . . . . . . . . . . 2019 Hyperbolic and parabolic unimodular random maps . . . . . . . . . . . . . . . . 2059.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2059.1.1 The Dichotomy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2079.1.2 Unimodular planar maps are sofic . . . . . . . . . . . . . . . . . . . . . . . . 2119.2 Unimodular maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2119.2.1 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2119.2.2 Unimodularity and the mass transport principle . . . . . . . . . . . . . . . . 213ixTable of Contents9.2.3 Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2169.2.4 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2169.2.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2179.3 Percolations and invariant amenability . . . . . . . . . . . . . . . . . . . . . . . . . 2189.3.1 Markings and percolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2189.3.2 Amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2199.3.3 Hyperfiniteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2209.3.4 Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2219.3.5 Unimodular couplings and soficity . . . . . . . . . . . . . . . . . . . . . . . . 2229.3.6 Vertex extremal length and recurrence of subgraphs . . . . . . . . . . . . . . 2269.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2279.4.1 Curvature of submaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2309.4.2 Invariance of the curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2339.5 Spanning forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2379.5.1 Uniform spanning forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2379.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2409.5.3 Proof of Theorem 9.5.13 and Theorem 9.5.16 . . . . . . . . . . . . . . . . . . 2429.5.4 Finite expected degree is needed . . . . . . . . . . . . . . . . . . . . . . . . . 2459.5.5 Percolation and minimal spanning forests . . . . . . . . . . . . . . . . . . . . 2469.5.6 Expected degree formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2489.6 The conformal type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2509.7 Multiply-connected and non-planar maps . . . . . . . . . . . . . . . . . . . . . . . . 2579.7.1 The topology of unimodular random rooted maps. . . . . . . . . . . . . . . . 2579.7.2 Theorem 9.1.1 in the multiply-connected planar case . . . . . . . . . . . . . 2619.8 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2639.8.1 Rates of escape of the random walk . . . . . . . . . . . . . . . . . . . . . . . 2639.8.2 Positive harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 26410 Uniform spanning forests of planar graphs . . . . . . . . . . . . . . . . . . . . . . . 26510.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26510.1.1 Universal USF exponents via circle packing . . . . . . . . . . . . . . . . . . . 26710.2 Background and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27110.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27110.2.2 Uniform Spanning Forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27110.2.3 Random walk, effective resistances . . . . . . . . . . . . . . . . . . . . . . . . 27410.2.4 Plane Graphs and their USFs . . . . . . . . . . . . . . . . . . . . . . . . . . 27710.2.5 Circle packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27810.3 Connectivity of the FUSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27910.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27910.3.2 Proof of Theorem 10.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282xTable of Contents10.3.3 The FUSF is connected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28610.4 Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28610.4.1 The Ring Lemma for double circle packings . . . . . . . . . . . . . . . . . . 28610.4.2 Good embeddings of planar graphs . . . . . . . . . . . . . . . . . . . . . . . 28810.5 Critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28910.5.1 The ring lemma for double circle packings . . . . . . . . . . . . . . . . . . . 28910.5.2 Good embeddings of planar graphs . . . . . . . . . . . . . . . . . . . . . . . 29110.5.3 Preliminary estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29310.5.4 Wilson\u2019s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29810.5.5 Proof of Theorems 10.1.3, 10.1.4 and 10.1.5 . . . . . . . . . . . . . . . . . . . 29910.5.6 The uniformly transient case . . . . . . . . . . . . . . . . . . . . . . . . . . . 30510.6 Closing remarks and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 30710.6.1 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30710.6.2 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309xiList of Figures1.1 Z2 and the 3-regular tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Free and wired boundary conditions on a 6\u00d7 6 square in Z2. . . . . . . . . . . . . . 131.3 Different maps with the same underlying graph. . . . . . . . . . . . . . . . . . . . . . 151.4 A finite planar map and its dual. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Circle packings of the 6-regular and 7-regular triangulations. . . . . . . . . . . . . . 171.6 A polyhedral planar map and its double circle packing. . . . . . . . . . . . . . . . . . 241.7 The grandparent graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.8 Two bounded degree, simple, proper plane triangulations for which the graph dis-tance is not comparable to the hyperbolic distance. . . . . . . . . . . . . . . . . . . . 461.9 Circle packings in multiply-connected circle domains. . . . . . . . . . . . . . . . . . . 471.10 The logical structure for the proof of Theorem 9.1.1 in the simply connected case. . 554.1 Updating an oriented spanning forest . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2 Illustration of the proof of Theorem 4.1.3. . . . . . . . . . . . . . . . . . . . . . . . . 805.1 A schematic illustration of the proof of Lemma 5.5.1. . . . . . . . . . . . . . . . . . . 995.2 An illustration of the graph G32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.1 Illustration of the proof of Lemma 6.2.6. . . . . . . . . . . . . . . . . . . . . . . . . . 1266.2 Illustration of the forest FR and the event CR,n. . . . . . . . . . . . . . . . . . . . . . 1367.1 A circle packing of a random hyperbolic triangulation. . . . . . . . . . . . . . . . . . 1507.2 Geodesic embeddings induced by circle packing. . . . . . . . . . . . . . . . . . . . . . 1647.3 An illustration of the mass transport used to show the exit measure has full support. 1747.4 Illustration of the proof of item 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1777.5 Extracting the core of a non-simple map. . . . . . . . . . . . . . . . . . . . . . . . . 1818.1 The square tiling and the circle packing of the 7-regular hyperbolic triangulation. . . 1868.2 Illustration of the proof of Theorem 8.1.1. . . . . . . . . . . . . . . . . . . . . . . . . 1959.1 The logical structure for the proof of Theorem 9.1.1 in the simply connected case. . 2099.2 The numbers of the theorems, propositions, lemmas and corollaries forming theindividual implications used to prove Theorem 9.1.1 in the simply connected case. . 2109.3 Different maps with the same underlying graph. . . . . . . . . . . . . . . . . . . . . . 213xiiList of Figures9.4 Examples of verices with positive, zero, and negative curvature. . . . . . . . . . . . . 2299.5 Illustration of the map M1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2329.6 The maps T4 (left) and M4 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2459.7 The covering of Ue by discs used in the proof of Lemma 9.6.2. . . . . . . . . . . . . . 2559.8 Possible topologies of a unimodular random map. . . . . . . . . . . . . . . . . . . . . 25710.1 A simple, 3-connected, finite plane graph and its double circle packing. . . . . . . . . 26810.2 Two bounded degree, simple, proper plane triangulations for which the graph dis-tance is not comparable to the hyperbolic distance. . . . . . . . . . . . . . . . . . . . 26910.3 Illustration of the proof of Theorem 10.1.2 in the case that T is CP hyperbolic. Left: On the eventBe\u03b5 , the paths \u03b7x and \u03b7y split V\u03b5 into two pieces, L and R. Right: We define a random set containinga path (solid blue) from \u03b7x to \u03b7y \u222a {\u221e} in G \\ C using a random circle (dashed blue). Here wesee two examples, one in which the path ends at \u03b7y, and the other in which the path ends at theboundary (i.e., at infinity). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28410.4 Proof of the Double Ring Lemma. If two dual circles are close but do not touch, there must bemany primal circles contained in the crevasse between them. This forces the two dual circles to eachhave large degree. The right-hand figure is a magnification of the left-hand figure. . . . . . . . . 28710.5 Proof of the Double Ring Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29110.6 The double circle packing of N\u00d7 Z4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 307xiiiAcknowledgementsFirst and foremost, I thank my advisors Omer Angel and Asaf Nachmias for guiding me throughthe early stages of my research, introducing me to the field, sharing problems and ideas, readingdrafts, and providing me with many opportunities to meet and collaborate with others. I wouldalso like to thank my Microsoft Research mentors Ander Holroyd and Yuval Peres for severalvery enjoyable and productive summers in Redmond, which were hugely beneficial to me. I alsothank Microsoft Research for supporting me financially in the final year of my doctoral studiesby awarding me the Microsoft Research PhD Fellowship. I also thank my collaborators OmerAngel, Nicolas Curien, Ander Holroyd, Avi Levy, Asaf Nachmias, Yuval Peres, and Gourab Rayfor all the stimulating conversations and discoveries we have had together. I hope for many moreto come. I also thank Russ Lyons and Yuval Peres for writing their wonderful book [173], whichI refer to at least once a week, and Itai Benjamini for sending me many interesting questions. Ialso thank everyone who has been a member of the UBC probability group during my time here,especially my fellow students and visiting students: Owen Daniel, Mallory Flynn, Spencer Frei,Tyler Helmuth, Sara\u00b4\u0131 Herna\u00b4ndez-Torres, Jieliang Hong, Thomas Hughes, Tim Jaschek, MatthiasKlo\u00a8ckner, Brett Kolesnik, Hongliang Lu, Guillermo Mart\u00b4\u0131nez Dibene, Gourab Ray, Saif Syed, AlexTomberg, Ben Wallace, Zichun Ye, Jing Yu, and Qingsan Zhu; I could not have asked for a bettergroup of colleagues, peers, and friends.Finally, although I never had the privilege to meet him, I would like to thank Oded Schramm,whose mathematical work, ingenuity, and vision has been a constant inspiration to me. His influenceis felt throughout essentially every subject touched on in this thesis, and it is no coincidence that hisname appears more than 250 times in this thesis, an average of once every 1.4 pages. Throughoutmy work, and especially in my joint work with Asaf, whenever we were unsure how best to approachsomething, it became routine to ask ourselves \u201cWhat would Oded do?\u201d Reflecting on this questionalways helped us cut right to the heart of the matter at hand, even if we could not always live upto the aspiration of doing things quite as elegantly as he would have.xivTo Meghan, my family and friends.xvChapter 1Introduction1.1 What is this thesis about?This thesis is a collection of papers about discrete probability on infinite graphs. The central themecan be summarised by the following question.What does the geometry of a graph tell us about the behaviour of probabilistic processes on thatgraph? Conversely, what does the behaviour of probabilistic processes on a graph tell us about theunderlying geometry?This question has formed an important strand of thought through modern probability theorysince its inception in the early 20th century. In this introduction, we will take a quick tour throughthe highlights of this tradition. Some of the results of this thesis will be presented as we go, althoughmost of our own contributions will be presented towards the end. Our aim is to present some ofthe main ideas underlying the field, and our own work, in a way that is hopefully accessible to ageneral mathematical audience, without getting too much into technical details.1.2 Random walks on graphsA graph G = (V,E) is a set of vertices V and a set of edges E. We think of vertices as points,and of edges as curves, each of which is either a loop starting and ending at the same point, orelse joins two points together. (More formally, we have an incidence relation which assigns toeach edge either one or two vertices, which are the endpoints of the edge.) For simplicity, we shallconsider in this section only simple graphs, that have no self-loops and at most one edge betweeneach two vertices. A graph is connected if we can pass from any vertex to any other vertex bymoving across a finite sequence of edges. The degree of a vertex is the number of edges incidentto it. We will assume throughout that our graphs have countably many vertices and edges, and arelocally finite, meaning that all the degrees are finite.The random walk on a graph is the process that, at each time n \u2265 0, chooses an edge uniformlyat random from among those incident to its current location, independently from everything it hasdone previously, and then moves to the vertex at the other endpoint of that edge. When we havesome graph in mind, we write pn(u, v) for the probability that a random walk started at the vertexu of the graph is at the vertex v at time n.11.2. Random walks on graphsPerhaps the first important work in the tradition we describe was George Po\u00b4lya\u2019s 1921 paper[196], in which he proved the following theorem. We say that a graph is recurrent if the randomwalk on the graph returns to its starting location almost surely (i.e., with probability one \u2013 we willhenceforth abbreviate this to a.s.), and transient if it has a positive probability never to return. Itis easy to see that the graph is recurrent if and only if the random walk returns to its starting pointinfinitely many times a.s., and transient if and only if it returns to its starting point only finitelymany times a.s. The hypercubic lattice Zd is the graph whose vertices are d-tuples of integers,and where two vertices are connected by an edge if they differ by exactly one in one coordinate andhave the same value in all other coordinates.Theorem (Po\u00b4lya 1921). The d-dimensional hypercubic lattice Zd is recurrent if d \u2264 2, and transientif d \u2265 3.This theorem can be summarised by the following famous aphorism, attributed to Shizuo Kaku-tani.A drunk man will find his way home,but a drunk bird may get lost forever.Po\u00b4lya\u2019s proof of his theorem was computational in nature. By computing the moment generatingfunction of the location of the random walk at time n, Po\u00b4lya showed that the random walk on Zdsatisfiesp2n(0, 0) \u2248 n\u2212d\/2. (1.2.1)On the other hand, it is easy to see that, for parity reasons, pn(0, 0) = 0 for every odd n. Thisestimate easily yields the theorem, since it is not hard to prove that the random walk on a graphis recurrent if and only if \u2211n\u22651pn(v, v) =\u221efor some (and hence every) vertex v of G \u2013 note that this sum is exactly the expected number oftimes the walk returns to v.A drawback to this approach is that it relies heavily upon the symmetries of the lattice, and isnot very robust. For example, the argument breaks down rather seriously if we take some arbitraryset of edges of Zd and replace each of the edges in this set with two edges in parallel with thesame endpoints. Besides this, it does not give us much insight into the geometric reasons that Z2is recurrent but Z3 is transient.1.2.1 The Nash-Williams Criterion and electrical networksThe first major step towards a geometric understanding of the recurrence\/transience problem wasthe Nash-Williams Criterion [186], which gives a sufficient condition for recurrence in terms ofthe isoperimetry of the graph. For each set of vertices W \u2286 V in a graph G, we write |W | =\u2211v\u2208W deg(v). We say that a set W is a cutset for v if every path from v to infinity must passthrough W .21.2. Random walks on graphsTheorem (Nash-Williams 1959). Let G be an infinite graph, let v be a vertex of G, and supposethat there exists a disjoint sequence of finite cutsets Wi for v such that\u2211i\u22651|Wi|\u22121 =\u221e.Then G is recurrent.This theorem easily recovers Po\u00b4lya\u2019s result in dimensions 1 and 2 by, for example, using thecutsets Wi = {\u2212i, i} in Z and the cutsets Wi = {(x, y) : |x|+ |y| = i} in Z2.Nash-Williams\u2019s proof was based on the correspondence between random walks and electricalnetworks, a tool which will be of great importance throughout this thesis. A detailed introduction,covering everything we discuss in this subsection, can be found in [173, Chapters 2 and 3]. We canthink of a graph as an electrical network in which each edge contains a unit resistor1. Supposethat we have such a network arising from a finite graph, and we attach the two ends of a batteryto two disjoint sets of vertices, A and B. Doing this will cause a current to flow through thenetwork. Mathematically, the current flow is a function \u03b8 from the set of oriented edges ofthe graph (that is, edges e together with a choice of one of the endpoints as the tail of the edge,denoted e\u2212, and the other as the head of the edge, denoted e+) that is antisymmetric in the sensethat \u03b8(e) = \u2212\u03b8(\u2212e) for every oriented edge e of G, where \u2212e denotes the reversal of e. Given anantisymmetric edge function \u03b8 and a vertex v, we write \u2207\u2217\u03b8(v) for the sum \u2211e\u2212=v \u03b8(e).Kirchoff\u2019s laws of electrical networks [154] are as follows:1. (The node law) For every vertex v of G, the sum \u2207\u2217\u03b8(v) of the currents along the orientededges emanating from v is positive if v \u2208 A, negative if v \u2208 B, and zero if v is not in A or B.2. (The cycle law) If e1, e2, . . . , en form a cycle, a path from A to itself, or a path from B toitself, then\u2211ni=1 \u03b8(ei) = 0.In general, we call an antisymmetric edge function satisfying the node law a flow from A toB. It is not hard to prove that, in a finite network, the antisymmetry condition and Kirchoff\u2019slaws determine the current flow up to a positive multiplicative constant. The strength of a flow isdefined to be the sum\u2211v\u2208A\u2207\u2217\u03b8(v), and we say that a flow is a unit flow if it has strength one.We define the unit current flow from A to B to be the unique current flow from A to B that hasstrength 1.Given a function f : V \u2192 R, we define the gradient \u2207f to be the antisymmetric edge function\u2207f(e) = f(e+)\u2212 f(e\u2212).1We can also consider networks with arbitrary resistances, for which there is a corresponding random walk thatchooses which edge to travel along weighted by its conductance, that is, the inverse of its resistance.31.2. Random walks on graphsThe basic connection between electrical network theory and random walks is that voltages (i.e.,antiderivatives of currents) are hitting probabilities. Given a graph G, we write Pv for the law ofthe random walk started at v. For each set of vertices A, we write \u03c4A for the first time that therandom walk hits A.Proposition. Let f(v) = Pv(\u03c4B < \u03c4A). Then the gradient \u2207f is a current flow from A to B.This is all well and good, but so far the electrical theory has not given us a useful way ofcalculating anything. However, the theory becomes very useful once we introduce effective resis-tances. The energy of an antisymmetric edge function \u03b8, and the Dirichlet energy of a functionf : V \u2192 R, are defined byE(\u03b8) = 12\u2211e\u2208E\u2192\u03b8(e)2 and E(f) = 12\u2211e\u2208E\u2192(f(e+)\u2212 f(e\u2212))2= E(\u2207f).The effective resistance between two disjoint sets A and B in a finite graph is defined to be theenergy of the unit current flow from A to B, and is denoted Reff(A \u2194 B) (or Reff(A \u2194 B;G) ifthe graph we are considering is not clear). Effective resistances are related to hitting probabilitiesas follows. Write \u03c4+A for the first time that the random walk hits A after time zero.Proposition. Reff(A\u2194 B)\u22121 =\u2211v\u2208A deg(v)Pv(\u03c4B < \u03c4+A ).In particular, the reciprocal of the effective resistance (a.k.a. the effective conductance)between a single vertex a and a set B is exactly the multiple of deg(a) with the probability thatthe random walk started at a hits B before it returns to a.The reason this is so useful is that we have the following two variational principles for theeffective resistance.Theorem (Thomson\u2019s Principle).Reff(A\u2194 B) = inf{E(\u03b8) : \u03b8 is a flow from A to B with strength at least one}.Theorem (Dirichlet\u2019s Principle).Reff(A\u2194 B)\u22121 =inf{E(f) : f : V \u2192 R such that f(v) \u2264 0 for all v \u2208 A and f(v) \u2265 1 for all v \u2208 B}.These allow us to get an upper bound on the effective resistance by constructing a low energyflow, or to get lower bounds by exhibiting low energy functions. (There are also several other closelyrelated variational formulas for the effective resistance - the key words are extremal length andthe method of random paths.)Exercise. Prove both Thomson\u2019s and Dirichlet\u2019s principles. Hint: Differentiate the energy withrespect to \u03b8 or f as appropriate and find the critical points.41.2. Random walks on graphsNow suppose that G is an infinite graph and A is a finite set of vertices in G, and let \u3008Vn\u3009n\u22650be an exhaustion of G, that is, an increasing sequence of finite sets of vertices such that\u22c3n\u22650 Vn.We define \u2202V Vn to be the set of vertices of Vn that have a neighbour not in Vn, and let Gn be thesubgraph of G induced by Vn, that is, the graph that has vertex set Vn and has all the edges of Gthat have both endpoints in Vn. We defineReff(A\u2192\u221e) = limn\u2192\u221eReff(A\u2192 \u2202Vn;Gn),so thatReff(A\u2192\u221e)\u22121 = limn\u2192\u221e\u2211v\u2208Adeg(v)Pv(\u03c4\u2202Vn < \u03c4+A ) =\u2211v\u2208Adeg(v)Pv(\u03c4+A =\u221e).Thus, we have that a graph is recurrent if and only if the effective resistance from some (andhence every) vertex to infinity is infinite. Both Thomson\u2019s and Dirichlet\u2019s principles extend toyield variational formulas for the effective conductance to infinity, with appropriate modifications:a flow from A to infinity is required to have \u2207\u2217\u03b8(v) positive if v \u2208 A and zero otherwise, while inthe Dirichlet principle we can take the infima over functions that are finitely supported (i.e. havef(v) = 0 for all but finitely many vertices v) and have f(v) \u2265 1 for all v \u2208 A.An important consequence of Thomson\u2019s principle is that the effective resistance is a monotonefunction of the edge set, a fact known as Rayleigh\u2019s monotonicity principle. In particular, thisimplies that every subgraph of a recurrent graph is recurrent. This is not obvious at all by directconsideration of the random walk.We are now ready to see how the Nash-Williams Criterion is proven. For each of the cutsetsWi, let Vi be the set of vertices u such that Wi is a cutset for u, and, for each n \u2265 1, define afinitely supported function fn with fn(v) = 1 byfn(u) =\u2211ni=1 |Wi|\u221211(u \u2208 Vi)\u2211ni=1 |Wi|\u22121.Since the gradients \u22071(u \u2208 Vi) have disjoint support, we can compute thatE(fn) =\u2211ni=1 |\u2202EVi| \u00b7 |Wi|\u22122(\u2211ni=1 |Wi|\u22121)2 \u2264\uf8eb\uf8ed n\u2211i=1|Wi|\u22121\uf8f6\uf8f8\u22121 .Thus, we haveReff(v \u2192\u221e) \u2265 E(fn)\u22121 \u2265n\u2211i=1|Wi|\u22121,and the claim follows since n was arbitrary.Exercise. Prove that Zd is transient for d \u2265 3 by exhibiting a flow from the origin to infinity thathas finite energy.51.2. Random walks on graphsFigure 1.1: Z2, left, is amenable, while the 3-regular tree, right, is nonamenable.1.2.2 The isoperimetric approachThe next milestone result in the theory was proven by Kesten, also in 1959. Nash-Williams gives usa good geometric criterion for recurrence, but what about transience? One rather extreme way fora graph to be transient, which turns out to have a simple geometric description, is for the returnprobabilities pn(v, v) to decay exponentially in n, meaning that there exists a constant \u03b1 < 1 andconstants Cv <\u221e such thatpn(v, v) \u2264 Cv\u03b1nfor every v \u2208 V and n \u2265 0.Kesten\u2019s theorem [151, 152] relates exponential decay of the return probabilities to the isoperime-try of the graph, that is, to the relationship between the sizes of sets in the graph to the sizes oftheir boundaries. We write |\u2202W | for the number of edges that have one endpoint in W and theother outside of W . A graph is said to be nonamenable if there exists a positive constant c > 0such that |\u2202W |\/|W | > c for every finite set W \u2282 V , and amenable otherwise. For example, Zd isamenable for every d \u2265 1 because\u2223\u2223\u2223[0, n]d\u2223\u2223\u2223 = nd but \u2223\u2223\u2223\u2202[0, n]d\u2223\u2223\u2223 \u2248 nd\u22121,so that |\u2202[0, n]d|\/|[0, n]d| \u2192 0 as n \u2192 \u221e. On the other hand, the d-regular tree2 is nonamenablefor every d \u2265 3: The average degree of a vertex in a finite tree with n vertices is easily seen to be2\u2212 2\/n \u2264 2, and it follows by an elementary calculation that for any finite subset W of a d-regulartree, we have|\u2202W | \u2265 (d\u2212 2)|W |.Theorem (Kesten 1959). Let G be a graph, and let v be a vertex of G. Then pn(v, v) decaysexponentially if and only if G is nonamenable.2A forest is a graph that does not contain any simple cycles. A tree is a connected forest. The d-regular tree isthe unique tree in which every vertex has degree d.61.2. Random walks on graphsKesten proved his theorem only for random walks on groups. The generalisation to arbitrarygraphs, as well as quantitative forms of the theorem, are due to many authors working in variouscontexts [11, 53, 65, 68, 80, 81, 101, 143, 224]. See [173, Chapter 6.10] for a detailed history. Thefact that the exponential decay of the return probabilities implies nonamenability is rather easy,the difficult part is the other direction.Nonamenability has many guises, and the amenability\/nonamenability dichotomy is fundamen-tal not only to the study of random walks, but also of percolation, uniform spanning forests, andmany other topics that we will not touch on here, such as the ergodic theory of group actions.Indeed, the Wikipedia page on amenable groups lists nine different properties of groups that areequivalent to amenability.We might hope for a nice isoperimetric criterion for transience, along the lines of Kesten\u2019sTheorem and the Nash-Williams Criterion, which also has an isoperimetric character. Note thatattaching an infinite path to a graph will not affect whether or not the graph is transient, but willcause the graph to have very bad isoperimetry, suggesting that the best we can hope for in generalis a sufficient condition for transience in terms of isoperimetry.We say that a graph G satisfies an \u03c6(t)-isoperimetric inequality if there exists a positiveconstant c > 0 such that |\u2202W |\/\u03c6(|W |) > c for every finite set W \u2282 V . In particular, G isnonamenable if and only if it satisfies a t-isoperimetric inequality. Similarly, we say that G satisfiesan anchored \u03c6(t)-isoperimetric inequality if there exists a positive constant c > 0 such that|\u2202W |\/\u03c6(|W |) > c for every finite connected set W \u2282 V containing some fixed vertex v (the choiceof which does not matter). The following theorem was proven by Thomassen [220]. The versionwe state here appears in the textbook of Lyons and Peres [173], and is adapted from a theorem ofLyons, Morris, and Schramm [170]. Similar theorems have also been obtained by He and Schramm[121], and Benjamini and Kozma [42].Theorem (Thomassen\u2019s Criterion). Suppose that G satisfies an anchored \u03c6(t)-isoperimetric in-equality for some increasing function \u03c6 such that\u221e\u2211n=1\u03c6(n)\u22122 <\u221e.Then G is transient.What if we are interested in the rate of decay of pn(u, u) (known in the jargon of the field as theon-diagonal heat kernel), rather than just the transience\/recurrence question? In particular,is there a geometric proof of Po\u00b4lya\u2019s estimate (1.2.1)? For d \u2265 2 an answer is provided by thefollowing results. We say that a graph satisfies a d-dimensional isoperimetric inequality if it satisfiesa t(d\u22121)\/d-isoperimetric inequality. Zd satisfies a d-dimensional isoperimetric inequality, so that wecan recover Po\u00b4lya\u2019s estimate from the following two theorems: See the textbooks [229] and [159]for history, proofs, and related theorems.71.2. Random walks on graphsTheorem. Let G be an infinite, bounded degree graph and let d \u2265 2. Then G satisfies a d-dimensional isoperimetric inequality if and only if there exists a constant C <\u221e such thatpn(u, u) \u2264 C n\u2212d\/2for every vertex u and n \u2265 1.Theorem. Let G be an infinite, bounded degree graph that satisfies a d-dimensional isoperimetricinequality for some d \u2265 2, and suppose that there exists a constant C <\u221e such that|B(u, r)| \u2264 C rdfor every vertex u and every r \u2265 1. Then there exists a constant c > 0 such thatpn(u, u) \u2265 cn\u2212d\/2for every vertex u and every n \u2265 1.It is possible to say much more. In particular, it is possible to recover the entire (off-diagonal)behaviour of the heat kernel pn(x, y) on Zd from geometric considerations. This understandinghas proven very important for developing a similar understanding for random walk on fractal-likegraphs such as the (pre-)Sierpinski gasket.1.2.3 Bounded harmonic functionsRecall that a function h on a graph is said to be harmonic if for each vertex u, the value of hat u is equal to the average value of h on the neighbours of u. The existence or nonexistence ofvarious types of non-constant harmonic functions (e.g. bounded, finite energy, positive, polynomialgrowth) on a graph is closely connected to the behaviour of the random walk on the graph.Perhaps the most important class of harmonic functions are the bounded harmonic functions.In probabilistic terms, bounded harmonic functions encode all possible \u2018behaviours at infinity\u2019 ofthe random walk that occur with positive probability. The correspondence, which goes back to thework of Blackwell [55], can be stated (more or less) precisely as follows: First, suppose we have a(measurable) set A of paths in G that is shift invariant in the sense that if we delete an initialsegment of any path in A , then the truncated path is also in A . We can define a function h on Gby setting h(v) to be the probability that if we start a random walk at v, then the resulting pathwill be in the set A . It is not hard to verify that, since A is shift invariant, the function h(v) isharmonic.More generally, we can obtain a bounded harmonic function on G from any (measurable)bounded shift invariant function on the set of paths in G, by taking the expectation of the functionapplied to the random walk. In fact, the function needs only be defined on some set that the ran-dom walk path is a.s. in, and two shift invariant functions will yield the same harmonic function if81.2. Random walks on graphsand only if they yield the same value on a.e. random walk path. It can be shown that in fact everybounded harmonic function on G arises this way. In particular, G admits non-constant boundedharmonic functions if and only if there is some shift-invariant set of paths A and a vertex v suchthat the probability that the random walk started at v is in the set A is strictly between 0 and 1(that is, if the invariant \u03c3-algebra is nontrivial). We say that a graph is Liouville if it does notadmit any non-constant bounded harmonic functions, and non-Liouville otherwise. See [173] fora detailed treatment of the theory of bounded harmonic functions and the Liouville property.For example, it can be shown that Zd is Liouville for every d \u2265 1 (see below). On the otherhand, if we attach together two copies of Z3 (with vertex sets {1} \u00d7 Z3 and {2} \u00d7 Z3) by a singleedge connecting their origins, then the resulting graph is non-Liouville, and in fact the vector spaceof bounded harmonic functions on this graph is equal to the linear span of the constant 1 functionand the functionh(x) = P(a random walk \u3008Xn\u3009n\u22650 started at v visits the set {1} \u00d7 Z3 infinitely often).In other words, which of the two copies of Z3 the random walk is eventually absorbed into is theonly non-trivial information there is about the random walk on this graph.How can we tell if a graph is Liouville or not? For transitive graphs, there is a nice char-acterisation, which follows in the special case of Cayley graphs from the work of Avez [26, 27],Derriennic [77], and Kaimanovich and Vershik [149, 226], and was generalised to transitive graphsby Kaimanovich and Woess [148]. Here, an automorphism of a (simple) graph G is a bijection\u03c6 : V \u2192 V such that \u03c6(u) is adjacent to \u03c6(v) if and only if u is adjacent to v, and a graph G istransitive if for every two vertices u and v of G, there is an automorphism of G sending u to v.Intuitively, a graph is transitive if it \u2018looks the same from every vertex\u2019.Theorem. Let G be a transitive graph. Then the following are equivalent.1. The random walk on G has positive speed, meaning thatlim infn\u2192\u221e1nd(X0, Xn) > 0almost surely when \u3008Xn\u3009n\u22650 is a random walk on G.2. The asymptotic entropylim infn\u2192\u221e1n\u2211v\u2208V\u2212pn(u, v) log pn(u, v) = lim infn\u2192\u221e1nEu[\u2212 log pn(X0, Xn)]is positive for every vertex u of G.3. G is non-Liouville.91.2. Random walks on graphs(In fact the limits infimum here can be replaced with limits: This follows from Kingman\u2019s sub-additive ergodic theorem for the speed, and Fekete\u2019s Lemma for the asymptotic entropy.) This the-orem allows us to immediately conclude that every transitive nonamenable graph is non-Liouville,since Kesten\u2019s theorem easily implies that both the speed and asymptotic entropy are positive forany bounded degree nonamenable graph. On the other hand, the theorem also implies that everytransitive graph that has subexponential growth, meaning thatlim infn\u2192\u221e1nlog |B(u, n)| = 0is Liouville, since an easy computation implies that\u2211v\u2208V\u2212pn(u, v) log pn(u, v) \u2264 log |B(u, n)|for every vertex u and n \u2265 0. Both of these conclusions fail for general bounded degree graphs.Unfortunately, there is still no good geometric understanding of the Liouville property, partic-ularly for the interesting case of amenable transitive graphs of exponential growth, some of whichare Liouville and some of which are not. In particular, it is a long-standing open problem to provethe widely believed claim that two Cayley graphs of the same group are either both Liouville orboth non-Liouville.The fact that item (1) implies item (2) in the previous theorem is a corollary of the Varopoulos-Carne inequality [66, 225], which is a beautiful theorem in its own right. (Its proof, due to Carne,is an extremely elegant piece of linear algebra involving an expansion of the n-step transition matrixPn in terms of Chebyshev polynomials of P ).Theorem (Varopoulos-Carne). Let G be a graph. Thenpn(u, v) \u2264 2\u221adeg(v)\/ deg(u) exp(\u2212d(u, v)2\/(2n)).Exercise. Use the Varopoulos-Carne Inequality to prove that the positivity of the speed and ofthe asymptotic entropy are equivalent for any bounded degree graph.A further nice consequence of the Varopoulos-Carne bound is that for any graph G which haspolynomial growth, meaning that there exist constants C, d <\u221e such that|B(u, n)| \u2264 C ndfor every vertex u and every n \u2265 0, we have thatlim supn\u2192\u221ed(X0, Xn)\u221an log(n)<\u221ealmost surely, so that graphs with polynomial growth are quite far away from having positive speed.101.2. Random walks on graphsExercise. Prove this claim.1.2.4 The Poisson boundaryIf we have a graph that we know to be non-Liouville, it is interesting to classify the space of allbounded harmonic functions in some geometric way. See [173, Chapter 14] for a detailed treatmentof the material covered in this subsection.The model situation is the classical Poisson integral formula, which states that for every boundedharmonic function u on the unit disc D, there exists a bounded measurable function f on \u2202D (uniqueup to almost-everywhere equivalence) such thatu(z) =12pi\u222b 2pi01\u2212 |z|2\u2223\u2223z \u2212 ei\u03b8\u2223\u22232 f(ei\u03b8) d\u03b8, (1.2.2)and moreover that any function u on D defined in this way is bounded and harmonic. The equation(1.2.2) therefore yields an isomorphism between the Banach spaces of bounded harmonic functionson the unit disc and bounded measurable functions on the unit circle, both with the uniform norm.Moreover, the right hand side of the expression is equal to the expectation of f(BT ), where B isBrownian motion started at z, and T is the first time B hits \u2202D.This example led Furstenberg, in his pioneering works [91\u201393], to define the notion of a Poissonboundary of a graph in the discrete setting. A compactification G of a graph G is a compact,metrisable topological space containing the vertex set of G as a dense, discrete subset. Given acompactification G, we write \u2202G = G \\ V . We say that the (boundary of the) compactification isa Poisson boundary of a transient graph G if the random walk on G converges in the compacti-fication a.s., and, for every bounded harmonic function h on G, there exists a bounded measurablefunction (unique up to almost everywhere equivalence) f : \u2202G\u2192 R such thath(v) = Ev f(limn\u2192\u221eXn)for every vertex v. (Note that the limit point of the walk is a shift invariant function of the walk,so that if we start with a bounded function f on the boundary then the function h defined as aboveis certainly bounded and harmonic.) For example, the one-point compactification is a Poissonboundary of a transient graph G if and only if G is Liouville. Probabilistically, a compactificationG of G is a Poisson boundary of G if the limit of a random walk in the compactification encapsulatesall of the walk\u2019s limiting behaviour.Although we have defined it topologically, the Poisson boundary is really a measure-theoreticnotion, and two compactification Poisson boundaries of the same graph need not be homeomorphic(this is in contrast to the Martin boundary, defined in terms of positive harmonic functions, whichis truly a topological object, see e.g. [201]).111.2. Random walks on graphsFor example, suppose that we have a transient tree. An end of the tree is an equivalenceclass of rays (i.e. infinite simple paths) in the tree, where two rays are equivalent if they have finitesymmetric difference. The ends compactification of the tree is a compact topological space whosepoints are the vertices and ends of the tree, and where a sequence of vertices \u3008vi\u3009i\u22650 converges toan end \u03be if, fixing some root vertex \u03c1 of the tree arbitrarily, the geodesics in the tree connecting\u03c1 to vi converge to a ray in the equivalence class of the end \u03be. In particular, a path converges toan end in the ends compactification of a tree if and only if the path is transient, i.e. visits eachvertex of the tree at most finitely often, and so the random walk on a tree converges a.s. in theends compactification if and only if the tree is transient. Now, if a function converges along atransient path, in a tree, it must also converge along any ray corresponding to the end that thatpath converges to. If h is a bounded harmonic function on a transient tree and X is a randomwalk started at \u03c1, then, by the Martingale convergence theorem, the limit of h along X exists a.s.,and it follows that h converges a.s. along the ray starting at \u03c1 that corresponds to the end that Xconverges to. This allows us to define a bounded function f on the boundary byf(\u03be) = limn\u2192\u221eh(vi)where \u3008vi\u3009i\u22650 is a ray corresponding to the end \u03be. We clearly have thath(v) = Ev f(limn\u2192\u221eXn),so that the ends compactification is a Poisson boundary of any transient tree.In general, it is not hard to show that a compactification Poisson boundary exists for anytransient graph. For example, the Martin boundary is also a Poisson boundary (but easier con-structions also exist). However, giving a geometric construction of the Poisson boundary, or showingthat some particular compactification is indeed a Poisson boundary, can be highly non-trivial. Inthe next section, we shall see that for bounded degree triangulations, there is a beautiful geometricconstruction of the Poisson boundary using circle packing.1.2.5 Infinite electrical networks and harmonic Dirichlet functionsIn Section 1.2.1, we saw how to define the effective resistance between two sets in a finite graph,and also the effective resistance from a finite set to infinity in an infinite graph. What about theeffective resistance between two finite sets in an infinite graph? It turns out that there is more thanone reasonable way to define this, leading to different quantities that have different significance,since unit currents in the graph might no longer be unique. See [173, Chapter 9] for a detailedtreatment of the material covered in this subsection.Perhaps the most obvious way to define effective resistances in an infinite graph is to use anexhaustion by induced subgraphs as we did before. This yields the free effective resistance. LetG be an infinite graph, let \u3008Vn\u3009n\u22651 be an exhaustion of G, and let Gn be the subgraph of G induced121.2. Random walks on graphsFigure 1.2: Free (left) and wired (right) boundary conditions on a 6\u00d7 6 square in Z2.by Vn for each n \u2265 1. The free effective resistance between two finite sets A and B in G is definedto beRFeff(A\u2194 B;G) = limn\u2192\u221eReff(A\u2194 B;Gn).Exercise. Prove that this limit exists and does not depend on the choice of exhaustion.We call an antisymmetric edge function on an infinite graphG a current if it satisfies Kirchhoff\u2019snode and cycle laws, and has finite energy. It turns out that the unit current flows from A to B inthe graphs Gn also converge to a current in G, called the free unit current flow from A to B,whose energy is exactly the free effective resistance.Free effective resistances also obey a version of the Thomson and Dirichlet principles:RFeff(A\u2194 B) =inf{E(\u03b8) : \u03b8 is a finitely supported flow from A to B with strength at least one}andRFeff(A\u2194 B)\u22121 =inf{E(f) : f : V \u2192 R is a function with f(v) \u2265 1 for all v \u2208 A and f(v) \u2264 0 for all v \u2208 B}.What if we instead take the infimal energy over all flows, not just those that are finitelysupported? This leads to the wired effective resistance, defined byRWeff (A\u2194 B) = inf{E(\u03b8) : \u03b8 is a flow from A to B with strength at least one}.In fact, it can be shown that this infimum is a minimum, that there is a unique flow obtaining thisminimum, and that this flow is a unit current flow from A to B, called the wired unit currentflow from A to B.Wired effective resistances and currents can also be obtained from an exhaustion as follows. Let\u3008Vn\u3009n\u22650 be an exhaustion of an infinite graph G. For each n, we also construct a graph G\u2217n from131.3. Circle packingG by gluing (wiring) every vertex of G \\ Vn into a single vertex, denoted \u2202n, and deleting all theself-loops that are created. We identify the set of edges of G\u2217n with the set of edges of G that haveat least one endpoint in Vn. Then we haveRWeff (A\u2194 B) = limn\u2192\u221eReff(A\u2194 B;G\u2217n),and the unit current flow from A to B in G\u2217n converges to the wired unit current flow from A to B.When are the free and wired effective resistances between any two sets the same? We say thata graph has unique currents if for any two finite sets A and B, there is a unique unit currentflow from A to B.Proposition. Let G be an infinite graph. Then the wired and free effective resistances between anytwo sets are equal if and only if G has unique currents, if and only if G does not admit non-constantharmonic functions of finite Dirichlet energy.This proposition tells us to expect non-constant harmonic functions of finite Dirichlet energyto play an important role in the electrical theory of infinite graphs. When do such functions exist?Proposition. Let G be an infinite graph. If G admits non-constant harmonic functions of finiteDirichlet energy, then G admits non-constant bounded harmonic functions of finite Dirichlet energy.In particular, G is non-Liouville.Proposition. Let G be an amenable transitive graph. Then G does not admit harmonic functionsof finite Dirichlet energy.It is more subtle to determine whether or not a nonamenable transitive graph admits non-constant harmonic functions of finite Dirichlet energy. (For Cayley graphs of groups, it is equivalentto the positivity of the first `2-Betti number of the group, see [173, Chapter 8.10].) Surprisingly, itis not monotone in how \u2018large\u2019 or \u2018expansive\u2019 the graph is. For example, if G is a bounded degreegraph rough isometric to d-dimensional hyperbolic space, then G admits non-constant harmonicDirichlet functions if and only if d = 2. Moreover, if G1 and G2 are any two infinite graphs, thenthe direct product of G1 and G2 does not admit any non-constant harmonic Dirichlet functions.1.3 Circle packing1.3.1 Planar graphs and mapsA planar graph is a graph that can be drawn in the plane so that no two edges intersect. Given agraph, a proper embedding of the graph into an oriented surface S is a drawing of the graph inthe surface such that each compact set in the domain intersects at most finitely many edges of thegraph, and each face of the drawing (i.e. connected component of the complement of the drawing)is homeomorphic to a disc. (For embeddings with infinite faces in multiply-connected surfacesthere is an additional technical condition required of proper embeddings which will not concern141.3. Circle packingFigure 1.3: Different maps with the same underlying graph. A map is determined by a graphtogether with a cyclic ordering of the oriented edges emanating from each vertex.us here.) A map is a graph together with an equivalence class of proper embeddings, where twoproper embeddings are considered equivalent if there is an orientation preserving homeomorphismmapping one surface to the other that sends one embedding to the other. The map is planar ifthe surface can be taken to be a domain D \u2282 C \u222a {\u221e}, and is simply connected if the surfacecan be taken to be the plane (or, equivalently, the open disc). Thus, a graph is planar if and onlyif it is the underlying graph of some planar map.It turns out (see [184]) that maps can be described combinatorially as graphs together withcyclic permutations {\u03c3v : v \u2208 V } of the oriented edges emanating from each vertex, which specifythe clockwise order of these edges in an embedding, so one does not need to be a topologist tostudy them.The dual of a map is the map M \u2020 that has the faces of M as vertices, the vertices of M asfaces, and for each edge e of M has an edge e\u2020 connecting the faces of M that are on either side ofe. We say that a map has bounded codegrees if its faces have a bounded number of sides, or,equivalently, if its dual has bounded degrees.Figure 1.4: A finite planar map (black, solid) and its dual (red, dashed).1.3.2 Circle packingThere are many ways to draw a planar map, which may or may not be useful for analyzing thebehaviour of random walk and other processes on the graph. One of the best ways is given bythe circle packing theorem. This theorem yields a canonical method of drawing planar graphs,closely connected to conformal mapping, and endows the graph with a geometry that, for many151.3. Circle packingpurposes, is better than the usual graph metric. Indeed, for bounded degree triangulations, acomprehensive theory has been developed by Angel, Barlow, Gurel-Gurevich, and Nachmias [17],and Chelkak [69], showing that the random walk on the circle packing behaves very similarly to aquasiconformal image of standard planar Brownian motion: Effective resistances, heat kernels, andharmonic measures on the graph can each be estimated in terms of the corresponding Brownianquantities. See [202] and [215] for introductions to the theory of circle packing. The interestedreader may also enjoy making their own circle packings using Ken Stephenson\u2019s CirclePack software[214].In this section, we will review the main theorems of circle packing as they relate to randomwalks and potential theory. Later, we will apply this theory to study uniform spanning forests ofplanar graphs, and will also study circle packings of unimodular random triangulations, two of themajor topics of original research in this thesis. Moreover, we will develop the dichotomy betweenparabolic and hyperbolic bounded degree planar maps, which will motivate the development of asimilar dichotomy for unimodular random planar maps, a further major topic of original work inthis thesis.A circle packing of a planar map G is a set of discs P = {P (v) : v \u2208 V } with disjoint interiorsin the Riemann sphere C \u222a {\u221e}, one for each vertex of G, such that two discs are tangent if andonly if their corresponding vertices are adjacent in G. The existence and uniqueness theorem forcircle packings, or simply the Circle Packing Theorem, is in our opinion one of the most beautifulresults in all of mathematics. It was first discovered by Koebe in 1936 [156] as a corollary to hiswork on the Riemann mapping theorem for multiply connected domains, but went largely forgottenuntil Thurston [221] (who was unaware of Koebe\u2019s proof) rediscovered it as a corollary to the workof Andreev [14] on convex polyehdra. Due to this storied history, the Circle Packing Theorem isoften called the Koebe-Andreev-Thurston Theorem. Like graphs, maps are said to be simple ifthey do not have any loops or multiple edges.Theorem. Every finite, simple planar map has a circle packing in the Riemann sphere. If the mapis a triangulation, then its circle packing is unique up to Mo\u00a8bius transformations.The circle packing theorem was extended to infinite, simple, simply connected triangulationsby He and Schramm [121, 122]. Their theorem can be thought of as a discrete analogue of theUniformization Theorem for Riemann Surfaces. (Indeed, a celebrated theorem of Rodin and Sulli-van [200] states that circle packings can be used to approximate conformal maps.) The carrier ofthe circle packing of a triangulation is the union of the discs in the triangulation together with thecurved triangular regions surrounded by three circles that correspond to the triangles of the map.A circle packing is said to be in a domain D \u2286 C \u222a {\u221e} if its carrier is D.Theorem (Schramm 1991, He and Schramm 1993). Let T be an infinite, simple, simply connectedtriangulation. Then T admits a circle packing either in the plane or in the disc, but not both, andthis circle packing is unique up to Mo\u00a8bius transformations of the plane or the disc as appropriate.161.3. Circle packingFigure 1.5: Circle packings of the 6-regular and 7-regular triangulations.In light of this theorem, we call a simply connected triangulation CP parabolic if it can becircle packed in the plane, and CP hyperbolic otherwise. A rather trivial compactness argument(using the Ring Lemma, below) shows that every simple triangulation can be circle packed in somedomain D \u2286 C \u222a {\u221e}. It is much harder to show that we can circle pack inside a domain that isgeometrically nice. (In fact, He and Schramm proved that, in the CP hyperbolic case, we can circlepack in any simply connected domain D strictly contained in the plane.)Recall that the unit disc can be identified with the hyperbolic plane through the Poincare\u00b4 discmodel, and that, under this identification, circles in the unit disc and circles in the hyperbolicplane are the same, but have different centres and radii. Moreover, the Mo\u00a8bius transformationsof the disc are exactly the orientation-preserving isometries of the hyperbolic plane. Thus, theabove theorem tells us to expect a strong connection between CP hyperbolic triangulations andhyperbolic geometry.He and Schramm [121] also pioneered the application of circle packing to probabilistic problems,proving the following remarkable theorem.Theorem (He and Schramm 1995). Let T be an infinite, simple, bounded degree, proper planetriangulation. Then T is CP parabolic if and only if it is recurrent.The He-Schramm Type Theorem follows as an immediate consequence of the following estimate,which also gives a good flavour of the kind of analysis we can do with circle packings. Let us firstintroduce some notation. Given a triangulation T and a circle packing P of T , we write z(v)and r(v) for the (Euclidean) centre and radius, respectively, of the disc P (v). Given z \u2208 C andR \u2265 r > 0, we write Bz(r) for the ball {w \u2208 C : |w \u2212 z| \u2264 r}, and Az(r,R) for the annulus{w \u2208 C : r \u2264 |w \u2212 z| \u2264 R}.171.3. Circle packingLemma (Resistances across annuli). Let T be a plane triangulation, and let P be a circle packingof T in some domain D \u2286 C \u222a {\u221e}.1. (Upper bound.) There exists a universal constant C such that the following holds. For everyclosed annulus Az0(r, \u03b1r) \u2282 D with \u03b1 \u2265 2 such that {v : z(v) \u2208 Bz0(r)} 6= \u2205, we haveReff({v : z(v) \u2208 Bz0(r)}\u2194 {v : z(v) \u2208 D \\Bz0(\u03b1 r)}) \u2264 C log\u03b1.2. (Lower bound.) Suppose that T has bounded degrees. Then there exists a constant C \u2032 dependingon the maximal degree of T such that the following holds. For every closed annulus Az0(r, \u03b1r)with \u03b1 \u2265 2 (not necessarily contained in D) such that {v : P (v) \u2286 Bz0(r)} 6= \u2205, we haveRFeff({v : P (v) \u2286 Bz0(r)}\u2194 {v : z(v) \u2208 D \\Bz0(\u03b1 r)}) \u2265 C \u2032 log\u03b1.The idea is that we can use the geometry of the circle packing to define a flow of sufficientlylow energy (to obtain the upper bound via Thomson\u2019s principle), and functions of sufficiently lowenergy (to obtain the lower bound via the Dirichlet principle).For the upper bound, we can define a flow that dissipates radially outwards in a roughly sym-metric fashion. (This is best done with the method of random paths, defining a random path bytaking the path in the graph that interpolates a radial line segment at a uniformly random anglefrom the centre of the annulus.)For the lower bound, the Ring Lemma of Rodin and Sullivan [200] is a crucial ingredient, whichgives us geometric control of the circle packing if we can control the degrees of vertices in thetriangulation (for instance, in the bounded degree case).Theorem (The Ring Lemma; Rodin and Sullivan 1987). There exists a sequence of constants\u3008kn : n \u2265 3\u3009 such that the following holds. If P is a circle packing of a planar triangulation in somedomain D \u2286 C \u222a {\u221e}, and v is a vertex of G such that P (v) does not contain \u221e, thenr(v)\/r(u) \u2264 kdeg(v)for every vertex u adjacent to v.In the setting of item 2. of the resistances across annuli lemma, the Ring Lemma implies thatthere exists some constant \u03b2 > 2 depending on the maximal degree such that if Bz0(r) contains acircle then the set of circles intersected by each of the annuli Az0(\u03b2nr, 2\u03b2nr) and Az0(\u03b2n+1r, 2\u03b2n+1r)are disjoint for every n. This allows us to reduce to the case \u03b1 = 2, since we can then add upthe resistances across each of the log\u03b2(\u03b1) many annuli Az0(\u03b2nr, 2\u03b2nr) by the series law. The case\u03b1 = 2 can then be handled by normalizing so that z = 0 and r = 1\/2, and then using the function|z| in Dirichlet\u2019s principle to obtain a lower bound on the free effective resistance.181.3. Circle packingThis resistance estimate has many further uses. For example, it can be used to prove our nextmilestone theorem about circle packing, the Benjamini-Schramm Convergence Theorem [47].Theorem (Benjamini and Schramm 1996). Let T be a CP hyperbolic, simply connected, boundeddegree triangulation, and let P be a circle packing of T in the disc. Then the random walk on Tconverges to a point in the boundary of the disc almost surely, and the law of the limit point hasfull support and no atoms.Here, \u2018the law of the limit point has full support\u2019 means that every interval in the boundary ofpositive length has a positive probability to contain the limit point of the random walk, while \u2018thelaw of the limit point has no atoms\u2019 means that for any particular point in the boundary, the limitof the walk does not equal that point a.s.In fact, there is an easy proof of the convergence and nonatomicity parts of this theorem usingthe resistances across annuli lemma above, first observed in the author\u2019s work with Yuval Peres[132], which we now sketch. If v is a vertex and A is a set of vertices in a transient graph G, thenwe have the estimatePv(\u03c4A <\u221e) \u2264 Reff(v \u2192\u221e)RFeff(v \u2194 A).Suppose we have a bounded degree triangulation circle packed in the disc, and that v is a fixedvertex of the triangulation. Then we have that there is some constant C <\u221e such thatPv(the random walk started at v ever hits {v : P (v) \u2286 B\u03be(\u03b5)}) \u2264 C log(1\/\u03b5)\u22121for all \u03be \u2208 \u2202D and all \u03b5 sufficiently small. In particular, for each point \u03be \u2208 \u2202D, we havePv(\u03be is in the closure of the set {z(Xn) : n \u2265 0})= 0. (1.3.1)We can now deduce convergence and nonatomicity: Observe that, for topological reasons, theintersection of \u2202D with the closure of the set {z(Xn) : n \u2265 0} is an interval a.s. (The key pointpowering this is that |z(Xn)| \u2192 1 and r(Xn) \u2192 0 a.s. as n \u2192 \u221e by transience.) The estimate(1.3.1) implies that the expected length of this interval is zero, and so the interval must be a pointa.s., establishing convergence. The fact that the law of the limit point does not have any atoms isobvious from (1.3.1).An immediate consequence of the Benjamini-Schramm convergence theorem is that every tran-sient, bounded degree, simple, proper plane triangulation is non-Liouville: For any bounded, mea-surable function f : \u2202D\u2192 V , we can define a harmonic function on T byh(v) = Ev f(limn\u2192\u221e z(Xn)),and this function will be non-constant provided that f is not (almost everywhere) constant. Infact, it is not too hard to reduce the case of a general bounded degree planar graph to this case,191.3. Circle packingyielding the following corollary.Corollary (Benjamini and Schramm 1996). Every transient, bounded degree planar graph is non-Liouville.We remark that Benjamini and Schramm also gave a different proof of this theorem using adifferent embedding, the square tiling, for which they also proved a convergence theorem [48].More recently, Angel, Barlow, Gurel-Gurevich, and Nachmias [17] showed that every harmonicfunction on a bounded degree, simple, simply connected triangulation can be represented by afunction on the unit circle in this way.Theorem (Angel, Barlow, Gurel-Gurevich, and Nachmias 2013). Let T be a bounded degree, CPhyperbolic, simple, simply connected triangulation, and let P be a circle packing of T in the unitdisc D. Then the circle packing compactification is a Poisson boundary of T .Recall from earlier that the probabilistic interpretation of this theorem is that the limit pointof the random walk on the unit circle encapsulates all the information about the limiting be-haviour of the random walk. The analogous theorem for square tiling was proven slightly earlierby Georgakopoulos [100].The original proof of this theorem was highly analytic. In order to prove it, the authors employedsophisticated machinery to prove various estimates for random walks on bounded degree circlepackings. Roughly speaking, these estimates allow one to compare the random walk on a boundeddegree circle packing in a domain to a (quasiconformal image of) standard planar Brownian motionin that domain. In particular, the authors proved heat kernel estimates, diffusivity-type estimates,and harmonic measure estimates. These estimates were further developed by Chelkak [69], whosework implies, among other things, that the harmonic measure on the unit disc of a boundeddegree CP hyperbolic triangulation is Ho\u00a8lder continuous. (He considered finite triangulations withboundary; the theorem below follows from his work by taking limits.) Here, given a bounded degreesimple triangulation circle packed in the unit disc, the harmonic measure from v, denoted \u03c9v, isthe law of the limit point in the unit circle of the random walk started at v.Theorem (Chelkak 2013). Let T be a bounded degree, CP hyperbolic, simple, simply connectedtriangulation, let P be a circle packing of T in the unit disc, and let v be such that z(v) = 0. Thenthere exist positive constants \u03b1 \u2264 \u03b2 and C (depending only on the maximal degree of T ) such thatC\u22121|I|\u03b1 \u2264 \u03c9v(I) \u2264 C|I|\u03b2for every interval I \u2282 \u2202D.These estimates are interesting in their own right, and have several further applications. In-deed, they also allowed those authors to prove that the unit circle is the Martin boundary of thetriangulation, a much stronger result that gives a representation formula for all positive harmonic201.3. Circle packingfunctions on the triangulation. Later, they also will play an important role in our calculation,carried out in joint work with Nachmias [130] (Chapter 10 of this thesis), of the critical exponentsfor uniform spanning forests of planar graphs.Theorem (Angel, Barlow, Gurel-Gurevich, and Nachmias 2013). Let T be a bounded degree, CPhyperbolic, simple, simply connected triangulation, and let P be a circle packing of T in the unitdisc D. Then the unit circle is a Martin boundary of T . That is, the following hold.1. For every two vertices u and v of T , the Radon-Nikodym derivative of the harmonic measuresd\u03c9vd\u03c9u: \u2202D\u2192 Ris continuous.2. For every positive harmonic function h on T and every vertex u of T , there exists a uniqueprobability measure \u00b5 on \u2202D such thath(v) = h(u)\u222bd\u03c9vd\u03c9u(\u03be) d\u00b5(\u03be)for every v \u2208 V .It turns out, however, that the heavy machinery developed by Angel, Barlow, Gurel-Gurevich,and Nachmias is not in fact required for the Poisson boundary result. In joint work with YuvalPeres in 2015 [132] (Chapter 8 of this thesis), we proved the following theorem by a much moreelementary argument, which implies the Poisson boundary results for both circle packing and squaretiling.Theorem (H. and Peres 2015). Let G be a planar map, and let z be an embedding of G into adomain D \u2286 D such that \u3008z(Xn)\u3009n\u22650 converges to a point in \u2202D almost surely, and the law of thelimit point is non-atomic. Then the compactification of G given by taking the closure of z(V ) in Dis a Poisson boundary of G.Thus, once one has established a.s. convergence of the random walk to a boundary point inan embedding of a planar graph in the disc, the identification of the Poisson boundary followsfor free: No further geometric control on the embedding is needed whatsoever. The proof of thistheorem was adapted from our work with Angel, Nachmias, and Ray [21] on unimodular randomtriangulations of unbounded degree, for which the analytic approach was not available.1.3.3 Harmonic Dirichlet functions on planar graphsBesides proving that every transient, bounded degree planar graph is non-Liouville, the Benjamini-Schramm convergence theorem also implies the even stronger result that every transient, boundeddegree planar graph admits non-constant harmonic functions of finite Dirichlet energy.211.3. Circle packingIf f : V \u2192 R is a function and Xt is the continuous time random walk3 on G, and we define ftbyft(v) = Ev[f(Xt)],then it turns out that E(ft) is a decreasing function of t. It follows that if f is a function of finiteenergy such that the limitlimn\u2192\u221e f(Xn)exists a.s. and is not concentrated on a point (the existence of this limit, and its law if it exists,does not depend on whether we use the continuous or discrete time random walk), thenf\u221e(v) := Ev[limn\u2192\u221e f(Xn)]= limt\u2192\u221eEvf(Xt)is a non-constant harmonic function with E(f\u221e) \u2264 E(f) < \u221e. (In fact, Ancona, Lyons, andPeres [12] proved that the limit of f(Xn) as n \u2192 \u221e always exists a.s. whenever f is a Dirichletfunction on a transient graph, from which it is possible to deduce the convergence part of theBenjamini-Schramm convergence theorem.)Now, if P is a circle packing of a bounded degree CP hyperbolic triangulation in the unit disc,then the function z assigning each vertex to its center has finite energy, sinceE(z) \u2264\u2211v\u2208Vdeg(v)r(v)2 \u2264 maxv\u2208Vdeg(v)\u2211v\u2208Vr(v)2 \u2264 maxv\u2208Vdeg(v).Thus, it follows from the Benjamini-Schramm convergence theorem and the above discussion thatevery transient, bounded degree, simply connected, simple plane triangulation admits non-constantharmonic functions of finite Dirichlet energy, namely z\u221e(v) = Ev[limn\u2192\u221e z(Xn)]. With a littleextra work to remove some of these hypotheses, we arrive at the following dichotomy.Corollary (Benjamini and Schramm 1996). Let G be a bounded degree planar graph. Then thefollowing are equivalent.1. G is transient.2. G admits non-constant positive harmonic functions.3. G is non-Liouville, i.e., admits non-constant bounded harmonic functions.4. G admits non-constant harmonic functions of finite Dirichlet energy.The implications (4) \u21d2 (3) \u21d2 (2) \u21d2 (1) hold on any graph. We shall later see that an evenmore far reaching dichotomy holds for unimodular random planar maps, without the assumptionof bounded degree.3This is the walk that has a semigroup of transition operators given by Pt = e\u2212t\u2206, where \u2206 = \u2207\u2217\u2207 is theLaplacian of G221.3. Circle packingLet us take a moment to mention a relevant work in progress. Now that we know that non-constant harmonic Dirichlet functions exist on any bounded degree, CP hyperbolic, simple, properplane triangulation, it is natural to wonder whether there is a geometric representation for the spaceof all such functions, as we had for the spaces of bounded harmonic functions and positive harmonicfunctions. In upcoming work, we give such a representation. For each function f : \u2202D \u2192 R, wedefineD(\u03c6) := 14pi2\u222b\u2202D\u222b\u2202D\u2223\u2223\u2223\u2223\u2223 \u03c6(\u03be)\u2212 \u03c6(\u03b6)2 sin (12 |\u03be \u2212 \u03b6|)\u2223\u2223\u2223\u2223\u22232d\u03be d\u03b6 <\u221e.where the integrals are taken with respect to arc length, and say f is Douglas-integrable ifD(f) <\u221e. The space of Douglas-integrable functions is a Hilbert space with the inner productf(o)g(o) +D(f, g) = f(o)g(o) + 14pi2\u222b\u2202D\u222b\u2202D(f(\u03be)\u2212 f(\u03b6))(g(\u03be)\u2212 g(\u03b6)))2 sin(12 |\u03be \u2212 \u03b6|) d\u03be d\u03b6,where o \u2208 V is an arbitrary root vertex. In fact, D(f) is exactly the Dirichlet energy of theharmonic extension of f to the unit disc.Theorem (H. 2017+). Let T be a transient, bounded degree, proper plane triangulation, and let Pbe a circle packing of T in the unit disc. Then the functionf 7\u2212\u2212\u2192 h, where h(v) = Ev f(limn\u2192\u221eXn)is a bounded linear isomorphism from the space of Douglas-integrable functions on \u2202D to the spaceof harmonic Dirichlet functions on T .By a bounded linear isomorphism we mean a bounded linear map with a bounded linear inverse,and not necessarily an isometry.A potentially surprising thing here is that the space of Douglas-integrable functions on theboundary is always the same; the expression for the Douglas energy does not involve the harmonicmeasure, or otherwise involve the triangulation in any way.1.3.4 Double circle packingBefore moving on, let us mention a generalisation of the circle packing theorem which, in ouropinion, is even more beautiful.Let M be a planar map with vertex set V and face set F . A double circle packing of M isa pair of circle packings P = {P (v) : v \u2208 V } and P \u2020 = {P \u2020(f) : f \u2208 F} satisfying the followingconditions (see Figure 1.6):1. (M is the tangency map of P .) For each pair of vertices u and v of M , the discs P (u)and P (v) are tangent if and only if u and v are adjacent in M . Moreover, for each vertex u,the discs corresponding to the vertices adjacent u appear around P (u) in the clockwise orderspecified by the map structure \u03c3u.231.3. Circle packingFigure 1.6: A polyhedral planar map and its double circle packing.2. (M \u2020 is the tangency map of P \u2020.) For each pair of faces f and g of G, the discs P \u2020(f) andP \u2020(g) are tangent if and only if f and g are adjacent in G\u2020. Moreover, for each face f , thediscs corresponding to the vertices adjacent f appear around P \u2020(f) in the clockwise orderspecified by the map structure \u03c3\u2020f .3. (Primal and dual circles are perpendicular.) For each vertex v and face f of M , thediscs P \u2020(f) and P (v) have non-empty intersection if and only if f is incident to v, and in thiscase the boundary circles of P \u2020(f) and P (v) intersect at right angles.Recall that a graph is 3-connected if the removal of any two vertices from the graph does not causethe graph to become disconnected. We call a map that is both simple and 3-connected polyhedral.Any embedding of a planar map that is not polyhedral must contain faces that are not convex,and it follows that any finite planar map admitting a double circle packing must be polyhedral.Conversely, Thurston\u2019s interpretation of Andreev\u2019s Theorem [179, 221] implies that every finite,polyhedral, planar map admits a double circle packing (see also [58]). The corresponding infinitetheory was developed by He [120], who proved that every infinite, polyhedral, simply connectedmap M with locally finite dual admits a double circle packing in either the Euclidean plane or thehyperbolic plane (but not both) and that this packing is unique up to Mo\u00a8bius transformations. Asbefore, we say that M is CP parabolic or CP hyperbolic as appropriate.In our work with Nachmias [130], we proved a form of the Ring Lemma for double circle packings.We write v \u22a5 f to mean that the vertex v is incident to the face f .Theorem (H. and Nachmias 2016). There exists a family of positive constants \u3008kn,m : n \u2265 3,m \u2265 3\u3009such that if (P, P \u2020) is a double circle packing of a polyhedral planar map M in a domain D \u2286 C\u222a{\u221e}and v is a vertex of M such that P (v) does not contain \u221e, thenr(v)\/r(f) \u2264 kdeg(v),maxg\u22a5v deg(g)for all f \u2208 F incident to v.241.3. Circle packingOnce the existence and uniqueness theorems and the Ring Lemma are in place, everything elsethat we have said about circle packings of simple triangulations extends more or less immediately todouble circle packings of polyhedral maps of bounded codegree. In particular, as in the He-SchrammTheorem [121], CP hyperbolicity is equivalent to transience for simply connected polyhedral mapswith bounded degrees and codegrees [120].1.3.5 Circle packing and the degreeAn interesting aspect of circle packing is that, in certain situations, we can deduce the CP type ofa triangulation from the degrees of its vertices. This was first observed by Beardon and Stephenson[34], who proved the following.Theorem (Beardon and Stephenson 1991). Let T be an infinite, simple, simply connected trian-gulation. If deg(v) \u2264 6 for every vertex v of T , then T is CP parabolic. If deg(v) \u2265 7 for everyvertex v of T , then T is CP hyperbolic.Exercise. Show that there exists a constant C > 1 such that the following holds. Suppose T is asimple triangulation circle packed in a domain D \u2286 C. Then1. for every vertex v of T with deg(v) \u2264 5, there is a neighbour u of v such that r(u)\/r(v) \u2265 C.2. for every vertex v of T with deg(v) = 6, there is a neighbour u of v such that r(u)\/r(v) \u2265 1and a neighbour w of v such that r(w)\/r(v) \u2264 1.3. for every vertex v of T with deg(v) \u2265 7, there is a neighbour u of v such that r(u)\/r(v) \u2264 C\u22121.Let rH(v) denote the hyperbolic radius of a disc P (v) \u2282 D.Exercise. Show that there exists a constant C > 1 such that the following holds. Suppose Tis a simple triangulation circle packed in a domain D \u2286 D. Then for every vertex v of T withdeg(v) \u2264 6, there is a neighbour u of v such that rH(u)\/rH(v) \u2265 C.Exercise. Deduce the above theorem of Beardon and Stephenson from the previous two exercises.We shall later see that for unimodular random simply connected triangulations, the circle pack-ing type is determined by the average degree.251.4. Percolation1.4 PercolationIn Bernoulli bond percolation, each edge of a (usually infinite) graph G is chosen randomly tobe either deleted with probability 1 \u2212 p, or else retained with probability p. The random graphobtained in this way is denoted G[p]. Connected components of G[p] are referred to as clusters.Although we will have relatively little new to say about percolation in this thesis, the theoryof percolation provides important motivation for our work on uniform spanning forests, and alsoleads naturally to the theory of unimodular random rooted graphs, another central topic of thisthesis. Percolation has become a topic of central importance to modern probability theory, andhas a huge literature surrounding it. The reader is referred to the beautiful, if slightly outdated,monograph of Grimmett [106] for a thorough treatment of the Euclidean (Zd) theory, and to [173]for the theory of percolation on more general graphs.The first basic result about percolation, without which the model would not be nearly asinteresting, is that for most graphs, percolation undergoes a non-trivial phase transition, meaningthat the critical probability,pc(G) = inf{p \u2208 [0, 1] : G[p] has an infinite cluster almost surely},is strictly between zero and one. Conjecturally, pc \u2208 (0, 1) for every transitive graph that is notrough isometric to Z: This has been resolved in most cases, and remains open only for transitivegraphs of intermediate volume growth (i.e., subexponential but superpolynomial). For example, asimple first moment argument shows that pc > 0 for any bounded degree graph, since the expectednumber of open simple paths of length n starting at a vertex v is at mostpn(maxu\u2208Vdeg(u))n,which yields the bound pc \u2265 (maxu\u2208V deg(u))\u22121. For Z2, a similar first moment argument withsimple curves surrounding the origin (known as a Peierls argument [189]) shows that pc(Z2) < 1,while for d \u2265 2 we clearly have that pc(Zd) \u2264 pc(Z2).The study of percolation can naturally be decomposed into the study of the subcritical (p < pc),critical (p = pc), and supercritical (p > pc) regimes. Here, we shall be interested mostly in thecritical and supercritical regimes.At criticality, the central question concerns the existence or nonexistence of an infinite cluster.Indeed, perhaps the best known open problem in modern probability theory is to prove that thereis no infinite cluster at criticality on Zd for all d \u2265 2. This problem was solved in two dimensionsby Russo in 1981 [203], and for all d \u2265 19 by Hara and Slade in 1994 [117]. More recently, Fitznerand van der Hoftstad [89] sharpened the methods of Hara and Slade to solve the problem for alld \u2265 11. (It is expected that this method can in principle, and with great effort, be pushed to handleall d \u2265 7. Dimensions 3, 4, 5, and 6 are expected to require new approaches.)261.4. PercolationIn 1996, Benjamini and Schramm [51] proposed a systematic study of percolation on generaltransitive (and quasi-transitive) graphs. They made several conjectures and posed many questions.Here is one of them.Conjecture (Benjamini and Schramm 1996). Let G be a transitive graph. If pc(G) < 1, then G[pc]does not have any infinite clusters almost surely.It quickly emerged that it is often substantially easier to study percolation on unimodulartransitive graphs than on general transitive graphs. A transitive graph G = (V,E) is said to beunimodular if for every function F : V 2 \u2192 [0,\u221e] that is automorphism equivariant in the sensethat F (\u03b3u, \u03b3v) = F (u, v) for every u, v \u2208 V and every automorphism \u03b3 of G, then, letting \u03c1 be afixed root vertex of G, \u2211v\u2208VF (\u03c1, v) =\u2211u\u2208VF (u, \u03c1). (1.4.1)The equality (1.4.1) is referred to as the Mass-Transport Principle (MTP)4. We think of F asa rule for sending a non-negative amount of mass from each vertex to each other vertex, so thatthe mass-transport principle says thatmass out of the root = mass into the root .It turns out that most transitive graphs one encounters \u2018in the wild\u2019 are unimodular. In particular,every amenable transitive graph, every Cayley graph of a finitely generated group, and everytransitive, simply connected, planar map with locally finite dual has the property of unimodularity.Figure 1.7: The grand-parent graph.An example of a transitive graph that is not unimodular is thegrandparent graph, defined as follows. Take a 3-regular tree, anddraw it in the plane so that every vertex has one \u2018parent\u2019 above it andtwo \u2018children\u2019 below it. The grandparent graph is formed by addingto the 3-regular tree an edge connecting each vertex to its grandpar-ent, that is, the parent of its parent. We can define an automorphismequivariant function on the grandparent graph byF (u, v) = 1(v is u\u2019s grandparent).Since every vertex has one grandparent but four grandchildren, we have\u2211v\u2208VF (\u03c1, v) = 1 but\u2211u\u2208VF (u, \u03c1) = 4,so that the Mass-Transport Principle does not hold for the grandparent graph.4The history of this definition is as follows: A locally compact group is said to be unimodular if its left and rightHaar measures coincide. A transitive graph is unimodular in our sense if and only if its group of automorphisms hasa transitive unimodular subgroup.271.4. PercolationA major achievement in the theory of critical percolation outside of the Euclidean setting wasmade in 1999 by Benjamini, Lyons, Peres, and Schramm [43]. This theorem came out of work [45]by a subset of the same authors on arbitrary (non-Bernoulli) automorphism invariant percolationprocesses, which we shall have more to say about later.Theorem (Benjamini, Lyons, Peres, and Schramm 1999). Let G be a nonamenable, unimodular,transitive graph. Then G[pc] has no infinite clusters almost surely.Partial progress was made in the nonunimodular case by Peres, Pete, and Scolnicov [192], whoproved there are no infinite clusters at criticality a.s. on certain well-known examples of nonuni-modular transitive graphs, including decorated trees (like the grandparent graph) and nonamenableDiestel-Leader graphs, and by Tima\u00b4r [222], who proved that there is at most one infinite cluster atcriticality on any nonunimodular transitive graph a.s.In 2016 [129] (Chapter 2 of this thesis), we found an elementary proof that the following estimateholds in any transitive graph:inf{P(x is connected to y in G[pc]): x, y \u2208 V, d(x, y) \u2264 n}\u2264(lim infr\u2192\u221e |B(x, r)|1\/r)\u2212n.This estimate readily implies that there cannot be a unique infinite cluster at pc in any transitivegraph of exponential growth. However, the existence of multiple infinite clusters at criticality hadalready been ruled out by previous work (the amenable and unimodular nonamenable cases arediscussed in the following subsection, while the nonunimodular case is handled by the work ofTima\u00b4r mentioned above). Thus, we obtained the following theorem.Theorem (H. 2016). Let G be a transitive graph with exponential growth. Then G[pc] has noinfinite clusters almost surely.1.4.1 The number of infinite clustersLet us now turn to the supercritical regime. In this regime, we are particularly interested inunderstanding the geometry of the infinite clusters of G[p]. The most basic question concerns thenumber of infinite clusters. For this question, the foundational result was proven by Newman andSchulman in 1981 [187].Theorem (Newman and Schulman 1981). Let G be a transitive graph. Then G[p] has either noinfinite clusters, a unique infinite cluster, or infinitely many infinite clusters almost surely for everyp \u2208 [0, 1].The key to this theorem is the insertion tolerance of Bernoulli percolation, which says thatfor any set of edges A and p > 0, the law of the subgraph G[p] \u222a A is absolutely continuous5 withrespect to the law of G[p].5Recall that the law of a random variable X is said to be absolutely continuous with respect to the law of arandom variable Y if whenever A is a set such that Y is a.s. not in A , then X is also a.s. not in A .281.4. PercolationThe proof is as follows: Given a configuration \u03c9 \u2208 {0, 1}E and an automorphism \u03b3 of G, define\u03b3\u03c9(e) = \u03c9(\u03b3\u22121(e))for every edge e. It is an easy fact that Bernoulli bond percolation on any transitive graph isergodic, meaning that if A \u2286 {0, 1}E is a set of configurations that is automorphism invariant inthe sense that\u03c9 \u2208 A \u21d2 \u03b3\u03c9 \u2208 Afor every configuration \u03c9 and automorphism \u03b3, then we must have that P(G[p] \u2208 A ) is eitherzero or one. In particular, the number of infinite components is automorphism invariant, and wededuce that the number of infinite components is not random. That is, for every p, there existsNp \u2208 N \u222a {\u221e} such that G[p] has Np infinite clusters a.s.Suppose for contradiction that Np \/\u2208 {0, 1,\u221e}. If we take a large connected set of edges A,then with high probability A is incident to more than one of the infinite clusters of G[p]. Thus,the configuration G[p]\u222aA has positive probability to have strictly fewer infinite clusters than G[p].But G[p] \u222a A is absolutely continuous with respect to G[p] by insertion-tolerance, hence it musthave Np infinite clusters a.s., yielding a contradiction.The Newman-Schulman Theorem can in fact be extended to any insertion-tolerant automor-phism invariant percolation on a transitive graph. Here, an automorphism-invariant percolationis a random subgraph \u03c9 of G such that \u03b3\u03c9 has the same distribution as \u03c9 for every automorphism \u03b3of G. (In the case that \u03c9 is also ergodic the Newman-Schulman Theorem follows exactly as above.In general, it is possible to reduce to the ergodic case by taking an ergodic decomposition, whichpreserves insertion tolerance.)The next major result on the number of clusters was due to Aizenmann, Kesten, and Newmanin 1987 [3], who proved that percolation on Zd has at most one infinite cluster a.s. for every d \u2265 1.A much simpler proof of this theorem was found by Burton and Keane in 1989 [64], which wasthen generalised to all transitive amenable graphs by Gandolfi, Keane, and Newman in 1992 [97].Like the Newman-Schulman Theorem, this theorem also extends to arbitrary insertion-tolerant,automorphism invariant percolations.Theorem. Let G be an amenable transitive graph, and let p \u2208 [0, 1]. Then G[p] has at most oneinfinite cluster almost surely.It is a major open problem, first stated by Benjamini and Schramm [51], to prove that theconverse holds. The uniqueness threshold is defined to bepu(G) = inf{p \u2208 [0, 1] : G[p] has a unique infinite cluster almost surely}.Conjecture (Benjamini and Schramm 1996). Let G be a transitive graph. Then pc < pu if andonly if G is nonamenable.291.4. PercolationNot much progress has been made on this question; see [112] and [173, Chapter 9] for an accountof what is known.Since the existence of a unique infinite cluster is not monotone with respect to the configuration,the following theorem, due to Schonmann [205], and Ha\u00a8ggstro\u00a8m, Peres, and Schonmann [114], isfar from obvious.Theorem (Ha\u00a8ggstro\u00a8m, Peres, and Schonmann 1999). Let G be a transitive graph, and let p2 > p1.If G[p1] has a unique infinite cluster almost surely, then G[p2] has a unique infinite almost surely.In the unimodular case, this theorem can also be deduced from the following beautiful theorem ofLyons and Schramm [176], which, informally, says that infinite clusters of Bernoulli bond percolationon a unimodular transitive graph all \u2018look the same\u2019. The theorem extends to insertion-tolerantautomorphism invariant percolations.Theorem (Lyons and Schramm 1999). Let G be a unimodular transitive graph, let p \u2208 [0, 1], andlet A \u2286 {0, 1}E be a measurable, shift invariant set. Then either every infinite cluster of G[p] isin A or no infinite clusters of G[p] are in A almost surely.For example, the theorem implies that either every infinite cluster of G[p] is transient or everyinfinite cluster of G[p] is recurrent a.s.The main idea of the proof of the Lyons-Schramm Indistinguishability Theorem can be sum-marised as follows. The coexistence of infinite clusters of different types (i.e., of infinite clusters inA and not in A ) can be shown to imply that for some infinite cluster, there exist infinitely manypivotal edges, that is, closed edges that change the type of the cluster if they are inserted. Heuristi-cally, this should contradicts the measurability of the property: The existence of pivotal edges faraway from the origin \u2013 which by insertion tolerance are in some sense indifferent to being insertedand hence changing the type of an infinite cluster \u2013 should imply that we cannot approximate theevent that the cluster at the origin is in A by an event depending only on finitely many edges ofthe configuration. (The ability to approximate a set in this way is the definition of the set beingmeasurable.)To see that the Lyons-Schramm Indistinguishability Theorem implies the Ha\u00a8ggstro\u00a8m-Peres-Schramm Uniqueness Monotonicty Theorem in the unimodular case, first observe that we cansample G[p1] by first sampling G[p2], and then performing Bernoulli p1\/p2 bond percolation onG[p2]. Consider the setA =\uf8f1\uf8f2\uf8f3\u03c9 \u2208 {0, 1}E : \u03c9 spans a connected subgraph H of G such that Bernoullip1\/p2 percolation on H has no infinite clusters almost surely\uf8fc\uf8fd\uf8feIf G[p2] has infinitely many infinite clusters a.s., then, by the Lyons-Schramm IndistinguishabilityTheorem, either all these clusters are in A , in which case G[p1] has infinitely many infinite clusters301.4. Percolationa.s. also, or else none of the clusters are in A , in which case G[p1] does not have any infinite clustersa.s., concluding the proof.Indistinguishability of infinite clusters can fail for nonunimodular transitive graphs. However,Ha\u00a8ggstro\u00a8m, Peres, and Schonmann [114] proved that indistinguishability still holds in this contextprovided that we restrict attention to what they called robust properties, of which the set A aboveis an example.1.4.2 The geometry of infinite clustersWhat can we say about the geometry of individual clusters? In general, we expect that if G[p] hasa unique infinite cluster, then this cluster will be similar to the original graph G in many ways.For example, it is conjectured that if G is a transient transitive graph, then every infinite cluster ofG[p] is transient a.s. for every p > pc. The first result of this form was due to Grimmett, Kesten,and Zhang [104].Theorem (Grimmett, Kesten, and Zhang 1993). Let p > pc(Zd). Then the unique infinite clusterof Zd[p] is transient almost surely.The original proof of this theorem was rather difficult. A much simpler proof was found byGabor Pete in 2008 [194], using isoperimetric techniques. An obstacle to applying isoperimetricmethods to percolation is that the infinite percolation cluster will necessarily contain \u2018bad regions\u2019.In particular, if G is a transitive graph and p < 1, it is easily seen that every infinite cluster of G[p]must contain arbitrarily large sets of vertices with only one edge in their boundary. This is becausesuch sets have positive probabilities to occur at the origin, and therefore must occur somewhere byergodicity.This difficulty led Benjamini, Lyons, and Schramm [45] to introduce the notion of the anchoredisoperimetric inequalities, whose definition we already saw in Section 1.2.2. These are less sensitiveto perturbations of the graph, and we can hope that if G is a sufficiently nice (e.g. transitive) graphthat satisfies an anchored \u03c6(t)-isoperimetric inequality, then the infinite clusters of G[p] will alsosatisfy an anchored \u03c6(t)-isoperimetric inequality for p > pc. Chen and Peres [70] proved that thisis indeed the case for graphs with anchored expansion, at least for sufficiently large p. A secondproof, due to Pete, appeared as an appendix to the same paper.Theorem (Chen, Peres, and Pete 2004). Let G be a graph with anchored expansion. Then thereexists p0 < 1 such that every infinite cluster of G[p] has anchored expansion almost surely for everyp \u2265 p0.A much more general form of this result was later proven by Pete [194], a special case of whichis as follows.Theorem (Pete 2008). Let G be a Cayley graph of a finitely presented group (e.g. Zd) that isnot rough isometric to Z, and suppose that G satisfies a \u03c6(t)-isoperimetric inequality for some311.5. Uniform spanning forestsincreasing \u03c6. Then there exists p0 < 1 such that every infinite cluster of G[p0] satisfies an anchored\u03c6(t)-isoperimetric inequality almost surely.By the Thomassen Criterion, these results imply that infinite clusters of G[p] are transient forsufficiently large p for a large class of transient graphs G. In the case of Zd, it is possible to recoverthe full Grimmett-Kesten-Zhang Theorem by a renormalization argument.Now suppose that G is non-Liouville. Does it follow that every infinite cluster of G[p] is non-Liouville for every p > pc. Conversely, if G is Liouville, does it follow that every infinite cluster ofG[p] is Liouville for every p > pc? In general, this question is very poorly understood. (In contrast,the existence of harmonic Dirichlet functions on percolation clusters of unimodular transitive graphsis very well understood since the work of Gaboriau [94].) Similarly to the case of transitive graphs,the non-Liouville property for percolation clusters on unimodular transitive graphs is equivalentto the random walk on the cluster having positive speed. Thus, we can answer the question inthe nonamenable unimodular case for large p by invoking the following beautiful theorem of Vira\u00b4g[227]. We shall see that much easier proofs, avoiding anchored expansion altogether, are possibleonce we introduce the notion of invariant nonamenability.Theorem (Vira\u00b4g 2000). Let G be a bounded degree graph with anchored expansion. Then therandom walk on G has positive speed almost surely, and for each vertex v there exist positiveconstants c and C such thatpn(v, v) \u2264 Ce\u2212cn1\/3for all n \u2265 1.1.5 Uniform spanning forestsThe Free Uniform Spanning Forest (FUSF) and the Wired Uniform Spanning Forest(WUSF) of an infinite graph G are defined as weak limits of the uniform spanning trees on largefinite subgraphs of G, taken with either free or wired boundary conditions respectively. Firststudied by Pemantle [190], the USFs are closely related many other areas of probability, includingelectrical networks [62, 154], Lawler\u2019s loop-erased random walk [44, 161, 228], sampling algorithms[197, 228], domino tiling [150], the Abelian sandpile model [137, 138, 178], the rotor-router model[126], and the Fortuin-Kasteleyn random cluster model [105, 109]. The USFs are also of interestin group theory, where the FUSFs of Cayley graphs are related to the `2-Betti numbers [94, 169]and to the fixed price problem of Gaboriau [95], and have also been used to approach the Dixmierproblem [87].Uniform spanning forests are also the major topic of this thesis, being the subject of four ofthe included papers and being of central importance in one other. One of our main goals will beto address the same questions for the uniform spanning forests that we asked about percolation inthe previous section.321.5. Uniform spanning forests1.5.1 Uniform spanning treesLet us start at the beginning. A uniform spanning tree of a finite graph G is simply a uniformrandom element from the set of spanning trees of G, that is, connected subgraphs of G that containevery vertex and no cycles. The connection between uniform spanning trees and electrical networksgoes back as far as it could, all the way to the 1847 work of Kirchhoff [154] in which the laws ofelectrical networks were introduced.Theorem (Kirchhoff\u2019s effective resistance formula). Let G be a finite graph. Then for every edgee of G,P(e is in a uniform spanning tree of G) = Reff(e\u2212 \u2194 e+)(Kirchhoff did not state this theorem probabilistically, but rather as a statement about the ratioof the number of spanning trees that include the edge to the total number of spanning trees.)Kirchhoff\u2019s interest in this theorem was computational: He also invented a method to computethe number of spanning trees of a graph, the famous Matrix-Tree Theorem.Theorem (Kirchhoff\u2019s Matrix-Tree Theorem). Let G be a finite connected graph with n vertices,and let \u03bb1, . . . , \u03bbn\u22121 be the non-zero eigenvalues of the Laplacian of G. Then|{spanning trees of G}| = 1nn\u22121\u220fi=1\u03bbi.Since the number of spanning trees that do not contain a given edge is equal to the numberof spanning trees of the graph in which that edge has been deleted, Kirchhoff\u2019s effective resistanceformula together with the Matrix-Tree theorem give a method for computing effective resistancesin finite graphs. From our perspective, we will be more interested in using the formula the otherway around, using the variational principles for the effective resistance to estimate the probabilitythat an edge is in a uniform spanning tree (or forest).How can we sample the uniform spanning tree of a finite graph? One very simple method isas follows. Pick some edge of the graph. We can compute the probability that the edge is in theuniform spanning tree (e.g. using the Matrix-Tree Theorem), and flip an appropriately biased cointo decide whether or not to include it. If we decide to include the edge, then we next contractit, i.e., identify the two endpoints of the edge. Otherwise, if we decided not to include the edge,we delete it from the graph. We then continue as in the first step, choosing another edge fromthe graph, choosing whether or not to include it by performing a Matrix-Tree calculation (in themodified graph) and flipping an appropriately biased coin, and then contracting or deleting theedge as appropriate. If we continue doing this until we have decided what to do with every edge,then the set of edges we chose to include will form a uniformly random spanning tree of G.Exercise. Prove that this algorithm works.331.5. Uniform spanning forestsA far-reaching extension of Kirchhoff\u2019s effective resistance formula, known as the Transfer-Current Theorem of Burton and Pemantle [62], allows one to calculate the probability that anyset of edges is included in the forest in terms of electrical quantities. (The case of two edgeshad previously been studied by Brooks, Smith, Stone, and Tutte [60], who were interested by theconnection to square tiling, which they invented.) Given two oriented edges e1 and e2, we defineY (e1, e2) to be the current flowing through e2 when a unit potential is placed on the endpoints ofe1. In other words,Y (e1, e2) = Pe+2(\u03c4+e+1< \u03c4+e\u22121)\u2212Pe\u22122(\u03c4+e+1< \u03c4+e\u22121).We fix arbitrarily an orientation of each edge e, and think of Y as a matrix indexed by E \u00d7 E,which we call the transfer-current matrix.Theorem (Burton and Pemantle 1993). Let G be an infinite graph. Then for any collection ofedges e1, . . . , en of G, we haveUST({e1, . . . , en} \u2286 T ) = det\u3008Y (ei, ej)\u30091\u2264i,j\u2264n.Note that the probabilities of all other events can be computed from the probabilities of theevents of the form {{e1, . . . , en} \u2286 T} using inclusion-exclusion.1.5.2 Sampling using random walksThe Aldous-Broder AlgorithmLet G be a finite graph, and let \u3008Xn\u3009n\u22650 be a random walk on G. For each vertex v of G, lete(v,X) be the oriented edge pointing into v that is crossed by the walk X as it enters v for thefirst time, and defineAB(X) = {\u2212e(v,X) : v 6= X0}to be the collection of (reversed) first entry edges. It is not hard to see that AB(X) is a spanningtree, where the edges of the tree are oriented so that every vertex other than X0 has exactly oneoriented edge in the tree emanating from it. The following theorem, however, is surprising at first.Theorem (Broder 1989, Aldous 1990). AB(X) is a uniform spanning tree of G.Note in particular that we can start the walk wherever we want, and this does not change thedistribution of the tree. This theorem gives another method of sampling the uniform spanning treeof a finite graph, named the Aldous-Broder algorithm after its inventors [8, 59].Exercise. Use the Aldous-Broder algorithm to prove Kirchhoff\u2019s effective resistance formula.Wilson\u2019s algorithmThe loop-erasure of a path in a graph is formed by erasing cycles from the path chronologicallyas they are created. (The loop-erasure is only defined for paths that are either finite or transient341.5. Uniform spanning forestsin the sense that they visit each vertex of the graph at most finitely often.) The loop-erasure ofsimple random walk is called loop-erased random walk, and was first studied by Lawler [161].Let {v0, v1, . . . , vn} be an enumeration of the vertices of a finite graph G and define a sequence oftrees Ti in G as follows:1. Let T0 have vertex set v0 and no edges.2. Given Ti, start an independent random walk from vi+1 stopped when it hits the set of verticesalready included in the tree Ti.3. Form the loop-erasure of this random walk path and let Ti+1 be the union of Ti with thisloop-erased path.4. Let T = Tn =\u22c3ni=1 Ti.Again, this algorithm clearly produces a spanning tree of G. Once again, however, the followingtheorem is very surprising. It is even surprising that the distribution of the tree T does not dependon the enumeration we chose.Theorem (Wilson 1996). The random tree T is a uniform spanning tree of G.Wilson\u2019s proof [228] uses an ingenious notion of \u2018cycle popping\u2019, of which the above algorithm ismerely one implementation. A major advantage of Wilson\u2019s algorithm over the Aldous-Broder al-gorithm is that it readily extends to generate the wired uniform spanning forest of an infinite graph.This led to a surge of progress on the wired uniform spanning forest following its introduction, inparticular the landmark work of Benjamini, Lyons, Peres, and Schramm [44].1.5.3 Uniform spanning forestsWe now define the free and wired uniform spanning forests of an infinite graph. These will bedefined as weak limits over exhaustions. These weak limits were both implicitly proven to exist byPemantle [190] in 1991, although the wired uniform spanning forest was not considered explicitlyuntil the work of Ha\u00a8ggstro\u00a8m [109] in 1995.We write USTG for the law of the uniform spanning forest of finite graph G. The free uniformspanning forest measure FUSFG is defined to be the weak limit of the sequence \u3008USTGn\u3009n\u22651, sothatFUSFG(S \u2282 F) = limn\u2192\u221eUSTGn(S \u2282 T ).for each finite set S \u2282 E, where F is a sample of the FUSF of G and T is a sample of the UST ofGn. The wired uniform spanning forest measure WUSFG is defined to be the weak limit of thesequence \u3008USTG\u2217n\u3009n\u22651, so thatWUSFG(S \u2282 F) = limn\u2192\u221eUSTG\u2217n(S \u2282 T )351.5. Uniform spanning forestsfor each finite set S \u2282 E, where F is a sample of the WUSF of G and T is a sample of the UST ofG\u2217n.The existence of these limits follow immediately from the Transfer-Current Theorem, togetherwith what we know about infinite electrical networks, since, for example, we know that currentson Gn converge to free currents on G and hence that the transfer-current matrix on Gn convergesto the free transfer current matrix on G (defined in the obvious way). The wired case is similar.In particular, Kirchhoff\u2019s effective resistance formula and the transfer-current theorem extend toboth the wired and free uniform spanning forests in the natural way. This point of view was notavailable to Pemantle at the time of his original work, but was available to Benjamini, Lyons, Peres,and Schramm [44], who reproved convergence in this way and deduced the following theorem.Theorem (Benjamini, Lyons, Peres, and Schramm 2001). Let G be an infinite graph. Then the freeand wired uniform spanning forests of G coincide if and only if G does not admit any non-constantharmonic functions of finite Dirichlet energy.In particular, we have that the free and wired forests coincide in any amenable transitive graphand that, by the theorems of Benjamini and Schramm [47, 48] stated in Section 1.3.3, we have thatthe free and wired uniform spanning forests of a bounded degree planar graph are different if andonly if the graph is transient.1.5.4 The number of trees in the wired forestAlthough they are defined as limits of trees, the uniform spanning forests of a graph need not beconnected. Indeed, we have the following theorem of Pemantle [190].Theorem (Pemantle 1991). The uniform spanning forest of Zd is connected if and only if d \u2264 4.Why is dimension four important? A much clearer picture emerged following the introductionof Wilson\u2019s algorithm. In their seminal paper [44], Benjamini, Lyons, Peres, and Schramm showedhow to extend Wilson\u2019s algorithm to generate the wired uniform spanning forest of any transientgraph. (It is obvious how to extend both the Aldous-Broder algorithm and Wilson\u2019s algorithmto infinite recurrent graphs.) This allowed them to prove many things. Their extension, calledWilson\u2019s algorithm rooted at infinity, can be described as follows. Let G be a transient graphand let \u3008vi\u3009i\u22651 be an enumeration of its vertices. Define a sequence of trees Fi in G as follows:1. Let F0 be the empty forest, that has no vertices or edges.2. Given Fi, start an independent random walk from vi+1 stopped when it hits the set of verticesalready included in the tree Fi. If the walk never hits this set, then it runs forever.3. Form the loop-erasure of this random walk path and let Fi+1 be the union of Fi with thisloop-erased path.4. Let F =\u22c3i\u22651 Fi.361.5. Uniform spanning forestsGiven the correctness of Wilson\u2019s algorithm, the following is fairly easy to verify from the definitions.(Simply consider running Wilson\u2019s algorithm on each G\u2217n with v0 = \u2202n and the other verticesappearing in the order specified, and take the limit as n\u2192\u221e.)Theorem (Benjamini, Lyons, Peres, and Schramm 2001). Let G be a transient graph. Then therandom forest F generated by Wilson\u2019s algorithm rooted at infinity is a wired uniform spanningforest of G.Using this algorithm, Benjamini, Lyons, Peres, and Schramm were able to prove the followingtheorem. We say that a graph G has the intersection property if whenever X and Y areindependent random walks on G, their traces {Xn : n \u2265 0} and {Yn : n \u2265 0} have non-emptyintersection a.s. (or, equivalently, if the traces have infinite intersection a.s.). We say that G hasthe non-intersection property if the traces instead have only finite intersection a.s.Theorem (Benjamini, Lyons, Peres, and Schramm 2001). Let G be an infinite graph. Then thewired uniform spanning forest of G is connected a.s. if and only if G has the intersection property.If G has the non-intersection property, then the wired uniform spanning forest of G has infinitelymany components almost surely.Perhaps this seems that it should be obvious from Wilson\u2019s algorithm, but it is not. Whatis obvious is that the WUSF is connected if and only if a random walk almost surely intersectsan independent loop-erased random walk. The fact that these two properties are equivalent wasproved in a companion paper by Lyons, Peres, and Schramm [174], using a clever second momentargument adapted from the work of Fitzsimmons and Salisbury [90, 204].It can be shown that every transitive graph either has the intersection property or the non-intersection property, yielding the following theorem, which is an analogue of the Newman-Schulmantheorem from percolation.Corollary (Benjamini, Lyons, Peres, and Schramm 2001). Let G be a transitive graph. Then thewired uniform spanning forest of G is either connected or has infinitely many connected componentsalmost surely.Moreover, it can be shown that a transitive graph has the intersection property if and only if\u2211n\u22651npn(v, v) =\u221efor some (and hence every) vertex v. Once all this is in place, Pemantle\u2019s theorem follows fromPo\u00b4lya\u2019s estimate (1.2.1). (The fact that Zd has the intersection property if and only if d \u2264 4 wasoriginally proved by Erdo\u00a8s and Taylor in 1960 [88].) More generally, we deduce that the wireduniform spanning forest is disconnected a.s. in any transitive graph satisfying a 5-dimensionalisoperimetric inequality.371.5. Uniform spanning forestsGreatly extending Pemantle\u2019s theorem, Benjamini, Kesten, Peres and Schramm [41] discoveredthat the transition between connectivity and disconnectivity in dimension four is merely the first ofan infinite family of related transitions occurring every four dimensions. In joint work with YuvalPeres [134] (not included in this thesis), we extended this theorem, and developed a detailed pictureof how the adjacency structure of the trees in the USF of Zd varies as a function of d. In particular,we showed that the adjacency structure of the forest undergoes a qualitative change every time thedimension increases and is above four, rather than just every four dimensions.1.5.5 Geometry of trees in the wired forestAfter connectivity, the most basic property of a forest is the number of ends its components have.Recall that we defined the space of ends of a tree in Section 1.2.4. Components of the WUSF areone-ended a.s. in many large classes of graphs. Generally speaking, we expect that components ofthe WUSF will be one-ended a.s. in any transient graph that we have not constructed to serve as acounterexample. The first generally applicable one-endedness theorem was proven using Wilson\u2019salgorithm by Benjamini, Lyons, Peres, and Schramm [44].Theorem (Benjamini, Lyons, Peres, and Schramm 2001). Let G be a unimodular transitive graphthat is not rough isometric to Z. Then every component of the wired uniform spanning forest of Gis one-ended almost surely.Their proof analysed the recurrent and transient cases separately. The proof in the transientcase involved an innovative use of the mass-transport principle, the discovery of which was recountedin the following memorable anecdote of Russ Lyons [1].To me, Oded\u2019s most distinctive mathematical talent was his extraordinary clarity ofthought, which led to dazzling proofs and results. Technical difficulties did not obscurehis vision. Indeed, they often melted away under his gaze. At one point when the fourof us [Oded Schramm, Russ Lyons, Itai Benjamini, and Yuval Peres] were working onuniform spanning forests, Oded came up with a brilliant new application of the Mass-Transport Principle. We were not sure it was kosher, and I still recall Yuval asking meif I believed it, saying that it seemed to be \u201csmoke and mirrors.\u201d However, when Odedexplained it again, the smoke vanished.Let us now briefly explain their proof. They begin with a relatively straightforward applicationof the mass transport principle to deduce that every component has at most two ends. For thispart of the proof, they observe that there is a natural (in particular, automorphism invariant)oriented version of the WUSF of a transient graph, in which every vertex has exactly one orientededge emanating from it in the forest. This can be defined by orienting towards the boundaryvertex when taking the weak limit, or equivalently by orienting edges of the forest according tothe direction they are traversed by the loop-erased random walks used to generate the forest whenrunning Wilson\u2019s algorithm. Define the core of a forest to be the set of vertices that lie on some381.5. Uniform spanning forestssimple bi-infinite path in the forest (which is empty if and only if every tree in the forest is finiteor one-ended). The core of a two-ended tree is a bi-infinite path which we call the trunk. Let Fbe the OWUSF of a transient unimodular transitive graph. Observe that the unique oriented edgein F emanating from a vertex in the core of F must have its other endpoint in the core also. Definea mass transportf(u, v)= P(u is in the core, v is the endpoint of the unique oriented edge emanating from u in F).Then we have that \u2211vf(o, v) = P(o is in the core)and \u2211vf(v, o) = E|{v in the core : the oriented edge emanating from v points into o}|= E[(|{v in the core : v is adjacent to o}| \u2212 1)1(o is in the core)]Thus, the mass-transport principle implies thatE[|{v in the core : v is adjacent to o}| | o is in the core] = 2.If F has a multiply-ended component with positive probability, then on this event its core isnonempty, and every vertex in the core has at least two other vertices in the core adjacent toit. If F has a component with more than two ends with positive probability, then on this eventthere exists a vertex in the core with at least three vertices in the core adjacent to it, and it fol-lows from automorphism invariance that the origin is such a vertex with positive probability. Thestatement about the expectation above implies that this cannot be the case, and so we deduce thatevery component of the WUSF has at most two ends as claimed.The authors then rule out the existence of a two ended component, considering separatelythe cases that the WUSF is connected or disconnected. The case that the WUSF is connectedis handled by a reasonably straightforward but still very elegant argument, a generalised versionof which appears in [127] (Chapter 4 of this thesis). For the disconnected case, the authors firstshow that if F contains two-ended components, then it is possible to sample F conditioned on theorigin being contained in the trunk of its component by first sampling the trunk, and then runningWilson\u2019s algorithm \u2018rooted at the trunk\u2019 to sample the rest of F, i.e., beginning the recursion inWilson\u2019s algorithm by setting F0 to be the trunk. The fact that this works, which the authors callthe trunk lemma, is intuitively plausible but unfortunately rather technical to prove.Once the trunk lemma is in place, we come to Schramm\u2019s \u201csmoke and mirrors\u201d mass-transportargument. Suppose for contradiction that F contains a two-ended component with positive proba-391.5. Uniform spanning forestsbility. Since F is disconnected, the trunk lemma implies that, conditional on both the trunk of thetree containing the origin, denoted T , and the event that this trunk contains the origin, there issome vertex w of G such that the random walk started at w does not hit T with positive probability.Let W be the set of such vertices. Then it is not hard to see that W must be adjacent to somevertex of T , and automorphsim-invariance implies that the origin is such a vertex with positiveprobability. it follows that, with positive probability, a random walk started at the origin does notreturn to T after time zero. In notation,E[\u03c4+T 1(o is in the trunk of a two-ended component)]=\u221eNow, for each vertex v in the trunk of a two-ended component of F, let Bv be the set of verticesu, including v itself, such that any simple infinite path starting at u in F must pass through v,called the bush at v. The trunk lemma implies that the probability that u is in Bo conditional onT and the event o \u2208 T is equal to the probability that a random walk started at u hits T , and doesso for the first time at o. By time reversal, this is equal to the expected number of times a randomwalk started at o hits u before returning to T for the first time, and so, summing over u,E[|Bo|1(o in the trunk of a two-ended component)]= E[\u03c4+T 1(o in the trunk of a two-ended component)]=\u221eOn the other hand, if we transport mass from each vertex of u of G in a two-ended component ofF to the unique vertex v such that u \u2208 Bv, we obtain from the mass-transport principle thatE|Bo|1(o is in the trunk of a two-ended component) = P(o is in a two-ended component) <\u221e,giving us the desired contradiction.A more geometric understanding of one-endedness in the transient case was developed byLyons, Morris, and Schramm [170] in 2008. These authors gave an isoperimetric criterion forone-endedness, very similar to the Thomassen criterion for transience. Their proof was based onelectrical techniques; see [173] for an updated exposition.Theorem (Lyons, Morris, and Schramm 2008). Let G be a graph that satisfies an f(t)-isoperimetricinequality for some increasing function f : (0,\u221e)\u2192 (0,\u221e) for which there exists a constant \u03b1 suchthat f(t) \u2264 t and f(2t) \u2264 \u03b1f(t) for all t \u2208 (0,\u221e). If\u222b \u221e11f(t)2dt <\u221e,then every component of the wired uniform spanning forest of G is one-ended almost surely.Corollary (Lyons, Morris, and Schramm 2008). Let G be a transitive transient graph. Then everycomponent of the wired uniform spanning forest of G is one-ended almost surely.401.5. Uniform spanning forestsTrees with one end (or finitely many ends) are clearly recurrent by the Nash-Williams Criterion.Recurrence of components in the WUSF holds even more generally than one-endedness, however,as was proven by Morris in 2003 [185] using electrical methods.Theorem (Morris 2003). Let G be an infinite graph. Then every component of the wired uniformspanning forest of G is recurrent almost surely.1.5.6 The interlacement Aldous-Broder algorithmUnlike Wilson\u2019s algorithm, it was for a long time not apparent how to extend the Aldous-Broderalgorithm to generate the wired uniform spanning forest of an infinite transient graph. In ourpaper [128] (Chapter 5 of this thesis), we showed that such an extension can be done by replacingthe random walk in the classical algorithm with the random interlacement process. Theinterlacement Aldous-Broder algorithm turns out, generally speaking, to be better suited thanWilson\u2019s algorithm for studying ends in the WUSF, and allowed us to prove several new results. Itis also of central importance in upcoming work in which we compute the critical exponents for theUSF in high dimensions.The interlacement process was originally introduced by Sznitman [216] to study the disconnec-tion of cylinders and tori by a random walk trajectory, and was generalised to arbitrary transientgraphs by Teixeira [218]. See the monographs [67, 82] for detailed introductions. Roughly speaking,the interlacement process I on a transient graph G is a \u2018Poissonian soup\u2019 of bi-infinite randomwalk trajectories in G. As we increase a real time parameter t, more and more of these trajectoriesappear. Formally, I is a random subset of W\/ \u223c \u00d7R, where W is the space of doubly infinitepaths in G that visit each vertex at most finitely often, and \u223c holds two such paths to be equivalentif and only if they are reparameterizations of each other.Theorem (H. 2015). Let G be a transient graph, let I be the interlacement process on G, and lett \u2208 R. For each vertex v of G, let \u03c4t(v) be the smallest time greater than t such that there existsa trajectory (W\u03c4t(v), \u03c4t(v)) \u2208 I passing through v, and let et(v) be the oriented edge of G that istraversed by the trajectory W\u03c4t(v) as it enters v for the first time. ThenABt(I ) :={\u2212et(v) : v \u2208 V }has the law of the oriented wired uniform spanning forest of G.We used this algorithm to prove an anchored isoperimetric condition for one-endedness, an-swering positively a question of Lyons, Morris, and Schramm [170].Theorem (H. 2015). Let G be a graph that satisfies an anchored f(t)-isoperimetric inequality forsome increasing function f : (0,\u221e)\u2192 (0,\u221e) for which there exists a constant \u03b1 such that f(t) \u2264 tand f(2t) \u2264 \u03b1f(t) for all t \u2208 (0,\u221e). Suppose that f also satisfies each of the following conditions:411.5. Uniform spanning forests1. \u222b \u221e11f(t)2dt <\u221eand2. \u222b \u221e1exp(\u2212\u03b5(\u222b \u221es1f(t)2dt)\u22121)ds <\u221efor every \u03b5 > 0.Then every component of the wired uniform spanning forest of G is one-ended almost surely.On the other hand, we also proved the following, which implies that it is not possible to tellwhether or not the components of the WUSF of a graph are one-ended a.s. from the coarse geometryof the graph alone. This answered negatively a question of Lyons, Morris, and Schramm [170].Theorem (H. 2015). There exist two bounded degree, rough isometric graphs G and G\u2032 such thatevery component of the wired uniform spanning forest of G is one-ended almost surely, but the wireduniform spanning forest of G\u2032 contains a component with uncountably many ends almost surely.What can we say about graphs that do have multiply ended components in their WUSF? Oneexample of a transient graph in which the WUSF has multiply-ended components is obtained fromZ5 by attaching an infinite path to each of the vertices u = (0, 0, 0, 0, 0) and v = (2, 0, 0, 0, 0). TheWUSF of this graph has the same distribution as the union of the WUSF of Z5 with each of the twoadded paths. If the vertices u and v are in the same component of the forest, then this componenthas exactly three ends, while all other components have one end. If not, then the componentcontaining u and the component containing v both have two ends, while all other components haveone end. In particular, the event that the WUSF contains a two-ended component has probabilitystrictly between 0 and 1. Nevertheless, the number of excessive ends of F, that is, the sum overall components of F of the number of ends minus 1, is equal to two a.s.In [128], we used the interlacement Aldous-Broder algorithm to prove that this is a generalphenomenon, answering positively a further question of Lyons, Morris, and Schramm [170].Theorem (H. 2015). Let G be an infinite graph. Then there exists a constant c \u2208 N \u222a {\u221e} suchthat the wired uniform spanning forest of G has c excessive ends almost surely.The key to all our results proved using interlacement Aldous-Broder is that it lets us view theWUSF as the stationary measure of the continuous time Markov process \u3008ABt(I )\u3009t\u2208R.In upcoming work, we use the interlacement Aldous-Broder algorithm to compute the criticalexponents for the WUSF of Zd, d \u2265 4, or more generally of any bounded degree graph satisfying a4-dimensional isoperimetric inequality. An example of one of the exponent theorems we obtain isthe following. The result is new even in the case of Zd, although related results have been obtained421.5. Uniform spanning forestsby Barlow and Ja\u00b4rai [30] and Bhupatiraju, Hanson, and Ja\u00b4rai [52] (who computed the exponent forthe extrinsic diameter up to a logarithmic correction). Here, the past of a vertex v in a sample ofthe wired uniform spanning forest F is defined to be the union of the finite connected componentsof F \\ {v}.Theorem (H. 2017). Let G be a bounded degree graph satisfying a d-dimensional isoperimetricinequality for some d > 4, and let F be the wired uniform spanning forest of G. Then there existsa positive constant C such thatC\u22121R\u22121 \u2264 P(the intrinsic diameter of pastF(v) is at least R) \u2264 CR\u22121andC\u22121R\u22121\/2 \u2264 P(|pastF(v)| \u2265 R) \u2264 CR\u22121\/2for every vertex v of G and every R \u2265 1.Using these exponents, we also show that every tree of the WUSF has spectral dimension 4\/3almost surely under the same conditions, meaning that the n-step return probability of a randomwalk on each of the trees decays like n\u22122\/3+o(1) almost surely.1.5.7 Indistinguishability of trees and the geometry of trees in the free forestAs we have seen, most of the basic properties of the WUSF are quite well understood. In com-parison, our understanding of the FUSF is relatively poor, and several very basic questions remainopen. For example, the following very basic property of the FUSF was proven only recently in jointwork with Asaf Nachmias [131] (Chapter 6 of this thesis), and independently by Tima\u00b4r [223]. (Thecorresponding result for the WUSF is an easy consequence of the work of Benjamini, Lyons, Peres,and Schramm [44]: see Lemma 9.5.4 of this thesis.)Theorem (H. and Nachmias 2015, Tima\u00b4r 2015). Let G be a unimodular random rooted graph withE[deg(\u03c1)] < \u221e and let F be a sample of the free uniform spanning forest of G. Then F is eitherconnected or has infinitely many components almost surely.The result remains open in the nonunimodular transitive case, where it is also expected tohold. Our proof was based on update-tolerance, a property of the USFs that we introduced inthe earlier work [127]. This property allows us to insert an edge of our choice into the uniformspanning forest, so long as we also delete some edge that depends on the forest and the edge thatwe chose to insert, in such a way that the resulting forest is absolutely continuous with respectto the uniform spanning forest that we started with. This property plays the role for USFs thatinsertion tolerance played for percolation. However, because we are forced to insert and deleteedges at the same time, update-tolerance is a much weaker property than insertion-tolerance. Inparticular, we cannot simply add an edge to merge two trees into one, so that we cannot simplyadapt the Newman-Schulman proof from percolation to show that there is either one or infinitelymany components.431.5. Uniform spanning forestsInstead, we first showed that every component has a well-defined frequency, meaning that ifX is a random walk on G independent of the forest F, then for each tree T of F the limitFreq(T ) := limn\u2192\u221e1nn\u2211i=11(Xn \u2208 T )exists and is almost surely equal to some constant depending on T but not on the walk X. If thereare finitely many components, their frequencies must add to 1, and ergodicity implies that the setof frequencies appearing in the forest is non-random. We then argue that, by performing updatesin an appropriate way, we can change the set of frequencies that appear, giving us a contradiction.(It is here that unimodularity is used in a crucial way to show that a tree with positive frequencycannot have a zero frequency branch.)The update-tolerance method allowed us to solve a conjecture of Benjamini, Lyons, Peres,and Schramm [44], which stated that both the WUSF and FUSF satisfy a Lyons-Schramm typeindistinguishability theorem.Theorem (H. and Nachmias 2015). Let G be a unimodular random rooted graph with E[deg(\u03c1)] <\u221e and let F be a sample of either the free uniform spanning forest or the wired uniform spanningforest of G. Then for each automorphism-invariant Borel-measurable set A of subgraphs of G,either every connected component of F is in A or every connected component of F is not in Aalmost surely.Partial progress on the conjecture was also made in the independent work of Tima\u00b4r [223], whoproved that components of the FUSF are indistinguishable from each other when G is a unimodulartransitive graph such that the FUSF and WUSF are different. Our proof in that case was handledby a similar approach to his, whereas the other case (of the WUSF or the FUSF when it is equalto the WUSF) was handled by a completely different method.Along the way, we also answered a further question of Benjamini, Lyons, Peres, and Schrammin the unimodular case. Those authors had proved that if the WUSF and FUSF of a transitivegraph are different, then, in contrast to the WUSF, the FUSF has at least one component that istransient and infinitely ended. We proved that in fact, if this occurs in the unimodular case, thenevery component of the FUSF is transient and infinitely-ended.Theorem (H. and Nachmias 2015, Tima\u00b4r 2015). Let G be a unimodular random rooted graph withE[deg(\u03c1)] <\u221e and let F be a sample of FUSFG. If the measures FUSFG and WUSFG are distinct,then every component of F is transient and has infinitely many ends almost surely.Although this would follow as a special case of the indistinguishability theorem above, we in facthad to prove it separately as part of the proof of the general theorem. The result is also expectedto hold in the nonunimodular case.441.5. Uniform spanning forests1.5.8 Uniform spanning forests of planar graphsUnlike for the WUSF, there is no simple criterion for the connectivity of the FUSF. Indeed, it isnot even known whether the number of components of the FUSF is nonrandom for any given fixedgraph. For planar graphs, the situation is much better due to the following duality between theFUSF and WUSF.Proposition (Benjamini, Lyons, Peres, and Schramm 2001). Let M be a simply connected planarmap with locally finite dual, and let F be the free uniform spanning forest of M . Then F\u2020 isdistributed as the wired uniform spanning forest of M \u2020. In particular, F is connected almost surelyif and only if every component of F\u2020 is one-ended almost surely.Using this duality, it follows easily from the theorems we have discussed previously that, forexample, the FUSF of any transitive, simply-connected planar map with locally finite dual isconnected almost surely. Benjamini, Lyons, Peres, and Schramm [44] asked whether the FUSFof any bounded degree, simply connected map is connected. In joint work with Nachmias [130](Chapter 10 of this thesis), we answered this question positively.Theorem. The free uniform spanning forest is almost surely connected in any bounded degree,simply-connected planar map.In light of the above duality, this theorem is equivalent in the case of a locally finite dual to thefollowing dual theorem. The general case is then easily deduced from the locally finite dual casevia an approximation argument.Theorem. Every component of the wired uniform spanning forest is one-ended almost surely inany bounded codegree, simply-connected planar map.Our proof uses circle packing in an essential way. Suppose T is a simple, simply-connectedtriangulation, circle packed in either the plane or the disc, and let e be a fixed edge of T . Let F bethe wired uniform spanning forest of T . We show that if the components of e\u2212 and e+ in F \\ {e}both have large diameter as measured in either the Euclidean or hyperbolic metric as appropriateon the circle packing, then with high probability e is not in F. This is done by exploring F in aMarkovian way, analyzing the circle packing to show that certain effective resistances are small,and then using Kirchoff\u2019s effective resistance formula.In the bounded degree case, the proof can be made quantitative, and led us to a computationof the critical exponents of the uniform spanning forest for transient, simply-connected, polyhedralplanar maps with bounded degrees and codegrees. Surprisingly, we found that, if one measuresdistances and areas using the hyperbolic geometry of the circle packings rather than the usualcombinatorial geometry, the exponents are universal over the entire class of graphs. For example,the probability that the past of a vertex has diameter at least r in the hyperbolic metric alwaysscales like 1\/r, even for triangulations with rather unusual circle packings such as those in the figureabove.451.5. Uniform spanning forestsFigure 1.8: Two bounded degree, simple, proper plane triangulations for which the graph distanceis not comparable to the hyperbolic distance. Similar examples are given in [215, Figure 17.7]. Left:In this example, rings of degree seven vertices (grey) are separated by growing bands of degree sixvertices (white), causing the hyperbolic radii of circles to decay. The bands of degree six verticescan grow surprisingly quickly without the triangulation becoming recurrent [211]. Right: In thisexample, half-spaces of the 8-regular (grey) and 6-regular (white) triangulations have been gluedtogether along their boundaries; the circles corresponding to the 6-regular half-space are containedinside a horodisc and have decaying hyperbolic radii.1.5.9 USFs of multiply-connected planar mapsIt is natural to ask whether our result concerning the connectivity of the FUSF extends to allbounded degree planar maps, without the assumption that they are simply connected. Indeed, ifsuch an extension were true, it would have interesting consequences for finite planar graphs. In[130] (Chapter 10) we present an example, which was analyzed in collaboration with Gady Kozma,to show that the theorem does not admit such an extension.Theorem. There exists a bounded degree planar graph G such that the free uniform spanning forestof G is disconnected with positive probability.In light of this example, it is an interesting problem to determine for which bounded degree planargraphs the FUSF is connected. As the final result of [130], we initiated progress on this questionby showing that the FUSF is connected almost surely on any bounded degree, countably-connectedplanar map. Here, countably connected means that the surface associated to the map is home-omorphic to a domain in C \u222a {\u221e} whose complement has at most countably many connectedcomponents.Theorem. Let M be a countably-connected, bounded degree planar map. Then the free uniformspanning forest of M is connected almost surely.The proof of this theorem requires two additional ingredients besides those appearing in thesimply-connected case. The first is the introduction of the transboundary uniform spanning forest461.5. Uniform spanning forestsFigure 1.9: Left: A circle packing in the multiply-connected circle domain D \\ {0}. Right: A circlepacking in a circle domain domain with several boundary components.(TUSF), which is dual to the FUSF of a (not necessarily proper) plane graph in the same way thatthe WUSF is dual to the FUSF of a proper plane graph. The second additional ingredient, whichwe use to prove this dual statement, is a transfinite induction inspired by He and Schramm\u2019s workon the Koebe Conjecture [122, 207].471.6. Unimodular random graphs1.6 Unimodular random graphsA rooted graph (G, \u03c1) is a connected, locally finite graph G = (V,E) together with a distinguishedvertex \u03c1, called the root. A graph isomorphism \u03c6 : G \u2192 G\u2032 is an isomorphism of rooted graphsif it maps the root to the root. The local topology (see [50]) is the topology on the set G\u2022 ofisomorphism classes of rooted graphs is defined so that two rooted graphs are close if they havelarge isomorphic balls around their roots. Formally, it is the topology induced by the metricdloc((G, \u03c1), (G\u2032, \u03c1\u2032))= e\u2212R,whereR = R((G, \u03c1), (G\u2032, \u03c1\u2032))= sup{R \u2265 0 : BR(G, \u03c1) \u223c= BR(G\u2032, \u03c1\u2032)},i.e., the maximal radius such that the balls BR(G, \u03c1) and BR(G\u2032, \u03c1\u2032) are isomorphic as rootedgraphs. A random rooted graph is a random variable taking values in the space G\u2022 endowedwith the local topology. Similarly, a doubly-rooted graph is a graph together with an orderedpair of distinguished (not necessarily distinct) vertices. Denote the space of isomorphism classes ofdoubly-rooted graphs equipped with this topology by G\u2022\u2022.A mass-transport is a Borel function f : G\u2022\u2022 \u2192 [0,\u221e]. A random rooted graph (G, \u03c1) is saidto be unimodular if it satisfies the Mass Transport Principle: for every mass transport f ,E[\u2211v\u2208Vf(G, \u03c1, v)]= E[\u2211u\u2208Vf(G, u, \u03c1)]. (MTP)Unimodularity of random rooted graphs was first defined by Benjamini and Schramm in theirseminal work [50], and was developed thoroughly by Aldous and Lyons [7], who showed that muchof what was known about, for example, percolation and uniform spanning forests on unimodulartransitive graphs generalised to the unimodular random rooted graph setting, often under theadditional assumption of bounded degree or of finite expected degree. Moreover, restricting oneselfto use only unimodularity rather than more delicate properties of a specific example often leadsto proofs that are not only more general, but also much more simple and elegant. Our work withYuval Peres on the collision property [133] (Chapter 3) is a good example of this.It is also possible to define, with similar definitions, the spaces of rooted and doubly-rootedmaps M\u2022 and M\u2022\u2022, and the corresponding notions of unimodularity and the Mass-TransportPrinciple. Similarly, we have unimodular random rooted marked graphs and maps, in which everyvertex and\/or edge has a random mark taking values in some Polish space. For example, givena unimodular transitive graph G and some automorphism-invariant percolation \u03c9 on G (such asBernoulli bond percolation, the FUSF, or the WUSF), we have that (G, \u03c1, \u03c9) is a unimodularrandom rooted marked graph, with edge marks in {0, 1}.Every unimodular transitive graph can be made into a unimodular random rooted graph bychoosing a root vertex arbitrarily. Furthermore, if G is a (possibly random) finite graph and \u03c1481.6. Unimodular random graphsis chosen uniformly from the vertex set of G, then the random rooted graph (G, \u03c1) is unimodu-lar. This leads to many further examples: It can be shown that the set of distributions of theunimodular random rooted graphs is closed under the topology of weak convergence, and hencethat if (Gn, \u03c1n) are a sequence of finite random graphs with uniform random roots converging tosome infinite random rooted graph (G, \u03c1) in distribution, then the limit (G, \u03c1) is itself unimodular.This procedure is known as a Benjamini-Schramm limit. A unimodular random rooted graphthat can be obtained as a Benjamini-Schramm limit of finite graphs is said to be sofic. It is amajor open problem, with important ramifications in group theory, to determine whether everyunimodular random rooted graph is sofic.Conjecture (Aldous and Lyons 2007). Every unimodular random rooted graph is sofic.It is the widespread belief, certainly among group theorists, that the conjecture is false. Wechoose to withhold judgment.1.6.1 ReversibilityUnimodularity is closely related to reversibility. A random rooted graph (G, \u03c1) with at least twovertices is said to be stationary if, when \u3008Xn\u3009n\u22650 is a simple random walk on G started at theroot,(G, \u03c1)d= (G,Xn)for all n and is said to be reversible if(G, \u03c1,Xn)d= (G,Xn, \u03c1)for all n. Note that this is not the same as the reversibility of the random walk on G, which holdsfor any graph6. Every reversible random rooted graph is clearly stationary, but the converse neednot hold in general: transitive graphs that are not unimodular are an important example.Aldous and Lyons proved the following correspondence between reversible random rooted graphsand unimodular random rooted graphs of finite expected degree: If (G, \u03c1) is a unimodular randomrooted network with E[deg \u03c1] <\u221e (that is almost surely not equal to a single degree zero vertex),then biasing the law of (G, \u03c1) by deg \u03c1 (that is, re-weighting the law of (G, \u03c1) by the Radon-Nikodymderivative deg \u03c1\/E[deg \u03c1]) yields the law of a reversible random rooted network. Conversely, if (G, \u03c1)is a reversible random rooted network, then biasing the law of (G, \u03c1) by deg\u22121 \u03c1 yields the law ofa unimodular random rooted network. This is summarized by the following bijection:{(G, \u03c1) unimodular with E[deg \u03c1] <\u221e, deg(\u03c1) \u2265 1 a.s.} bias by deg \u03c1\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2192\u2190\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212bias by deg\u22121 \u03c1{(G, \u03c1) reversible}.6Rather, a random rooted graph (G, \u03c1) is reversible if and only if the G\u2022\u2022-valued Markov process\u3008(G,Xn, Xn+1)\u3009n\u22650 is reversible.491.6. Unimodular random graphs1.6.2 Invariant amenability and nonamenabilityGiven a unimodular random rooted graph (G, \u03c1) and a random subgraph \u03c9 of G, we say that \u03c9 is apercolation on (G, \u03c1) if (G, \u03c1, \u03c9) is unimodular when considered as a random rooted marked graph,with edge marks in {0, 1}. A primary example is given by (G, \u03c1) a unimodular transitive graphand \u03c9 an automorphism-invariant random subgraph of G. The following property of nonamenableunimodular transitive graphs was discovered for regular trees by Ha\u00a8ggstro\u00a8m [111], and was extendedto all unimodular transitive graphs by Benjamini, Lyons, Peres, and Schramm [36].Theorem. Let G be a nonamenable unimodular transitive graph. Then there exists \u03b1 > 0 suchthat every automorphism-invariant percolation \u03c9 on G with Edeg\u03c9(\u03c1) \u2265 deg \u03c1 \u2212 \u03b1 has an infiniteconnected component almost surely.This theorem gives a great deal of insight about percolation on nonamenable proofs, and isproven very easily. We call an automorphism-invariant percolation \u03c9 on G finitary if all itsclusters are finite almost surely. We write K\u03c9(v) for the connected component of \u03c9 containing v.Suppose that \u03c9 is a finitary percolation on G, and define a mass-transportF (G, u, v, \u03c9) =deg u\u2212 deg\u03c9 u|K\u03c9(u)| 1(v \u2208 K\u03c9(u))ThenE\u2211vF (G, \u03c1, v, \u03c9) = E [deg \u03c1\u2212 deg\u03c9 \u03c1]andE\u2211uF (G, u, \u03c1, \u03c9) = E[ |\u2202EK\u03c9(\u03c1)||K\u03c9(\u03c1)|],so that the Mass-Transport Principle yields thatE [deg \u03c1\u2212 deg\u03c9 \u03c1] = E[ |\u2202EK\u03c9(\u03c1)||K\u03c9(\u03c1)|].Since G is nonamenable, the right hand side is bounded by a positive constant, and the resultfollows.Aldous and Lyons [7] observed that Ha\u00a8ggstro\u00a8m\u2019s theorem can be used as a definition for anotion of nonamenability for unimodular random graphs, which we call invariant nonamenability.If (G, \u03c1) is a unimodular random rooted graph and \u03c9 is a random subgraph of G, we say that \u03c9is a percolation on G if (G, \u03c1, \u03c9) is a unimodular random rooted marked graph. We say that aunimodular random rooted graph (G, \u03c1) is invariantly nonamenable if there exists a positiveconstant \u03b5 such thatE[ |\u2202EK\u03c9(\u03c1)||K\u03c9(\u03c1)|]> \u03b5for every finitary percolation \u03c9. By the mass-transport argument above, if Edeg \u03c1 < \u221e this is501.6. Unimodular random graphsequivalent to there existing a positive constant \u03b1 such thatEdeg\u03c9 \u03c1 \u2264 Edeg \u03c1\u2212 \u03b1for every finitary percolation \u03c9. In other words, (G, \u03c1) is invariantly nonamenable if the conclusionof Ha\u00a8ggstro\u00a8m\u2019s theorem holds for it. If (G, \u03c1) is not invariantly nonamenable we say that it isinvariantly amenable.Invariant nonamenability turns out to be the right notion of nonamenability for unimodularrandom graphs in many ways. For example, it is much easier to prove that supercritical percolationclusters on a transitive unimodular random graph are invariantly nonamenable than to prove theyhave anchored expansion. Moreover, it is often easier to use invariant nonamenability to deduceproperties of the graph than it is to use anchored expansion. The following theorem, originallystated for percolations on unimodular random graphs by Benjamini, Lyons, and Schramm [45] andobserved to generalize to unimodular random rooted graphs by Aldous and Lyons [7], allows usto easily deduce many of the properties of invariantly nonamenable graphs from their classicallynonamenable analogues. It plays for invariant nonamenability a role analogous to that Virag\u2019s\u2018ocean and islands\u2019 construction [227] plays in the context of anchored expansion.Theorem (Benjamini, Lyons, and Schramm 1999). Let (G, \u03c1) be an invariantly nonamenable uni-modular random rooted graph. Then there exists a percolation \u03c9 on (G, \u03c1) such that deg(v) isbounded on \u03c9, \u03c9 is (classically) nonamenable, and \u03c9 is almost surely a forest.This theorem easily implies, for example, that the random walk on a bounded degree, invariantlynonamenable random graph has positive speed.1.6.3 Unimodular random planar mapsBenjamini and Schramm were motivated to introduce the notion of unimodular random graphs bytheir work on random planar maps. They proved the following remarkable theorem.Theorem (Benjamini and Schramm 2001). Let (T, \u03c1) be an infinite unimodular random rootedsimple triangulation that is a Benjamini-Schramm limit of finite simple triangulations of the plane.Then T can be circle packed in either the plane C or the punctured plane C \\ {0} almost surely.In light of the work of He and Schramm [121] (who also prove that bounded degree circlepackings in C \\ {0} are recurrent), the following is an immediate corollary of this theorem forsimple triangulations. The general case follows by a straightforward reduction.Corollary (Benjamini and Schramm 2001). Let (M,\u03c1) be a bounded degree unimodular randomrooted planar map that is a Benjamini-Schramm limit of finite planar maps. Then M is recurrentalmost surely.In joint work with Angel, Nachmias, and Ray [21] (Chapter 7), we established a stronger versionof this theorem via a much simpler proof, based on a simple mass-transport argument. Our theorem511.6. Unimodular random graphsshould be compared with the results of Beardon and Stephenson [35] and He and Schramm thatwe mentioned earlier, which also pertained to the connection between the circle packing type andthe average degree in some sense.Theorem. Let (G, \u03c1) be an infinite, simply-connected, ergodic unimodular random rooted planartriangulation. Then eitherE[deg(\u03c1)] = 6, in which case (G, \u03c1) is invariantly amenable and almost surely CPparabolic,or elseE[deg(\u03c1)] > 6, in which case (G, \u03c1) is invariantly nonamenable and almost surely CPhyperbolic.Benjamini-Schramm limits of finite planar maps are easily seen to have average degree at mostsix by Euler\u2019s formula. The case of the Benjamini-Schramm theorem in which the limit is notsimply-connected can be reduced to the simply-connected case by taking universal covers.The hypothesis that (M,\u03c1) is ergodic means that its law is an extreme point of the set oflaws of unimodular random rooted maps. Equivalently, it means that for any event A \u2286M\u2022 suchthat does not depend on the choice of root, (M,\u03c1) has probability either zero or one to be in A .Ergodicity rules out, for example, taking the unimodular random rooted triangulation that is the6-regular triangular lattice with probability 1\/2 and the 7-regular hyperbolic triangulation withprobability 1\/2.Random maps play an important role in the theory of Liouville quantum gravity, which is far toobig a subject to go into on seriously here. The reader is referred to the survey [98] for a summary ofthe state of that field as it stood in 2013, although they should note that substantial progress hasbeen made since then. A central player in this theory is the uniform infinite planar triangulationof Angel and Schramm [24].Theorem (Angel and Schramm 2003). Let Tn be chosen uniformly from the set of triangulationsof the sphere with n vertices, and let \u03c1n be a uniformly chosen random vertex of Tn. Then thereexists a simply-connected random rooted triangulation (T, \u03c1) such that(Tn, \u03c1n)d\u2212\u2212\u2212\u2192n\u2192\u221e (T, \u03c1).The random rooted triangulation (T, \u03c1) is known as the uniform infinite planar triangula-tion (UIPT). A similar result for quadrangulations (i.e., maps in which every face has degree four)is due to Krikun [157].Since the UIPT does not have bounded degrees, its recurrence does not follow from the He-Schramm theorem. Indeed, it remained an open problem to prove that it was recurrent for sometime until it was finally solved by Gurel-Gurevich and Nachmias in 2013 [107], using a strengthened,quantitative version of the original Benjamini-Schramm proof from the bounded degree case.521.6. Unimodular random graphsTheorem (Gurel-Gurevich and Nachmias 2013). Let (M,\u03c1) be a unimodular random rooted planarmap that is a Benjamini-Schramm limit of finite planar maps, and suppose that there exists apositive constant c such thatP(deg \u03c1 \u2265 n) \u2264 exp(\u2212cn)for all n \u2265 1. Then M is recurrent almost surely.Corollary (Gurel-Gurevich and Nachmias 2013). The UIPT and UIPQ are recurrent almost surely.A completely different new proof of this theorem, not using circle packing, has appeared in arecent work of Lee [164].A distinguishing feature of the UIPT is that it enjoys a certain form of spatial Markov property.In [75], following similar work on half-planar models by Angel and Ray [23], Curien showed thatset of all random rooted triangulations with this property form a one parameter family (T\u03ba, \u03c1),with \u03ba \u2208 (0, 2\/27]. Each of these random triangulations is unimodular, and the UIPT is given bythe extremal value \u03ba = 2\/27. Curien also showed that the other (\u03ba < 2\/27) triangulations in thefamily are hyperbolic in various senses. These triangulations were a major motivation for our workon unimodular random planar maps.1.6.4 The dichotomy theoremWhat about maps that are not triangulations? There, the appropriate quantity to look at is notthe average degree but the average curvature. Recall that the internal angles of a regular k-gonare given by (k \u2212 2)pi\/k. We define the angle sum at a vertex v of a map M to be\u03b8(v) = \u03b8M (v) =\u2211f\u22a5vdeg(f)\u2212 2deg(f)pi,where we write f \u22a5 v if the vertex v is incident to the face f , and interpret the sum with multi-plicities if v is multiply incident to f . This definition extends to maps with infinite faces, with theconvention that (\u221e\u22122)\/\u221e = 1. In the case that every face of M has degree at least 3, we interpret\u03b8(v) as the total angle of the corners at v if we form M by gluing together regular polygons, wherewe consider the upper half-space {x + iy \u2208 C : y > 0} with edges {[n, n + 1] : n \u2208 Z} to be aregular \u221e-gon. Of course, M cannot necessarily be drawn in the plane with regular polygons, andthe angle sum at a vertex of M need not be 2pi. We define the curvature of M at the vertex v tobe the angle sum deficit\u03ba(v) = \u03baM (v) = 2pi \u2212 \u03b8(v)531.6. Unimodular random graphsand define the average curvature of a unimodular random rooted map (M,\u03c1), denoted K(M,\u03c1),to be the expected curvature at the rootK(M,\u03c1) = E[\u03ba(\u03c1)] = 2pi \u2212 E\uf8ee\uf8f0\u2211f\u22a5\u03c1deg(f)\u2212 2deg(f)pi\uf8f9\uf8fb .Note that if E[deg(\u03c1)] is finite then K(M,\u03c1) is also finite.In joint work with Angel, Nachmias, and Ray [20] (Chapter 9), we used this notion to greatlyexpand and generalise the dichotomy from the previous subsection, showing that many diverseproperties of a unimodular random planar map are determined by the average curvature. Thetheorem complements, but does not exactly parallel, the dichotomy for deterministic boundeddegree planar maps that we saw earlier.Theorem (Angel, H., Nachmias, and Ray 2016: The Dichotomy Theorem). Let (M,\u03c1) be aninfinite, ergodic, unimodular random rooted planar map and suppose that E[deg(\u03c1)] <\u221e. Then theaverage curvature of (M,\u03c1) is non-positive and the following are equivalent:1. (M,\u03c1) has average curvature zero.2. (M,\u03c1) is invariantly amenable.3. Every bounded degree subgraph of M is amenable almost surely.4. Every subtree of M is amenable almost surely.5. Every bounded degree subgraph of M is recurrent almost surely.6. Every subtree of M is recurrent almost surely.7. (M,\u03c1) is a Benjamini-Schramm limit of finite planar maps.8. (M,\u03c1) is a Benjamini-Schramm limit of a sequence \u3008Mn\u3009n\u22650 of finite maps such thatgenus(Mn)#{vertices of Mn} \u2212\u2212\u2212\u2192n\u2192\u221e 0.9. The Riemann surface associated to M is conformally equivalent to either the plane C or thecylinder C\/Z almost surely.10. M does not admit any non-constant bounded harmonic functions almost surely.11. M does not admit any non-constant harmonic functions of finite Dirichlet energy almostsurely.12. The laws of the free and wired uniform spanning forests of M coincide almost surely.13. The wired uniform spanning forest of M is connected almost surely.14. Two independent random walks on M intersect infinitely often almost surely.15. The laws of the free and wired minimal spanning forests of M coincide almost surely.16. Bernoulli(p) bond percolation on M has at most one infinite connected component for everyp \u2208 [0, 1] almost surely (in particular, pc = pu).17. M is vertex extremal length parabolic almost surely.In light of this theorem, we call a unimodular random rooted map (M,\u03c1) with E[deg(\u03c1)] < \u221e541.6. Unimodular random graphsConnectivityoftheFUSFLipton\u2013TarjanBLPSBLPSAldous\u2013LyonsMasstransport+dualityBenjamini,Curien,GeorgakopoulosAldous\u2013LyonsInvariantly amenableAverage curvature zeroConformally parabolicWMSF = FMSF pc = puHarmonicDirichletfunctions areconstantLiouvilleIntersectionPropertyWUSF connectedBenjamini-SchrammLPSBLS,Aldous-LyonsAldous\u2013LyonsBenjamini\u2013SchrammBenjamini-Schrammlimit of finitelow-genus mapsBenjamini-Schrammlimit of finiteplanar mapsWUSF = FUSFVELparabolicSubgraphconditionsFigure 1.10: The logical structure for the proof of Theorem 9.1.1 in the simply connected case.Implications new to this paper are in red. Blue implications hold for arbitrary graphs; the orangeimplication holds for arbitrary planar graphs, and green implications hold for unimodular randomrooted graphs even without planarity. A few implications between items that are known but notused in the proof are omitted.parabolic if its average curvature is zero (and, in the planar case, clauses (1)\u2013(17) all hold), andhyperbolic if its average curvature is negative (and, in the planar case, the clauses all fail).A central connection between the average curvature and the behaviour of random processes onthe map is given by the following formula for the expected degree of the FUSF. It is known thatthe expected degree at the root of the WUSF of a unimodular random rooted graph is always equalto 2, and since the FUSF stochastically dominates the WUSF it follows that they are equal if andonly the expected degree of the FUSF at the root is 2 also.Theorem (Angel, H., Nachmias, and Ray 2016). Let (M,\u03c1) be an infinite, simply connected uni-modular random map, and suppose that E[deg(\u03c1)] < \u221e, and let F be the free uniform spanningforest of M . ThenE[degF(\u03c1)] =1piE[\u03b8(\u03c1)] = 2\u2212 1piK(M,\u03c1). (1.6.1)In particular, the FUSF and WUSF of M coincide if and only if (M,\u03c1) has average curvature zero.Secondly, we have the following theorem, which is also a major piece of the Dichotomy Theoremabove, and complements our results with Nachmias that held for deterministic bounded degree551.6. Unimodular random graphsplanar graphs [130].Theorem (Angel, H., Nachmias, and Ray 2016). Let (M,\u03c1) be a simply connected unimodularrandom rooted map with E[deg(\u03c1)] <\u221e. Then the free uniform spanning forest of M is connectedalmost surely.The case in which the dual of M is recurrent and locally finite followed easily from previousresults. In the case that the dual of M is transient and locally finite, this theorem follows from ourearlier work [127, 128], the main purpose of which was to remove the bounded degree hypothesiswhich was required by the earlier result of Aldous and Lyons [7].Theorem (H. 2015). Let (G, \u03c1) be a transient unimodular random rooted graph, and let F be thewired uniform spanning forest of G. Then every component of F is one-ended almost surely.The case that the dual of M \u2020 is not locally finite was handled by a separate argument basedon a variation of Wilson\u2019s algorithm.Using the fact that the FUSF is connected almost surely, we were able to deduce the followingtheorem, which verifies the Aldous-Lyons conjecture in the case of simply-connected planar maps.Theorem (Angel, H., Nachmias, and Ray 2016). Every simply-connected unimodular randomrooted map is sofic, that is, a Benjamini-Schramm limit of finite maps.Note that, by the Dichotomy Theorem, the sequence of approximating maps are necessarilynon-planar in the hyperbolic case.1.6.5 Boundary theory of unimodular random triangulationsIn our joint work with Angel, Nachmias, and Ray [21], we also developed the boundary theory ofcircle packings of unimodular random triangulations. When P is a circle packing of a graph G =(V,E) in the unit disc, we define zh(v) to be the hyperbolic centre of the circle of P correspondingto the vertex v.Theorem. Let (G, \u03c1) be a simple, one-ended, CP hyperbolic unimodular random planar triangu-lation with E[deg2(\u03c1)] < \u221e. Let P be a circle packing of G in the unit disc, and let \u3008Xn\u3009n\u22650 be asimple random walk on G. The following hold conditional on (G, \u03c1) almost surely:1. z(Xn) and zh(Xn) both converge to a (random) point denoted \u039e \u2208 \u2202D,2. The law of \u039e has full support \u2202D and no atoms.3. \u2202D is a realisation of the Poisson boundary of G. That is, for every bounded harmonicfunction h on G there exists a bounded measurable function g : \u2202D\u2192 R such thath(v) = Ev[g(\u039e)].561.7. Open problemsAlthough the results are similar to those proved in the bounded degree case by Benjaminiand Schramm [47] and by Angel, Barlow, Gurel-Gurevich, and Nachmias [17], the proofs are verydifferent. Indeed, the analytic methods of [17, 47] fail badly in the absence of the bounded degreeassumption, and were unavailable to us. This required us to develop an entirely different, moreprobabilistic, approach to the boundary theory of planar graphs. (Georgakopoulos\u2019s square tilinganalysis [100] is less reliant on bounded degrees, but the assumption is still crucial to Benjaminiand Schramm\u2019s proof that the random walk converges to a point in the boundary of the squaretiling.) In joint work with Peres [132], we later adapted these methods to give simpler proofs inthe bounded degree case, as we mentioned earlier. It is worth noting, however, that the analyticmethods give stronger results when applicable: for instance, we do not know that the boundaryof the unit disc is a realization of the Martin boundary for circle packings of unimodular randomhyperbolic triangulations.Finally, our methods also yielded results concerning the speed of the random walk as measuredin the hyperbolic metric.Theorem. Let (G, \u03c1) be a simple, one-ended, CP hyperbolic unimodular random rooted planartriangulation with E[deg2(\u03c1)] <\u221e and let P be a circle packing of G in the unit disc. Then almostsurelylimn\u2192\u221edhyp(zh(\u03c1), zh(Xn))n= limn\u2192\u221e\u2212 log r(Xn)n> 0.In particular, both limits exist. Moreover, the limits do not depend on the choice of packing, and if(G, \u03c1) is ergodic then this limit is an almost sure constant.Unlike the previous theorem, this result does not have an analogue in the deterministic boundeddegree case.1.7 Open problemsWe conclude the introduction with a collection of our favourite open problems that appear in therest of the thesis. Many other interesting open problems on similar topics, several of which we havealready mentioned, appear in [173].Question. Let G be a bounded degree proper plane graph.1. Let H be a finite graph. Is the free uniform spanning forest of the product graph G \u00d7 Hconnected almost surely?2. Let G\u2032 be a bounded degree graph that is rough isometric to G. Is the the free uniform spanningforest of G connected almost surely?Conjecture. Let D be a domain. Then either every simple, bounded degree triangulation admittinga circle packing in D has an almost surely connected free uniform spanning forest, or every simple,bounded degree triangulation admitting a circle packing in D has an almost surely disconnected freeuniform spanning forest.571.7. Open problemsQuestion. Let G be a unimodular transitive graph, and let F be the transboundary uniform spanningforest of G. Is F almost surely a topological spanning tree of G?Question. Let T be a bounded degree triangulation that admits a circle packing in the complementof a domain D which is quasi-homogeneous in the sense that the group of conformal automorphismsof D acts cocompactly on D. Is the free uniform spanning forest of T connected almost surely?Question. Let T be a bounded degree triangulation that admits a circle packing in the complementof a positive-length Cantor set. Is the free uniform spanning forest of T disconnected almost surely?Question. Let G be a uniformly transient network with infe c(e) > 0. Does it follow that everycomponent of the wired uniform spanning forest of G is one-ended almost surely?Question. Let d \u2265 3, let I be the interlacement process on Zd, and let \u3008Ft\u3009t\u2208R = \u3008ABt(I )\u3009t\u2208R.If d = 3, 4, do there exist times at which Ft is disconnected? If d \u2265 5, do there exist times at whichFt is connected?Conjecture. Let (T, \u03c1) be a simple, CP hyperbolic, unimodular random rooted triangulation withE[deg2 \u03c1] < \u221e. Then the boundary of the circle packing of T in D is a realization of the Martinboundary of T almost surely.58Part ITwo Short Papers59Chapter 2Critical percolation on anyquasi-transitive graph of exponentialgrowth has no infinite clustersSummary. We prove that critical percolation on any quasi-transitive graph of exponential volumegrowth does not have a unique infinite cluster. This allows us to deduce from earlier results thatcritical percolation on any graph in this class does not have any infinite clusters. The result is newwhen the graph in question is either amenable or nonunimodular.2.1 IntroductionIn Bernoulli bond percolation, each edge of a graph G = (V,E) (which we will always assumeto be connected and locally finite) is either deleted or retained at random with retention probabilityp \u2208 [0, 1], independently of all other edges. We denote the random graph obtained this way byG[p]. Connected components of G[p] are referred to as clusters. Given a graph G, the criticalprobability, denoted pc(G) or simply pc, is defined to bepc(G) = sup{p \u2208 [0, 1] : G[p] has no infinite clusters almost surely} .A central question concerns the existence or non-existence of infinite clusters at the critical proba-bility p = pc. Indeed, proving that critical percolation on the hypercubic lattice Zd has no infiniteclusters for every d \u2265 2 is perhaps the best known open problem in modern probability theory.Russo [203] proved that critical percolation on the square lattice Z2 has no infinite clusters, whileHara and Slade [117] proved that critical percolation on Zd has no infinite clusters for all d \u2265 19.More recently, Fitzner and van der Hofstad [89] improved upon the Hara-Slade method, provingthat critical percolation on Zd has no infinite clusters for every d \u2265 11. See e.g. [106] for furtherbackground.In their highly influential paper [51], Benjamini and Schramm proposed a systematic study ofpercolation on general quasi-transitive graphs; that is, graphs G = (V,E) such that the action ofthe automorphism group Aut(G) on V has only finitely many orbits (see e.g. [173] for more detail).They made the following conjecture.602.1. IntroductionConjecture 2.1.1 (Benjamini and Schramm). Let G be a quasi-transitive graph. If pc(G) < 1,then G[pc] has no infinite clusters almost surely.Benjamini, Lyons, Peres, and Schramm [43, 45] verified the conjecture for nonamenable, uni-modular, quasi-transitive graphs, while partial progress has been made for nonunimodular, quasi-transitive graphs (which are always nonamenable [173, Exercise 8.30]) by Tima\u00b4r [222] and by Peres,Pete, and Scolnicov [192]. In this note, we verify the conjecture for all quasi-transitive graphs ofexponential growth.Theorem 2.1.2. Let G be a quasi-transitive graph with exponential growth. Then G[pc] has noinfinite clusters almost surely.A corollary of Theorem 2.1.2 is that pc < 1 for all quasi-transitive graphs of exponential growth,a result originally due to Lyons [168].We prove Theorem 2.1.2 by combining the works of Benjamini, Lyons, Peres, and Schramm [43]and Tima\u00b4r [222] with the following simple connectivity decay estimate. Given a graph G, we writeB(x, r) to denote the graph distance ball of radius r around a vertex x of G. Recall that a graphG is said to have exponential growth ifgr(G) := lim infr\u2192\u221e |B(x, r)|1\/ris strictly greater than 1 whenever x is a vertex of G. It is easily seen that gr(G) does not dependon the choice of x. Let \u03c4p(x, y) be the probability that x and y are connected in G[p], and let\u03bap(n) := inf{\u03c4p(x, y) : x, y \u2208 V, d(x, y) \u2264 n}.Theorem 2.1.3. Let G be a quasi-transitive graph with exponential growth. Then\u03bapc(n) := inf{\u03c4pc(x, y) : x, y \u2208 V, d(x, y) \u2264 n} \u2264 gr(G)\u2212nfor all n \u2265 1.Remark 2.1.4. The upper bound on \u03bapc(n) in Theorem 2.1.3 is attained when G is a regular tree.There are many amenable groups of exponential growth, and, to our knowledge, the conclusionof Theorem 2.1.2 was not previously known for any of their Cayley graphs. Among probabilists,the best known examples are the lamplighter groups [173, 192]. See e.g. [33, 72, 140, 172, 183] forfurther interesting examples.Following the work of Lyons, Peres, and Schramm [175, Theorem 1.1], Theorem 2.1.2 has thefollowing immediate corollary, which is new in the amenable case. The reader is referred to [175]and [173] for background on minimal spanning forests.Corollary 2.1.5. Let G be a unimodular quasi-transitive graph of exponential growth. Then everycomponent of the wired minimal spanning forest of G is one-ended almost surely.612.2. Proof2.2 ProofProof of Theorem 2.1.2 given Theorem 2.1.3. Let us recall the following results:Theorem (Newman and Schulman [187]). Let G be a quasi-transitive graph. Then G[p] has eitherno infinite clusters, a unique infinite cluster, or infinitely many infinite clusters almost surely forevery p \u2208 [0, 1].Theorem (Burton and Keane [64]; Gandolfi, Keane, and Newman [97]). Let G be an amenablequasi-transitive graph. Then G[p] has at most one infinite cluster almost surely for every p \u2208 [0, 1].Theorem (Benjamini, Lyons, Peres, and Schramm [36, 43]). Let G be a nonamenable, unimodular,quasi-transitive graph. Then G[pc] has no infinite clusters almost surely.Theorem (Tima\u00b4r [222]). Let G be a nonunimodular, quasi-transitive graph. Then G[pc] has atmost one infinite cluster almost surely.The statements given for the first two theorems above are not those given in the original papers;the reader is referred to [173] for a modern account of these theorems and for the definitions ofunimodularity and amenability. Similarly, Tima\u00b4r\u2019s result is stated for transitive graphs, but hisproof easily extends to the quasi-transitive case. For our purposes, the significance of the abovetheorems is that, to prove Theorem 2.1.2, it suffices to prove that if G is a quasi-transitive graph ofexponential growth, then G[pc] does not have a unique infinite cluster almost surely. This followsimmediately from Theorem 2.1.3, since if G[pc] contains a unique infinite cluster then\u03c4pc(x, y) \u2265 Ppc(x and y are both in the unique infinite cluster)\u2265 Ppc(x is in the unique infinite cluster)Ppc(y is in the unique infinite cluster)for all x, y \u2208 V by Harris\u2019s inequality [118]. Quasi-transitivity implies that the right hand side isbounded away from zero if G[pc] contains a unique infinite cluster almost surely, and it follows thatlimn\u2192\u221e \u03bapc(n) > 0 in this case.Lemma 2.2.1. Let G be any graph. Then \u03bap(n) is a supermultiplicative function of n. That is,for every p, n and m, we have that \u03bap(m+ n) \u2265 \u03bap(m)\u03bap(n).Proof. Let u and v be two vertices with d(u, v) \u2264 m + n. Then there exists a vertex w suchthat d(u,w) \u2264 m and d(w, v) \u2264 n. Since the events {u \u2194 w} and {w \u2194 v} are increasing and{u\u2190\u2192 v} \u2287 {u\u2194 w} \u2229 {w \u2194 v}, Harris\u2019s inequality [118] implies that\u03c4p(u, v) \u2265 \u03c4p(u,w)\u03c4p(w, v) \u2265 \u03bap(m)\u03bap(n).The claim follows by taking the infimum.622.2. ProofLemma 2.2.2. Let G be a quasi-transitive graph. Then supn\u22651(\u03bap(n))1\/n is left continuous in p.That is,lim\u03b5\u21920+supn\u22651(\u03bap\u2212\u03b5(n))1\/n= supn\u22651(\u03bap(n))1\/nfor every p \u2208 (0, 1]. (2.2.1)Proof. Recall that an increasing function is left continuous if and only if it is lower semi-continuous,and that lower semi-continuity is preserved by taking minima (of finitely many functions) andsuprema (of arbitrary collections of functions). Now, observe that \u03c4p(x, y) is lower semi-continuousin p for each pair of fixed vertices x and y: This follows from the fact that \u03c4p(x, y) can be writtenas the supremum of the continuous functions \u03c4 rp (x, y), which give the probabilities that x and y areconnected in G[p] by a path of length at most r. (See [106, Section 8.3].) Since G is quasi-transitive,there are only finitely many isomorphism classes of pairs of vertices at distance at most n in G, andwe deduce that \u03bap(n) is also lower semi-continuous in p for each fixed n. Thus, supn\u22651(\u03bap(n))1\/nis a supremum of lower semi-continuous functions and is therefore lower semi-continuous itself.We will require the following well known theorem.Theorem 2.2.3. Let G be a quasi-transitive graph, and let \u03c1 be a fixed vertex of G. Then theexpected cluster size is finite for every p < pc. That is,\u2211x\u03c4p(\u03c1, x) <\u221e for every p < pc.This theorem was proven in the transitive case by Aizenmann and Barsky [4], and in the quasi-transitive case by Antunovic\u00b4 and Veselic\u00b4 [25]; see also the recent work of Duminil-Copin and Tassion[83] for a beautiful new proof in the transitive case.Proof of Theorem 2.1.3. Let \u03c1 be a fixed root vertex of G. For every p \u2208 [0, 1] and every n \u2265 1, wehave\u03bap(n) \u00b7\u2223\u2223B(\u03c1, n)\u2223\u2223 \u2264 \u2211x\u2208B(\u03c1,n)\u03c4p(\u03c1, x) \u2264\u2211x\u03c4p(\u03c1, x).Thus, it follows from Theorem 2.2.3, Lemma 2.2.1, and Fekete\u2019s Lemma thatsupn\u22651(\u03bap(n))1\/n = limn\u2192\u221e(\u03bap(n))1\/n \u2264 lim supn\u2192\u221e(\u2211x \u03c4p(\u03c1, x)|B(\u03c1, n)|)1\/n= gr(G)\u22121for every p < pc. We conclude by applying Lemma 2.2.2.63Chapter 3Collisions of random walks inreversible random graphs3.1 IntroductionLet G be an infinite, connected, locally finite graph. G is said to have the infinite collisionproperty if for every vertex v of G, two independent random walks \u3008Xn\u3009n\u22650 and \u3008Yn\u3009n\u22650 startedfrom v collide (i.e. occupy the same vertex at the same time) infinitely often almost surely (a.s.).Although transitive recurrent graphs such as Z and Z2 are easily seen to have the infinite collisionproperty, Krishnapur and Peres [158] showed that the infinite collision property does not hold forthe comb graph, a subgraph of Z2.Chen and Chen [71] proved that the infinite cluster of supercritical Bernoulli bond percolationin Z2 a.s. has the infinite collision property. Barlow, Peres and Sousi [32] gave a sufficient conditionfor the infinite collision property in terms of the Green function. They deduced that severalclassical random recurrent graphs have the infinite collision property, including the incipient infinitepercolation cluster in dimensions d \u2265 19.However, the methods of [71] and [32] both require precise estimates on the graphs underconsideration, and the infinite collision property was still not known to hold for several importantrandom recurrent graphs \u2013 see Corollary 3.1.2.In this note we prove that the infinite collision property holds a.s. for a large class of randomrecurrent graphs. Recall that a rooted graph (G, \u03c1) is a graph G together with a distinguishedroot vertex \u03c1, and that a random rooted graph (G, \u03c1) is said to be reversible if (G, \u03c1,X1) and(G,X1, \u03c1) have the same distribution, where X1 is the first step of a simple random walk on Gstarted at \u03c1 (see Section 3.1.1 for more details).Theorem 3.1.1. Let (G, \u03c1) be a recurrent reversible random rooted graph. Then G has the infinitecollision property almost surely.The assumption of reversibility can be replaced either by the assumption that (G, \u03c1) is stationaryor by the assumption that (G, \u03c1) is unimodular and the root \u03c1 has finite expected degree (seeSection 3.1.1 for definitions of these terms).Corollary 3.1.2. Each of the following graphs has the infinite collision property almost surely.(This is by no means an exhaustive list.)643.1. Introduction1. The Uniform Infinite Planar Triangulation (UIPT) and Quadrangulation (UIPQ).2. The Incipient Infinite Cluster (IIC) of Bernoulli bond percolation in Z2.3. Every component of each of the Wired Uniform and Minimal Spanning Forests (WUSF andWMSF) of any Cayley graph.The UIPT was introduced by Angel and Schramm [24] and the UIPQ was introduced byKrikun [157]. They are unimodular by construction and were shown to be recurrent by Gurel-Gurevich and Nachmias [107]. The IIC is an infinite random subgraph of Z2 introduced by Kesten[153]. For each n, let Cn be the largest cluster of a critical Bernoulli bond percolation on the box[0, n]2 and let \u03c1n be a uniformly random vertex of Cn. Ja\u00b4rai [136] showed that the IIC can bedefined as the weak limit of the random rooted graphs (Cn, \u03c1n), and is therefore unimodular [7,\u00a72]. For background on the Wired Uniform and Minimal Spanning Forests, see Chapters 10 and 11of [173] and Section 7 of [7].In Section 3.3 we provide extensions of Theorem 3.1.1 to networks and to the continuous-timerandom walk.Remark 3.1.3. If G is a non-bipartite graph with the infinite collision property, it is easy to seethat two independent random walks started from any two vertices of G will collide infinitely oftena.s. On the other hand, if G is a bipartite graph with the infinite collision property, then twoindependent random walks on G will collide infinitely often if and only if their starting points areat an even distance from each other.3.1.1 DefinitionsIn this section we give concise definitions of stationary, reversible and unimodular random rootedgraphs. We refer the reader to Aldous and Lyons [7] for more details.A rooted graph (G, \u03c1) is a connected, locally finite (multi)graph G = (V,E) together witha distinguished vertex \u03c1, the root. An isomorphism of graphs \u03c6 : G \u2192 G\u2032 is an isomorphism ofrooted graphs \u03c6 : (G, \u03c1)\u2192 (G\u2032, \u03c1\u2032) if \u03c6(\u03c1) = \u03c1\u2032. The set of isomorphism classes of rooted graphs isendowed with the local topology [50], in which, roughly speaking, two (isomorphism classes of)rooted graphs are close to each other if and only if they have large isomorphic balls around theroot. A random rooted graph is a random variable taking values in the space of isomorphismclasses of rooted graphs endowed with the local topology. Similarly, a doubly-rooted graph isa graph together with an ordered pair of distinguished (not necessarily distinct) vertices. Denotethe space of isomorphism classes of doubly-rooted graphs equipped with this topology by G\u2022\u2022.Recall that the simple random walk on a locally finite (multi)graph G = (V,E) is the Markovprocess \u3008Xn\u3009n\u22650 on the state space V with transition probabilities p(u, v) defined to be the fractionof edges emanating from u that end in v. A random rooted graph (G, \u03c1) is said to be stationary653.1. Introductionif, when \u3008Xn\u3009n\u22650 is a simple random walk on G started at the root,(G, \u03c1)d= (G,Xn)for all n and is said to be reversible if(G, \u03c1,Xn)d= (G,Xn, \u03c1)for all n. Note that this is not the same as the reversibility of the random walk on G, which holdsfor any graph7. Every reversible random rooted graph is clearly stationary, but the converse neednot hold in general [36, Examples 3.1 and 3.2]. However, Benjamini and Curien [38, Theorem4.3] showed that every recurrent stationary random rooted graph is necessarily reversible, so thatTheorem 3.1.1 also applies under the apparently weaker assumption of stationarity.Reversibility is closely related to the property of unimodularity. A mass transport is a functionf : G\u2022\u2022 \u2192 [0,\u221e]. A random rooted graph (G, \u03c1) is said to be unimodular if it satisfies the Mass-Transport Principle: for every mass transport f ,E[\u2211vf(G, \u03c1, v)]= E[\u2211uf(G, u, \u03c1)]. (MTP)That is,Expected mass out equals expected mass in.The Mass-Transport Principle was first introduced by Ha\u00a8ggstro\u00a8m [110] to study dependent perco-lation on Cayley graphs. The current formulation of the Mass-Transport Principle was suggestedby Benjamini and Schramm [50] and developed systematically by Aldous and Lyons in [7].As noted in [38], if (G, \u03c1) is a unimodular random rooted graph with E[deg(\u03c1)] < \u221e, thenbiasing the law of (G, \u03c1) by deg(\u03c1) gives an equivalent law of a reversible random rooted graph.Conversely, if (G, \u03c1) is a reversible random rooted graph, then biasing the law of (G, \u03c1) by deg(\u03c1)\u22121gives an equivalent law of a reversible random rooted graph. For example, if (G, \u03c1) is a finiterandom rooted graph then it is unimodular if and only if \u03c1 is uniformly distributed on G, and isreversible if and only if \u03c1 is distributed according to the stationary measure of simple random walkon G.In light of the above correspondence, Theorem 3.1.1 may be stated equivalently as follows.Theorem 3.1.4. Let (G, \u03c1) be a recurrent unimodular random rooted graph with E[deg(\u03c1)] < \u221e.Then G has the infinite collision property almost surely.We provide two variations on the proof of Theorem 3.1.1. The first uses the Mass-TransportPrinciple. The second, given in Section 3.3, uses reversibility, and applies in the network settingalso.7Rather, a random rooted graph (G, \u03c1) is reversible if and only if the G\u2022\u2022-valued Markov process\u3008(G,Xn, Xn+1)\u3009n\u22650 is reversible.663.2. Proof of Theorem 3.1.13.2 Proof of Theorem 3.1.1Proof. Let G be a graph and let pn( \u00b7 , \u00b7 ) denote the n-step transition probabilities for simplerandom walk on G. For each vertex u of G, let qfin(u) denote the probability that two independentrandom walks started at u collide only finitely often, and let q0(u) denote the probability that twoindependent random walks started at u do not collide at all after time zero. Finally, for each pairof vertices u and v let qlast(u, v) be the probability that two independent random walks started atu collide for the last time at v, so thatqfin(u) =\u2211vqlast(u, v).Decomposing according to the time of the last collision givesqlast(u, v) =\u2211n\u22650pn(u, v)2q0(v) (3.2.1)and henceqfin(u) =\u2211v\u2211n\u22650pn(u, v)2q0(v). (3.2.2)Suppose that (G, \u03c1) is a recurrent unimodular random rooted graph with E[deg(\u03c1)] < \u221e.Consider the mass transportf(G, u, v) = deg(u)qlast(u, v).Each vertex u sends a total mass of deg(u)qfin(u), while, by (3.2.1), each vertex v receives a totalmass of \u2211uf(G, u, v) =\u2211n\u22650\u2211udeg(u)pn(u, v)2q0(v)= q0(v)\u2211n\u22650\u2211udeg(v)pn(v, u)pn(u, v)= q0(v) deg(v)\u2211n\u22650p2n(v, v).Since G is recurrent, the sum\u2211n\u22650 p2n(v, v) is infinite a.s. for every vertex v of G. By the Mass-Transport Principle,E[deg(\u03c1)qfin(\u03c1)] = E[q0(\u03c1) deg(\u03c1)\u2211n\u22650p2n(\u03c1, \u03c1)].Since the left-hand expectation is finite by assumption, we must have that q0(\u03c1) = 0 a.s., andconsequently that qfin(v) = 0 for every vertex v in G a.s.673.3. Extensions3.3 Extensions3.3.1 NetworksRecall that a network (G, c) is a connected locally finite graph G = (V,E) together with afunction c : E \u2192 (0,\u221e) assigning to each edge e of G a positive conductance c(e). Graphs maybe considered to be networks by setting c(e) \u2261 1. Write c(u) for the sum of the conductances ofthe edges emanating from u and c(u, v) for the sum of the conductances of the edges joining uand v. The random walk \u3008Xn\u3009n\u22650 on a network (G, c) is the Markov chain on V with transitionprobabilities p(u, v) = c(u, v)\/c(u). Unimodular and reversible random rooted networks are definedsimilarly to the unweighted case [7]. In particular, a random rooted network (G, c, \u03c1) is defined tobe reversible if, letting \u3008Xn\u3009n\u22650 be a random walk on (G, c) started from \u03c1,(G, c, \u03c1,Xn)d= (G, c,Xn, \u03c1) for all n \u2265 1.We now extend Theorem 3.1.1 to the setting of reversible random rooted networks. The proofgiven also yields an alternative proof of Theorem 3.1.1.Theorem 3.3.1. Let (G, c, \u03c1) be a recurrent reversible random rooted network. Then (G, c) hasthe infinite collision property almost surely.Proof. Let qfin and q0 be defined as in the proof of Theorem 3.1.1. Taking expectations on bothsides of (3.2.2) with u = \u03c1,E[qfin(\u03c1)]= E[\u2211v\u2211n\u22650pn(\u03c1, v)2q0(v)]=\u2211n\u22650E[pn(\u03c1,Xn)q0(Xn)].Applying reversibility,E[qfin(\u03c1)] =\u2211n\u22650E[pn(Xn, \u03c1)q0(\u03c1)]= E[q0(\u03c1)\u2211n\u22650p2n(\u03c1, \u03c1)].Since (G, c) is recurrent,\u2211n\u22650 p2n(v, v) =\u221e a.s. for every vertex v of G. Thus, since the left-handexpectation is finite, we must have that q0(\u03c1) = 0 a.s. and hence also that qfin(\u03c1) = 0 a.s.3.3.2 Continuous-time random walkLet (G, c, \u03c1) be a recurrent unimodular random rooted network, and let \u3008Xt\u3009t\u22650 denote the continuous-time random walk on (G, c) started from \u03c1, which jumps across each edge e with rate c(e) (see e.g.[188]). Corollary 4.3 of [7] states that if (G, c, \u03c1) is a unimodular random rooted network such thatthe continuous-time random walk on (G, c) a.s. does not explode (i.e. make infinitely many jumpsin a finite amount of time), then (G, c, \u03c1) is reversible for the continuous-time random walk in thesense that(G, c, \u03c1,Xt)d= (G, c,Xt, \u03c1) (3.3.1)683.3. Extensionsfor all t \u2265 0. It is clear that the continuous-time walk on any graph can make at most finitely manyvisits to any fixed vertex in a finite amount of time, and it follows that the continuous-time walk ona recurrent network does not explode a.s. Thus, we deduce that eq. (3.3.1) holds whenever (G, c, \u03c1)is a recurrent reversible random rooted network (in particular, we do not require the assumptionthat E[c(\u03c1)] <\u221e).The natural analogue of the infinite collision property also holds a.s. for the continuous-timerandom walk on recurrent unimodular random rooted networks.Theorem 3.3.2. let (G, c, \u03c1) be a recurrent unimodular random rooted network and let \u3008Xt\u3009t\u22650 and\u3008Yt\u3009t\u22650 be two independent continuous-time random walks on (G, c) started from any two vertices.Then the set of times {t : Xt = Yt} has infinite Lebesgue measure almost surely.Proof. Let \u3008Xt\u3009t\u22650 and \u3008Yt\u3009t\u22650 be independent continuous-time random walks starting at the samevertex u of G. By considering the walks only at integer times t = n, the proof of Theorem 3.3.1readily shows that the set of integer collision times {n \u2208 N : Xn = Yn} is infinite a.s.For every s \u2265 0 there is a positive probability that neither \u3008Xt\u3009t\u22650 nor \u3008Yt\u3009t\u22650 has made anyjumps by time s. Thus, the law of the sequence \u3008(Xn+s, Yn+s)\u3009n\u22651 is absolutely continuous withrespect to the law of \u3008(Xn, Yn)\u3009n\u22651. It follows that for every s \u2265 0, the walks \u3008Xt\u3009t\u22650 and \u3008Yt\u3009t\u22650collide at an infinite set of times of the form {n+ s : n \u2208 N} a.s., and consequently thatLeb({t : Xt = Yt}) = \u222b 10\u2223\u2223{n : Xn+s = Yn+s}\u2223\u2223ds =\u221e a.s.For every other vertex v, there is a positive probability that X1 = u and Y1 = v, so that the lawof two independent continuous-time random walks started from u and v is absolutely continuouswith respect to the law of \u3008(Xt+1, Yt+1)\u3009t\u22650. Thus, the set of collision times of two independentcontinuous-time random walks started from u and v has infinite Lebesgue measure a.s.Remark 3.3.3. Theorem 3.3.2 has consequences for the voter model on unimodular random recurrentnetworks. For every network (G, c) in which two independent continuous-time random walks collidea.s., duality between the voter model and continuous-time coalescing random walk ([166, \u00a75] and[6, \u00a714]) implies that the only ergodic stationary measures for the voter model on (G, c) are theconstant (a.k.a. consensus) measures. Thus, a consequence of Theorem 3.3.2 is that this holds forthe voter model on recurrent unimodular random rooted networks.69Part IIUniform Spanning Forests70Chapter 4Wired cycle-breaking dynamics foruniform spanning forestsSummary. We prove that every component of the wired uniform spanning forest (WUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degreehypothesis required by earlier results. We deduce that every component of the WUSF is one-endedalmost surely in every supercritical Galton-Watson tree, answering a question of Benjamini, Lyons,Peres and Schramm.Our proof introduces and exploits a family of Markov chains under which the oriented WUSFis stationary, which we call the wired cycle-breaking dynamics.4.1 IntroductionThe uniform spanning forests (USFs) of an infinite, locally finite, connected graph G are definedas infinite-volume limits of uniformly chosen random spanning trees of large finite subgraphs ofG. These limits can be taken with respect to two extremal boundary conditions, free and wired,giving the free uniform spanning forest (FUSF) and wired uniform spanning forest (WUSF)respectively (see Section 4.2 for detailed definitions). The study of uniform spanning forests wasinitiated by Pemantle [190], who, in addition to showing that both limits exist, proved that thewired and free forests coincide in Zd for all d and that they are almost surely a single tree if andonly if d \u2264 4. The question of connectivity of the WUSF was later given a complete answer byBenjamini, Lyons, Peres and Schramm (henceforth referred to as BLPS) in their seminal work [44],in which they proved that the WUSF of a graph is connected if and only if two independent randomwalks on the graph intersect almost surely [44, Theorem 9.2].After connectivity, the most basic topological property of a forest is the number of ends itscomponents have. An infinite connected graph G is said to be k-ended if, over all finite sets ofvertices W , the graph G \\W formed by deleting W from G has a maximum of k distinct infiniteconnected components. In particular, an infinite tree is one-ended if and only if it does not containany simple bi-infinite paths and is two-ended if and only if it contains a unique simple bi-infinitepath.Components of the WUSF are known to be one-ended for several large classes of graphs. Again,this problem was first studied by Pemantle [190], who proved that the USF on Zd has one end for2 \u2264 d \u2264 4 and that every component has at most two ends for d \u2265 5. (For d = 1 the forest714.1. Introductionis all of Z and is therefore two-ended.) A decade later, BLPS [44, Theorem 10.1] completed andextended Pemantle\u2019s result, proving in particular that every component of the WUSF of a Cayleygraph is one-ended almost surely if and only if the graph is not itself two-ended. Their proof wasthen adapted to random graphs by Aldous and Lyons [7, Theorem 7.2], who showed that all WUSFcomponents are one-ended almost surely in every transient reversible random rooted graph withbounded vertex degrees. Taking a different approach, Lyons, Morris and Schramm [170] gave anisoperimetric condition for one-endedness, from which they deduced that all WUSF componentsare one-ended almost surely in every transient transitive graph and every non-amenable graph.In this paper, we remove the bounded degree assumption from the result of Aldous and Lyons [7].We state our result in the natural generality of reversible random rooted networks. Recall that anetwork is a locally finite, connected (multi)graph G = (V,E) together with a function c : E \u2192(0,\u221e) assigning a positive conductance c(e) to each unoriented edge e of G. For each vertex v,the conductance c(v) of v is defined to be the sum of the conductances of the edges adjacent to v,where self-loops are counted twice. Locally finite, connected graphs without specified conductancesare considered to be networks by setting c \u2261 1. The WUSF of a network is defined in Section 4.2and reversible random rooted networks are defined in Section 4.5.Theorem 4.1.1. Let (G, \u03c1) be a transient reversible random rooted network and suppose thatE[c(\u03c1)\u22121] < \u221e. Then every component of the wired uniform spanning forest of G is one-endedalmost surely.The condition that the expected inverse conductance of the root is finite is always satisfied bygraphs, for which c(\u03c1) = deg(\u03c1) \u2265 1. In Example 4.5.1 we show that the theorem can fail in theabsence of this condition.Theorem 4.1.1 applies (indirectly) to supercritical Galton-Watson trees conditioned to survive,answering positively Question 15.4 of BLPS [44].Corollary 4.1.2. Let T be a supercritical Galton-Watson tree conditioned to survive. Then everycomponent of the wired uniform spanning forest of T is one-ended almost surely.Previously, this was known only for supercritical Galton-Watson trees with offspring distributioneither bounded, in which case the result follows as a corollary to the theorem of Aldous and Lyons [7],or supported on a subset of [2,\u221e), in which case the tree is non-amenable and we may apply thetheorem of Lyons, Morris and Schramm [170].Our proof introduces a new and simple method, outlined as follows. For every transient network,we define a procedure to \u2018update an oriented forest at an edge\u2019, in which the edge is added tothe forest while another edge is deleted. Updating oriented forests at randomly chosen edgesdefines a family of Markov chains on oriented spanning forests, which we call the wired cycle-breaking dynamics, for which the oriented wired uniform spanning forest measure is stationary(Proposition 4.3.2). This stationarity allows us to prove the following theorem, from which weshow Theorem 4.1.1 to follow by known methods.724.2. The wired uniform spanning forestTheorem 4.1.3. Let G be any network. If the wired uniform spanning forest of G contains morethan one two-ended component with positive probability, then it contains a component with three ormore ends with positive probability.The case of recurrent reversible random rooted graphs remains open, even under the assumptionof bounded degree. In this case, it should be that the single tree of the WUSF has the same numberof ends as the graph (this prediction appears in [7]). BLPS proved this for transitive recurrentgraphs [44, Theorem 10.6].4.1.1 ConsequencesThe one-endedness of WUSF components has consequences of fundamental importance for theAbelian sandpile model. Ja\u00b4rai and Werning [138] proved that the infinite-volume limit of the sandpilemeasures exists on every graph for which every component of the WUSF is one-ended almost surely.Furthermore, Ja\u00b4rai and Redig [137] proved that, for any graph which is both transient and has one-ended WUSF components, the sandpile configuration obtained by adding a single grain of sandto the infinite-volume random sandpile can be stabilized by finitely many topplings (their proofis given for Zd but extends to this setting, see [135]). Thus, a consequence of Theorem 4.1.1 isthat these properties hold for the Abelian sandpile model on transient reversible random graphs ofunbounded degree.Theorem 4.1.1 also has several interesting consequences for random plane graphs, which weaddress in upcoming work with Angel, Nachmias and Ray [20]. In particular, we deduce fromTheorem 4.1.1 that every Benjamini-Schramm limit of finite planar graphs is almost surely Liouville,i.e. does not admit non-constant bounded harmonic functions.4.2 The wired uniform spanning forestIn this section we briefly define the wired uniform spanning forest and introduce the propertiesthat we will need. For a comprehensive treatment of uniform spanning trees and forests, as well asa detailed history of the subject, we refer the reader to Chapters 4 and 10 of [173].Notation and orientation Throughout this paper, the graphs on which the USFs and USTsare defined will be connected and locally finite unless stated otherwise. We do not distinguishnotationally between oriented and unoriented trees, forests or edges. Whether or not a tree, forestor edge is oriented will be clear from context. Edges e are oriented from their tail e\u2212 to their heade+, and have reversal \u2212e. An oriented tree or forest is a tree or forest together with an orientationof its edges. Given an oriented tree or forest in a graph, we define the past of each vertex v to bethe set of vertices u for which there is a directed path from u to v in the oriented tree or forest.For a finite connected graph G, we write USTG for the uniform measure on the set of spanningtrees (i.e. connected cycle-free subgraphs containing every vertex) of G, considered for measure-734.2. The wired uniform spanning foresttheoretic purposes to be functions from E to {0, 1}. More generally, if G is a finite network, wedefine USTG to be the probability measure on spanning trees of G for which the measure of a treet is proportional to the product of the conductances of its edges.There are two extremal (with respect to stochastic ordering) ways to define infinite volume limitsof the uniform spanning tree measures. Let G be an infinite network and let Vn be an increasingsequence of finite connected subsets of V such that\u22c3Vn = V , which we call an exhaustion of G.For each n, let the network Gn be the subgraph of G induced by Vn together with the conductancesinherited from G. The weak limit of the measures USTGn is known as the free uniform spanningforest: for each finite subset S \u2282 E,FUSFG(S \u2286 F) := limn\u2192\u221eUSTGn(S \u2286 T ),where F is a sample of the FUSF of G and T is a sample of the UST of Gn. Alternatively, at eachstep of the exhaustion we define a network G\u2217n by identifying (\u2018wiring\u2019) V \\ Vn into a single vertex\u2202n and deleting all the self-loops that are created, and define the wired uniform spanning forestto be the weak limitWUSFG(S \u2286 F) := limn\u2192\u221eUSTG\u2217n(S \u2286 T ),where F is a sample of the WUSF of G and T is a sample of the UST of G\u2217n.Both limits were shown (implicitly) to exist for every network and every choice of exhaustion byPemantle [190], although the WUSF was not defined explicitly until the work of Ha\u00a8ggstro\u00a8m [109].As a consequence, the limits do not depend on the choice of exhaustion. Both measures aresupported on spanning forests (i.e. cycle-free subgraphs containing every vertex) of G for whichevery connected component is infinite. The WUSF is usually much more tractable, thanks in partto Wilson\u2019s algorithm rooted at infinity, which both connects the WUSF to loop-erased randomwalk and allows us to sample the WUSF of an infinite network directly rather than by passing toan exhaustion.Wilson\u2019s algorithm [228] is a remarkable method of generating the UST on a finite or recurrentnetwork by joining together loop-erased random walks. It was extended to generate the WUSF oftransient networks by BLPS [44]. Let G be a network, and let \u03b3 be a path in G that is either finiteor transient, i.e. visits each vertex of G at most finitely many times. The loop-erasure LE(\u03b3) isformed by erasing cycles from \u03b3 chronologically as they are created. Formally, LE(\u03b3)i = \u03b3ti wherethe times ti are defined recursively by t0 = 0 and ti = 1 + max{t \u2265 ti\u22121 : \u03b3t = \u03b3ti\u22121}. (In thepresence of multiple edges, a path is not determined by its vertex-trajectory. However, the definitionof the loop-erasure extends to this setting in the obvious way. Similarly, when performing Wilson\u2019salgorithm in the presence of multiple edges, we consider the random walks and their loop-erasuresto be random paths in the graph.) Let {vj : j \u2208 N} be an enumeration of the vertices of G anddefine a sequence of forests in G as follows:1. If G is finite or recurrent, choose a root vertex v0 and let F0 include v0 and no edges (in whichcase we call the algorithm Wilson\u2019s algorithm rooted at v0). If G is transient, let F0 = \u2205744.3. Wired cycle-breaking dynamics(in which case we call the algorithm Wilson\u2019s algorithm rooted at infinity).2. Given Fi, start an independent random walk from vi+1 stopped if and when it hits the set ofvertices already included in Fi.3. Form the loop-erasure of this random walk path and let Fi+1 be the union of Fi with thisloop-erased path.4. Let F =\u22c3Fi.This is Wilson\u2019s algorithm: the resulting forest F has law USTG in the finite case [228] and WUSFGin the infinite case [44], and is independent of the choice of enumeration.We also consider oriented spanning trees and forests. Let OUSTG\u2217n denote the law of the uniformspanning tree of G\u2217n oriented towards the boundary vertex \u2202n, so that every vertex of G\u2217n other than\u2202n has exactly one oriented edge emanating from it in the tree, while \u2202n does not have any orientededges emanating from it. Wilson\u2019s algorithm on G\u2217n rooted at \u2202n may be modified to produce anoriented tree with law OUSTG\u2217n by considering the loop-erased paths in step (2) to be orientedchronologically. If G is transient, making the same modification to Wilson\u2019s algorithm rootedat infinity yields a random oriented forest, known as the oriented wired uniform spanningforest [44] of G and denoted OWUSFG. The proof of the correctness of Wilson\u2019s algorithm rootedat infinity [44, Theorem 5.1] also shows that, when Gn is an exhaustion of a transient network G,the measures OUSTG\u2217n converge weakly to OWUSFG.4.3 Wired cycle-breaking dynamicsLet G be an infinite transient network and let F(G) denote the set of oriented spanning forests f ofG such that every vertex has exactly one oriented edge emanating from it in f . For each f \u2208 F(G)and oriented edge e of G, the update U(f, e) \u2208 F(G) of f is defined by the following procedure:Definition 4.3.1 (Updating f at e). If e or its reversal \u2212e is already included in f , or is a self-loop,let U(f, e) = f . Otherwise,\u2022 If e+ is in the past of e\u2212 in f , so that there is a directed path \u3008e1, . . . , ek, d\u3009 from e+ to e\u2212in f , letU(f, e) = f \u222a {\u2212e,\u2212e1, . . . ,\u2212ek} \\ {d, ek, . . . , e1}.\u2022 Otherwise, if e+ is not in the past of e\u2212 in f , let d be the unique oriented edge of f withd\u2212 = e\u2212 and let U(f, e) = f \u222a {e} \\ {d}.See Figure 1 for examples. Note that in either case, as unoriented forests, we have simply thatU(f, e) = f \u222a {e} \\ {d}; the change in orientation in the first case ensures that every vertex hasexactly one oriented edge emanating from it in U(f, e), so that U(f, e) \u2208 F(G).754.3. Wired cycle-breaking dynamicse(a) In this example, e+ is not in the past of e\u2212 in the forest.e(b) In this example, e+ is in the past of e\u2212 in the forest.Figure 4.1: Updating an oriented spanning forest (left, solid black) of Z2 (dashed black) at anoriented edge e (left, blue) to obtain a new oriented spanning forest (right, solid black). Arrowheads represent orientations of edges.Let v be a vertex of G. We define the wired cycle-breaking dynamics rooted at v to bethe Markov chain on F(G) with transition probabilitiespv(f0, f1) =1c(v)c({e : e\u2212 = v and U(f0, e) = f1}).That is, we perform a step of the dynamics by choosing an oriented edge randomly from the set{e : e\u2212 = v} with probability proportional to its conductance, and then updating at this edge.Dynamics of this form for the UST on finite graphs are well-known, see [173, \u00a74.4].To explain our choice of name for these dynamics, as well as our choice to consider orientedforests, let us give a second, equivalent, description of the update rule.If e or its reversal\u2212e is already included in f , or is a self-loop, let U(f, e) = f . Otherwise,\u2022 If e+ and e\u2212 are in the same component of f , then f \u222a e contains a (not necessarilyoriented) cycle. Break this cycle by deleting the unique edge d of f that is both764.3. Wired cycle-breaking dynamicscontained in this cycle and adjacent to e\u2212, letting U\u02dc(f, e) = f \u222a {e} \\ {d}.\u2013 If e+ was not in the past of e\u2212 in f , let U(f, e) = U\u02dc(f, e).\u2013 Otherwise, if e+ was in the past of e\u2212 in f , then there exists an oriented path frome\u2212 to d+ in U\u02dc(f, e). Let U(f, e) be the oriented forest obtained by by reversingeach edge in this path.\u2022 If e+ and e\u2212 are not in the same component of f , we consider e together with thetwo infinite directed paths in f beginning at e\u2212 and e+ to constitute a wired cycle,or \u2018cycle through infinity\u2019. Break this wired cycle by deleting the unique edge d in fsuch that d\u2212 = e\u2212, letting U(f, e) = f \u222a {e} \\ {d}.The benefit of taking our forests to be oriented is that it allows us to define these wired cyclesunambiguously. If every component of the WUSF of G is one-ended almost surely, then there isa unique infinite simple path from each of e\u2212 and e+ to infinity, so that wired cycles are alreadydefined unambiguously and the update rule may be defined without reference to an orientation.Proposition 4.3.2. Let G be an infinite transient network. Then for each vertex v of G, OWUSFGis a stationary measure for the wired cycle-breaking dynamics rooted at v, i.e. for pv( \u00b7 , \u00b7 ).Proof. Let \u3008Vn\u3009n\u22651 be an exhaustion of G. We may assume that Vn contains v and all of itsneighbours for all n \u2265 1.Let T (G\u2217n) denote the set of spanning trees of G\u2217n oriented towards the boundary vertex \u2202n.For each t \u2208 T (G\u2217n) and oriented edge e with e\u2212 = v, we define the update U(t, e) of t at e by thesame procedure (Definition 4.3.1) as for f \u2208 F(G).Proposition 4.3.3. U(Tn, E)d= Tn for every n \u2265 1.Proposition 4.3.3 is a slight variation on the classical Markov Chain-Tree Theorem [10, 165, 173]:Define a Markov chain on T (G\u2217n), as we did on F(G), bypv(t0, t1) =1c(v)c({e : e\u2212 = v and U(t0, e) = t1}).The claimed equality in distribution is equivalent to OUSTG\u2217n being a stationary measure forpv( \u00b7 , \u00b7 ), and so it suffices to verify that OUSTG\u2217n satisfies the detailed balance equations for pv( \u00b7 , \u00b7 ).This verification, which is both straightforward and similar to that of the classical Markov Chain-Tree Theorem, is omitted.To complete the proof, we show that U(Tn, E) converges to U(F, E) in distribution. It mightat first seem that this convergence holds trivially, but in fact some work is required: Updating For Tn at E requires knowledge of whether or not E+ is in the past of E\u2212, which cannot necessarilybe obtained by observing the tree or forest only within a finite set. A priori, it is therefore possiblethat E+ is in the past of E\u2212 in Tn due to the existence of a very long oriented path from E+ toE\u2212 in Tn that disappears in the limit, obstructing the claimed convergence in distribution. Thisbehaviour will be ruled out by Lemma 4.3.4.774.3. Wired cycle-breaking dynamicsBy the Skorokhod representation theorem, there exist random variables \u3008Tn\u3009n\u22651 and F, definedon some common probability space, such that Tn has law OUSTG\u2217n for each n, F has law OWUSFG,and Tn converges to F almost surely as n tends to infinity. Let E be an oriented edge chosenrandomly from the set {e : e\u2212 = v} with probability proportional to its conductance, independentlyof \u3008Tn\u3009n\u22651 and F. We write P for the probability measure under which \u3008Tn\u3009n\u22651, F and E are sampledas indicated. It suffices to prove that U(Tn, E) converges to U(F,E) in probability with respect toP.Given F, let R be the length of the longest finite simple path in F connecting v to one of itsneighbours in G that is in the same component as v in F. Since Tn converges to F almost surely,there exists a random N such that Tn and F coincide on the ball BR(v) of radius R about v in Gfor all n \u2265 N .We claim that, with probability tending to one, F and Tn agree about whether or not E+ is inthe past of v.Lemma 4.3.4. Consider the eventsP = {E+ is in the past of v in F} and Pn = {E+ is in the past of v in Tn}.The probability of the symmetric difference P4Pn converges to zero as n\u2192\u221e.Proof of lemma. Given E, the probability that E+ is in the past of v in Tn is, by Wilson\u2019s algorithm,the probability that v is contained in the loop-erasure of a random walk from E+ to \u2202n in G\u2217n. SinceG is transient, this probability converges to the probability that v is contained in the loop-erasedrandom walk from E+ in G. This probability is exactly the probability that E+ is in the past of vin F, and soP(Pn) \u2212\u2212\u2212\u2192n\u2192\u221e P(P).If P(P) \u2208 {0, 1}, we are done. Otherwise, on the event P, there is by definition a finite directedpath from E+ to v in F. This directed path is also contained in Tn for all n \u2265 N and soP(Pn |P) \u2212\u2212\u2212\u2192n\u2192\u221e 1.Combining these two above limits givesP(Pn | \u00acP) = P(Pn)\u2212 P(Pn |P)P(P)P(\u00acP) \u2212\u2212\u2212\u2192n\u2192\u221e 0.and henceP(P4Pn) = P(P)\u2212 P(P \u2229Pn) + P(Pn \u2229 \u00acP)= P(P)\u2212 P(Pn |P)P(P) + P(Pn | \u00acP)P(\u00acP)\u2212\u2212\u2212\u2192n\u2192\u221e P(P)\u2212 P(P) + 0 = 0.784.4. Proof of Theorem 4.1.3Let r \u2265 1. Observe that on the event{Tn and F coincide on the ball of radius max{R, r} about v} \\ (P4Pn),U(F, E) and U(Tn, E) coincide on the ball of radius r about v. By Lemma 4.3.4 and the definitionof P, the probability of this event converges to 1 as n\u2192\u221e, and consequently U(Tn, E) convergesto U(F, E) in probability with respect to P.4.3.1 Update-toleranceLet G be a transient network and let F be a sample of OWUSFG. An immediate consequence ofProposition 4.3.2 is that for each oriented edge e of G, the law of U(F, e) is absolutely continuouswith respect to the law of F.Corollary 4.3.5. Let G be a transient network and let e be an oriented edge of G. Then for everyevent A \u2282 F(G),OWUSFG(F \u2208 A ) \u2265 c(e)c(e\u2212)OWUSFG(U(F, e) \u2208 A ).Proof. By Proposition 4.3.2,OWUSFG(F \u2208 A ) =\u2211e\u02c6\u2212=e\u2212c(e\u02c6)c(e\u2212)OWUSFG(U(F, e\u02c6) \u2208 A )\u2265 c(e)c(e\u2212)OWUSFG(U(F, e) \u2208 A ).We refer to this property as update-tolerance by analogy to the well-established theories ofinsertion- and deletion-tolerant invariant percolation processes [173, Chapters 7 and 8].4.4 Proof of Theorem 4.1.3Proof. Let G be a network such that the WUSF of G contains at least two two-ended connectedcomponents with positive probability. Since G\u2019s WUSF is therefore disconnected with positiveprobability, Wilson\u2019s algorithm implies that G is necessarily transient. The trunk of a two-endedtree is defined to be the unique bi-infinite simple path contained in the tree, or equivalently theset of vertices and edges in the tree whose removal disconnects the tree into two infinite connectedcomponents.Let F0 be a sample of OWUSFG. By assumption, there exists a (non-random) path \u3008\u03b3i\u3009ni=0 inG such that, with positive probability, \u03b30 and \u03b3n are in distinct two-ended components of F0, \u03b3nis in the trunk of its component, and \u03b3i is not in the trunk of \u03b3n\u2019s component for i < n. Write A\u03b3for this event.For each 1 \u2264 i \u2264 n, let ei be an edge with e\u2212i = \u03b3i and e+i = \u03b3i\u22121, and let Fi \u2208 F(G) be definedrecursively byFi = U(Fi\u22121, ei) for 1 \u2264 i \u2264 n.794.4. Proof of Theorem 4.1.3e1 e2 e3 e4Figure 4.2: When we update along a path (blue arcs) connecting a two-ended component to thetrunk of another two-ended component (with each edge oriented backwards), a three-ended com-ponent is created. Edges whose removal disconnects their component into two infinite connectedcomponents are bold.We claim that on the event A\u03b3 , the component containing \u03b3n in the updated forest Fn has atleast three ends. Applying update-tolerance (Corollary 4.3.5) iteratively will then imply that theprobability of the WUSF containing a component with three or more ends is at leastOWUSFG(A\u03b3)n\u220fi=1c(ei)c(\u03b3i)which is positive as claimed.First, notice that \u03b3i\u2019s component in Fi has at least two ends for each 0 \u2264 i \u2264 n. This may beseen by induction on i. The component of \u03b30 in F0 is two-ended by assumption, while for each0 \u2264 i < n:\u2022 If \u03b3i+1 is in the same component as \u03b3i in Fi, then the component containing \u03b3i+1 in theupdated forest Fi+1 has the same number of ends and the same vertex set as the componentof \u03b3i in Fi.\u2022 If \u03b3i+1 is in a different component to \u03b3i in Fi, then the component containing \u03b3i+1 in Fi+1 isequal to the union of the component of \u03b3i in Fi, the edge ei, and the past of \u03b3i+1 in Fi. Thus,the component of \u03b3i+1 in Fi+1 has at least as many ends as the component of \u03b3i in Fi.This induction also shows that for every 0 \u2264 i \u2264 n, the component of Fi containing \u03b3i has vertexset equal to the union of the vertices in the component of F0 containing \u03b30, and the pasts of thevertices \u03b3j in Fj for 0 \u2264 j < i. By definition of the event A\u03b3 , the vertex \u03b3i is not in the trunkof \u03b3n\u2019s component in F0 for any i < n, and so in particular \u03b3n is not in the past of \u03b3i in Fi\u22121 forany i < n, so that \u03b3n\u22121 and \u03b3n are in different components of Fn\u22121. Furthermore, since neitherendpoint of ei is contained in the trunk of \u03b3n\u2019s component in F0 for any 0 \u2264 i \u2264 n\u2212 1, the trunk of\u03b3n\u2019s component in F0 is still contained in Fn\u22121. From this, we see that \u03b3n\u2019s component in Fn hasat least three ends as claimed. See Figure 1 for an illustration.804.5. Reversible random networks and the proof of Theorem 4.1.14.5 Reversible random networks and the proof of Theorem 4.1.1A rooted network (G, \u03c1) is a network G together with a distinguished vertex \u03c1, the root. Anisomorphism of graphs is an isomorphism of rooted networks if it preserves the conductances andthe root. A random rooted network (G, \u03c1) is a random variable taking values in the space ofisomorphism classes of random rooted networks (see [7] for precise definitions, including that of thetopology on this space). Similarly, we define doubly-rooted networks to be networks togetherwith an ordered pair of distinguished vertices. Let (G, \u03c1) be a random rooted network and let\u3008Xn\u3009n\u22650 be simple random walk on G started at \u03c1. We say that (G, \u03c1) is reversible if the randomdoubly-rooted networks (G, \u03c1,Xn) and (G,Xn, \u03c1) have the same distribution(G, \u03c1,Xn)d= (G,Xn, \u03c1)for every n, or equivalently for n = 1. Be careful to note that this is not the same as the reversibilityof the random walk on G, which holds for any network. Reversibility is essentialy equivalent to therelated property of unimodularity. We refer the reader to [7] for a systematic development andoverview of the beautiful theory of reversible and unimodular random rooted graphs and networks,as well as many examples.We now deduce Theorem 4.1.1 from Theorem 4.1.3. Our proof that the WUSF cannot have aunique two-ended component is adapted closely from Theorem 10.3 of [44].Proof of Theorem 4.1.1. Let (G, \u03c1) be a reversible random rooted network such that E[c(\u03c1)\u22121] <\u221e.Biasing the law of (G, \u03c1) by the inverse conductance c(\u03c1)\u22121 (that is, reweighting the law of (G, \u03c1)by the Radon-Nikodym derivative c(\u03c1)\u22121\/E[c(\u03c1)\u22121]) gives an equivalent unimodular random rootednetwork, as can be seen by checking involution invariance of the biased measure [7, Proposition 2.2].This allows us to apply Theorem 6.2 and Proposition 7.1 of [7] to deduce that every component ofthe WUSF of G has at most two ends almost surely. Theorem 4.1.3 then implies that the WUSFof G contains at most one two-ended component almost surely.Suppose for contradiction that the WUSF contains a single two-ended component with positiveprobability. Recall that the trunk of this component is defined to be the unique bi-infinite path inthe component, which consists exactly of those edges and vertices whose removal disconnects thecomponent into two infinite connected components.Let \u3008Xn\u3009n\u22650 be a random walk on G started at \u03c1, and let F be an independent random spanningforest of G with law WUSFG, so that (since WUSFG does not depend on the choice of exhaustion ofG) the sequence \u3008(G,Xn, F )\u3009n\u22650 is stationary. If the trunk of F is at some distance r from \u03c1, thenXr is in the trunk with positive probability, and it follows by stationarity that \u03c1 is in the trunkof F with positive probability. We will show for contradiction that in fact the probability that theroot is in the trunk must be zero.Recall that, for each n, the forest F may be sampled by running Wilson\u2019s algorithm rooted atinfinity, starting with the vertices \u03c1 and Xn. If we sample F in this way and find that both \u03c1 and814.5. Reversible random networks and the proof of Theorem 4.1.1Xn are contained in F\u2019s unique trunk, we must have had either that the random walk started from\u03c1 hit Xn, or that the random walk started from Xn hit \u03c1. Taking a union bound,P(\u03c1 and Xn in trunk) \u2264 P(random walk started at Xn hits \u03c1)+P(random walk started at \u03c1 hits Xn).By reversibility, the two terms on the right hand side are equal and henceP(\u03c1 and Xn in trunk) \u2264 2P(random walk started at Xn hits \u03c1).The probability on the right hand side is now exactly the probability that simple random walkstarted at \u03c1 returns to \u03c1 at time n or greater, and by transience this converges to zero. Thus,P(\u03c1 and Xn in trunk) = E[1(\u03c1 in trunk)1(Xn in trunk)] \u2212\u2212\u2212\u2192n\u2192\u221e 0and soE\uf8ee\uf8f01(\u03c1 in trunk)1nn\u221111(Xi in trunk)\uf8f9\uf8fb \u2212\u2212\u2212\u2192n\u2192\u221e 0. (?)Let I be the invariant \u03c3-algebra of the stationary sequence \u3008(G,Xn, F )\u3009n\u22650. The Ergodic Theoremimplies that1nn\u221111(Xi in trunk)a.s.\u2212\u2212\u2212\u2192n\u2192\u221e P(\u03c1 in trunk | I ).Finally, combining this with (?) and the Dominated Convergence Theorem givesE[1(\u03c1 in trunk) \u00b7 P(\u03c1 in trunk | I )] = E [P(\u03c1 in trunk | I )2] = 0.It follows that P(\u03c1 in trunk) = 0, contradicting our assumption that F had a unique two-endedcomponent with positive probability.Proof of Corollary 4.1.2. Given a probability distribution \u3008pk; k \u2265 0\u3009 on N, the augmentedGalton-Watson tree T with offspring distribution \u3008pk\u3009 is defined by taking two independentGalton-Watson trees T1 and T2, both with offspring distribution \u3008pk\u3009, and then joining them by asingle edge between their roots. Lyons, Pemantle and Peres [171] proved that T is reversible whenrooted at the root of the first tree T1; See also [7, Example 1.1].If the distribution \u3008pk\u3009 is supercritical (i.e. has expectation greater than 1), then the asso-ciated Galton-Watson tree is infinite with positive probability and on this event is almost surelytransient [173, Chapter 16]. Thus, Theorem 4.1.1 implies that every component of T \u2019s WUSF isone-ended almost surely on the event that either T1 or T2 is infinite.Recall that for every connected graph G and every edge e of G which has a positive probabilityof not being included in G\u2019s WUSF, the law of G\u2019s WUSF conditioned not to contain e is equal toWUSFG\\{e} [44, Proposition 4.2], where, if G \\ {e} is disconnected, WUSFG\\{e} is defined to be the824.5. Reversible random networks and the proof of Theorem 4.1.1union of independent samples of WUSFs of the two connected components of G \\ {e}. Let e be theedge between the roots of T1 and T2 that was added to form the augmented tree T . On the positiveprobability event that T1 and T2 are both infinite, running Wilson\u2019s algorithm on T started fromthe roots of T1 and T2 shows, by transience of T1 and T2, that e has positive probability not tobe included in T \u2019s WUSF. On this event, T \u2019s WUSF is distributed as the union of independentsamples of WUSFT1 and WUSFT2 . It follows that every component of T1\u2019s WUSF is one-endedalmost surely on the event that T1 is infinite.Example 4.5.1 (E[c(\u03c1)\u22121] <\u221e is necessary). Let (T, o) be a 3-regular tree with unit conductancesrooted at an arbitrary vertex o. Form a network G by adjoining to each vertex v of T an infinitepath, and setting the conductance of the nth edge in each of these paths to be 2\u2212n\u22121. Let onbe the nth vertex in the added path at o. Define a random vertex \u03c1 of G which is equal to owith probability 4\/7 and equal to the nth vertex in the path at o with probability 3\/(7 \u00b7 2n) foreach n \u2265 1. The only possible isomorphism classes of (G, \u03c1,X1) are of the form (G, on, on+1),(G, on+1, on), (G, o, o1), (G, o1, o), or (G, o, o\u2032), where o\u2032 is a neighbour of o in T . This allows us toeasily verify that (G, \u03c1) is a reversible random rooted network:P((G, \u03c1,X1) = (G, on, on+1)) = P((G, \u03c1,X1) = (G, on+1, on)) =17 \u00b7 2nfor all n \u2265 1 andP((G, \u03c1,X1) = (G, o, o1)) = P((G,X1, \u03c1) = (G, o, o1)) =17.When we run Wilson\u2019s algorithm on G started from a vertex of T , every excursion of the randomwalk into one of the added paths is erased almost surely. It follows that the WUSF of G is simplythe union of the WUSF of T with each of the added paths, and hence every component has infinitelymany ends almost surely.83Chapter 5Interlacements and the wired uniformspanning forestSummary. We extend the Aldous-Broder algorithm to generate the wired uniform spanning forests(WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in theclassical algorithm with Sznitman\u2019s random interlacement process. We then apply this algorithmto study the WUSF, showing that every component of the WUSF is one-ended almost surely inany graph satisfying a certain weak anchored isoperimetric condition, that the number of \u2018excessiveends\u2019 in the WUSF is non-random in any graph, and also that every component of the WUSF isone-ended almost surely in any transient unimodular random rooted graph. The first two of theseresults answer positively two questions of Lyons, Morris and Schramm, while the third extends arecent result of the author.Finally, we construct a counterexample showing that almost sure one-endedness of WUSF com-ponents is not preserved by rough isometries of the underlying graph, answering negatively a furtherquestion of Lyons, Morris and Schramm.5.1 IntroductionThe uniform spanning forests (USFs) of an infinite, locally finite, connected graph G are definedas weak limits of uniform spanning trees (USTs) of large finite subgraphs of G. These weak limitscan be taken with either free or wired boundary conditions (see Section 5.3.1), yielding the freeuniform spanning forest (FUSF) and wired uniform spanning forest (WUSF) respectively.The USFs are closely related to several other topics in probability theory, including loop-erasedrandom walks [161, 228], potential theory [44, 62], conformally invariant scaling limits [163, 208],domino tiling [150] and the Abelian sandpile model [78, 135]. In this paper, we develop a newconnection between the wired uniform spanning forest and Sznitman\u2019s interlacement process[216, 218].A key theoretical tool in the study of the UST and USFs is Wilson\u2019s algorithm [228], whichallows us to sample the UST of a finite graph by joining together loop-erasures of random walkpaths. In their seminal work [44], Benjamini, Lyons, Peres, and Schramm (henceforth referredto as BLPS) extended Wilson\u2019s algorithm to infinite transient graphs and used this extension toestablish several fundamental properties of the WUSF. For example, they proved that the WUSFof an infinite, locally finite, connected graph is connected almost surely (a.s.) if and only if the845.1. Introductionsets of vertices visited by two independent random walks on the graph have infinite intersectiona.s. This recovered the earlier, pioneering work of Pemantle [190], who proved that the FUSF andWUSF of Zd coincide for all d and are a.s. connected if and only if d \u2264 4. Wilson\u2019s algorithmhas also been instrumental in the study of scaling limits of uniform spanning trees and forests[28, 163, 193, 208, 209].Prior to the introduction of Wilson\u2019s algorithm, the best known algorithm for sampling the USTof a finite graph was the Aldous-Broder algorithm [8, 59], which generates a uniform spanningtree of a finite connected graph G as the collection of first-entry edges of a random walk on G. Wenow describe this algorithm in detail. Let \u03c1 be a fixed vertex of G, and let \u3008Xn\u3009n\u22650 be a simplerandom walk on G started at \u03c1. For each vertex v of G, let e(v) be the edge of G incident to v that istraversed by the random walk Xn as it enters v for the first time, and let T ={e(v) : v \u2208 V \\ {\u03c1}}be set of first-entry edges. Aldous [8] and Broder [59] proved independently that the resultingrandom spanning tree T is distributed uniformly on the set of spanning trees of G (see also [173,\u00a74.4]). If we orient the edge in the direction opposite to that in which it was traversed by therandom walk, then the spanning tree is oriented towards \u03c1, meaning that every vertex of G otherthan \u03c1 has exactly one oriented edge emanating from it in the tree.While the algorithm extends without modification to generate USTs of recurrent infinite graphs,the collection of first entry edges of a random walk on a transient graph might not span thegraph. Thus, naively running the Aldous-Broder on a transient graph will not necessarily producea spanning forest of the graph. Moreover, unlike in Wilson\u2019s algorithm, we cannot simply continuethe algorithm by starting another random walk from a new location. As such, it has hitherto beenunclear how to extend the Aldous-Broder algorithm to infinite transient graphs and, as a result,the Aldous-Broder algorithm has been of limited theoretical use in the study of USFs of infinitegraphs.In this paper, we extend the Aldous-Broder algorithm to infinite, transient graphs by replacingthe random walk with the random interlacement process. The interlacement process was originallyintroduced by Sznitman [216] to study the disconnection of cylinders and tori by a random walktrajectory, and was generalised to arbitrary transient graphs by Teixeira [218]. The interlacementprocess I on a transient graph G is a point process on the spaceW\u2217\u00d7R, whereW\u2217 is the space ofdoubly-infinite paths in G modulo time-shift (see Section 5.3.3 for precise definitions), and shouldbe thought of as a collection of random walk excursions from infinity. We refer the reader to themonographs [82] and [67] for an introduction to the extensive literature on the random interlacementprocess.We state our results in the natural generality of networks. Recall that a network (G, c) isa connected, locally finite graph G = (V,E), possibly containing self-loops and multiple edges,together with a function c : E \u2192 (0,\u221e) assigning a positive conductance c(e) to each edge e ofG. The conductance c(v) of a vertex v is defined to be the sum of the conductances of the edgesemanating from v. Graphs without specified conductances are considered as networks by settingc(e) \u2261 1. We will usually suppress the notation of conductances, and write simply G for a network.855.2. ApplicationsSee Section 5.3.1 for detailed definitions of the USFs on general networks.Oriented edges e are oriented from their tail e\u2212 to their head e+. The reversal of an orientededge e is denoted \u2212e.Theorem 5.1.1 (Interlacement Aldous-Broder). Let G be a transient, connected, locally finitenetwork, let I be the interlacement process on G, and let t \u2208 R. For each vertex v of G, let \u03c4t(v)be the smallest time greater than t such that there exists a trajectory (W\u03c4t(v), \u03c4t(v)) \u2208 I passingthrough v, and let et(v) be the oriented edge of G that is traversed by the trajectory W\u03c4t(v) as itenters v for the first time. ThenABt(I ) :={\u2212et(v) : v \u2208 V }has the law of the oriented wired uniform spanning forest of G.A useful feature of the interlacement Aldous-Broder algorithm is that it allows us to considerthe wired uniform spanning forest of an infinite transient graph as the stationary measure of theergodic Markov process \u3008ABt(I )\u3009t\u2208R. Indeed, it is with this stationarity in mind that we considerthe interlacement process to be a point process on W\u2217 \u00d7 R rather than the more usual W\u2217 \u00d7 R+.For example, a key step in proving that the number of excessive ends of the WUSF is non-randomis to show that the number of indestructible excessive ends is a.s. monotone in the time evolutionof the process \u3008ABt(I )\u3009t\u2208R.5.2 Applications5.2.1 EndsOther than connectivity, the most basic topological property of a forest is the number of ends itscomponents have. Here, an infinite, connected graph G = (V,E) is said to be k-ended if, over allfinite subsets W of V , the subgraph of G induced by V \\W has a supremum of k infinite connectedcomponents. In particular, an infinite tree is k-ended if and only if there exist exactly k distinctinfinite simple paths starting at each vertex of the tree. Components of the WUSF are known tobe one-ended a.s. in several large classes of graphs. The first result of this kind is due to Pemantle[190], who proved that the WUSF of Zd is one-ended a.s. for 2 \u2264 d \u2264 4, and that every componentof the WUSF of Zd has at most two ends a.s. for every d \u2265 5 (the WUSF of Z is the whole ofZ and is therefore two-ended). BLPS [44] later completed this work, showing in particular thatevery component of the WUSF is one-ended a.s. in any transient Cayley graph. We note thatone-endedness of WUSF components has important consequences for the Abelian sandpile model[135, 137, 138].Taking a different approach, Lyons, Morris and Schramm [170] gave an isoperimetric criterionfor one-endedness of WUSF components, from which they deduced that the every component ofthe WUSF is one-ended in every transitive graph not rough isometric to Z, and also every non-amenable graph. Unlike the earlier results of BLPS, the results of Lyons, Morris and Schramm are865.2. Applicationsrobust in the sense that their assumptions depend only upon the coarse geometry of the graph anddo not require any kind of homogeneity. They asked [170, Question 7.9] whether the isoperimetricassumption in their theorem could be replaced by the anchored version of the same condition, andin particular whether every WUSF component is one-ended a.s. in any graph with anchored expan-sion (defined below). Unlike classical isoperimetric conditions, anchored isoperimetric conditionsare often preserved under random perturbations such as supercritical Bernoulli percolation [70, 194].Given a network G and a set K of vertices of G, we write \u2202EK for the set of edges of G withexactly one endpoint in K, and write |K| for the sum of the conductances of the vertices in K.Similarly, if W is a set of edges in G, we write |W | for the sum of the conductances of the edgesin W . Given an increasing function f : (0,\u221e) \u2192 (0,\u221e), we say that G satisfies an anchoredf(t)-isoperimetric inequality ifinf{|\u2202EK|f(|K|) : K \u2282 V connected, v \u2208 K, |K| <\u221e}> 0for every vertex v of G. (In contrast, the graph is said to satisfy a (non-anchored) f(t)-isoperimetricinequality if the infimum inf |\u2202EK|\/f(|K|) is positive when taken over all sets of vertices K with|K| < \u221e.) In particular, G is said to have anchored expansion if and only if it satisfies an an-chored t-isoperimetric inequality, and is said to satisfy a d-dimensional anchored isoperimetricinequality if it satisfies an anchored t(d\u22121)\/d-isoperimetric inequality. Such anchored isoperimetricinequalities are known to hold on, for example, supercritical percolation clusters on Zd and relatedgraphs, such as half-spaces and wedges [194].Theorem 5.2.1. Let G be a network with c0 := infe c(e) > 0, and suppose that G satisfies ananchored f(t)-isoperimetric inequality for some increasing function f : (0,\u221e) \u2192 (0,\u221e) for whichthere exists a constant \u03b1 such that f(t) \u2264 t and f(2t) \u2264 \u03b1f(t) for all t \u2208 (0,\u221e). Suppose that falso satisfies each of the following conditions:1. \u222b \u221ec01f(t)2dt <\u221eand2. \u222b \u221ec0exp(\u2212\u03b5(\u222b \u221es1f(t)2dt)\u22121)ds <\u221efor every \u03b5 > 0.Then every component of the wired uniform spanning forest of G is one-ended almost surely.In particular, Theorem 5.2.1 applies both to every graph with anchored expansion and to everygraph satisfying a d-dimensional anchored isoperimetric inequality with d > 2. The graph formedby joining two copies of Z2 together with a single edge between their origins satisfies a 2-dimensional875.2. Applicationsisoperimetric inequality but has a two-ended WUSF. The theorem can fail if edge conductancesare not bounded away from zero, as can be seen by attaching an infinite path with exponentiallydecaying edge conductances to the root of a 3-regular tree.Theorem 5.2.1 comes very close to giving a complete answer to [170, Question 7.9]. The isoperi-metric condition of [170] is essentially that G satisfies an f(t)-isoperimetric inequality for some fsatisfying all conditions of Theorem 5.2.1 with the possible exception of (2); the precise conditionrequired is slightly weaker than this but also more technical. Our formulation of Theorem 5.2.1 isadapted from the presentation of the results of [170] given in [173, Theorem 10.43]. The differencein requirements on the function f(t) between Theorem 5.2.1 and [173, Theorem 10.43] can be seenby considering f(t) of the form t1\/2 log\u03b1(1 + t): In particular, we observe that [173, Theorem 10.43]applies to graphs satisfying a t1\/2 log\u03b1(1 + t)-isoperimetric inequality for some \u03b1 > 1\/2, while ourtheorem applies to graphs satisfying an anchored t1\/2 log\u03b1(1 + t)-isoperimetric inequality only if\u03b1 > 1.In Section 5.6, we give an example of two bounded degree, rough-isometric graphs G and G\u2032such that every component of the WUSF of G is one-ended, while the WUSF of G\u2032 a.s. contains acomponent with uncountably many ends. This answers negatively Question 7.6 of [170], and showsthat the behaviour of the WUSF of a graph cannot always be determined from the coarse geometricproperties of the graph alone.5.2.2 Excessive endsOne example of a transient graph in which the WUSF has multiply-ended components is thesubgraph of Z6 spanned by the vertex set(Z5 \u00d7 {0})\u222a({(0, 0, 0, 0, 0), (2, 0, 0, 0, 0)}\u00d7 N) ,which is obtained from Z5 by attaching an infinite path to each of the vertices u = (0, 0, 0, 0, 0) andv = (2, 0, 0, 0, 0). The WUSF of this graph, which we denote F, is equal in distribution to the unionof the WUSF of Z5 with each of the two added paths. If u and v are in the same component ofF, then there is a single component of F with three ends and all other components are one-ended.Otherwise, u and v are in different components of F, so that there are exactly two components ofF that are two-ended and all other components are one-ended. Each of these events has positiveprobability, so that the event that there exists a two-ended component of the WUSF has probabilitystrictly between 0 and 1. Nevertheless, the number of excessive ends of F, that is, the sum overall components of F of the number of ends minus 1, is equal to two a.s.In light of this example, Lyons, Morris and Schramm [170, Question 7.8] asked whether thenumber of excessive ends of the WUSF is non-random (i.e., equal to some constant a.s.) forany graph. Our next application of the interlacement Aldous-Broder algorithm is to answer thisquestion positively.885.2. ApplicationsTheorem 5.2.2. Let G be a network. Then the number of excessive ends of the wired uniformspanning forest of G is non-random.When combined with the spatial Markov property of the wired uniform spanning forest, The-orem 5.2.2 has the following immediate corollary, which states that a natural weakening of [170,Question 7.6] has a positive answer. (As mentioned above, we show the original question to havea negative answer in Section 5.6).Corollary 5.2.3. Let G be a network, and suppose that G\u2032 is a network obtained from G by addingand deleting finitely many edges. Then the wired uniform spanning forests of G and G\u2032 have thesame number of excessive ends almost surely. In particular, if every tree of the wired uniformspanning forest of G is one-ended a.s., then the same is true of G\u2032.5.2.3 Ends in unimodular random rooted graphsAnother generalisation of the one-endedness theorem of BLPS [44] concerns transient unimodularrandom rooted graphs. A rooted graph is a connected, locally finite graph G together witha distinguished vertex \u03c1, the root. An isomorphism of graphs is an isomorphism of rooted graphsif it preserves the root. The local topology on the space G\u2022 of isomorphism classes of rootedgraphs is defined so that two rooted graphs are close if they have large graph distance balls aroundtheir respective roots that are isomorphic as rooted graphs. Similarly, a doubly rooted graph is agraph together with an ordered pair of distinguished vertices, and the local topology on the spaceG\u2022\u2022 of isomorphism classes of doubly rooted graphs is defined similarly to the local topology onG\u2022. A random rooted graph (G, \u03c1) is said to be unimodular if it satisfies the Mass-TransportPrinciple, which states that for every Borel function f : G\u2022\u2022 \u2192 [0,\u221e] (which we call a masstransport),E[\u2211v\u2208Vf(G, \u03c1, v)]= E[\u2211v\u2208Vf(G, v, \u03c1)].In other words, the expected mass sent by the root is equal to the expected mass received by theroot. Unimodular random rooted networks are defined similarly by allowing the mass-transport todepend on the edge conductances. We refer the reader to Aldous and Lyons [7] for a systematicdevelopment and overview of the theory of unimodular random rooted graphs and networks, aswell as several examples.Aldous and Lyons [7] proved that every component of the WUSF is one-ended a.s. in anybounded degree unimodular random rooted graph, and the author of this article [127] later extendedthis to all transient unimodular random rooted graphs with finite expected degree at the root,deducing that every component of the WUSF is one-ended a.s. in any supercritical Galton Watsontree. (The assumption of finite expected degree was implicit in [127] since there we consideredreversible random rooted graphs, which correspond to unimodular random rooted graphs withfinite expected degree.) Our final application of the interlacement Aldous-Broder algorithm is toextend the main result of [127] by removing the condition that the expected degree of the root is895.3. Background and definitionsfinite.Theorem 5.2.4. Let (G, \u03c1) be a transient unimodular random rooted network. Then every com-ponent of the wired uniform spanning forest of G is one-ended almost surely.5.3 Background and definitions5.3.1 Uniform spanning forestsFor each finite graph G = (V,E), let USTG denote the uniform measure on the set of spanning treesof G (i.e. connected, cycle-free subgraphs of G containing every vertex), which are considered formeasure-theoretic purposes to be functions E \u2192 {0, 1}. More generally, for each finite network G,let USTG denote the probability measure on the set of spanning trees of G such that the probabilityof a tree is proportional to the product of the conductances of its edges.Let G be an infinite network, and let \u3008Vn\u3009n\u22650 be an exhaustion of V by finite connected subsets,i.e. an increasing sequence of finite connected subsets of V such that\u22c3Vn = V . For each n, letGn denote the subnetwork of G induced by Vn, and let G\u2217n denote the finite network obtained byidentifying (wiring) every vertex of G in V \\ Vn into a single vertex \u2202n, and deleting the infinitelymany self-loops from \u2202n to itself. The wired uniform spanning forest measure is defined as theweak limit of the uniform spanning tree measures on G\u2217n. That is, for every finite set S \u2282 E,WUSFG(S \u2286 F) := limn\u2192\u221eUSTG\u2217n(S \u2286 T ),where F is a sample of the WUSF of G and T is a sample of the UST of G\u2217n. In contrast, thefree uniform spanning forest measure is defined as the weak limit of the uniform spanningtree measures on the finite induced subnetworks Gn. It is easily seen that both the free and wiredmeasures are supported on the set of essential spanning forests of G, that is, the set of cycle-freesubgraphs of G that contain every vertex and do not have any finite connected components.It will also be useful to consider oriented trees and forests. Given an infinite network G withexhaustion \u3008Vn\u3009n\u22650, let OUSTG\u2217n denote the law of a uniform spanning tree of G\u2217n that has beenoriented towards the boundary vertex \u2202n, meaning that every vertex of G\u2217n other than \u2202n hasexactly one oriented edge emanating from it in the tree. BLPS [44] proved that if G is transient,then the measures OUSTG\u2217n converge weakly to a measure OWUSF, the oriented wired uniformspanning forest (OWUSF) measure. This measure is supported on the set of essential spanningforests of G that are oriented so that every vertex of G has exactly one oriented edge emanatingfrom it in the forest. The WUSF of a transient graph can be obtained from the OWUSF of thegraph by forgetting the orientations of the edges.5.3.2 The space of trajectoriesLet G be a graph. For each \u2212\u221e \u2264 n \u2264 m \u2264 \u221e, let L(n,m) be the graph with vertex set{i \u2208 Z : n \u2264 i \u2264 m} and with edge set {(i, i + 1) : n \u2264 i \u2264 m \u2212 1}. We define W(n,m) to be the905.3. Background and definitionsset of multigraph homomorphisms from L(n,m) to G such that the preimage of each vertex in Gis finite, and define W to be the unionW :=\u22c3{W(n,m) : \u2212\u221e \u2264 n \u2264 m \u2264 \u221e} .For each set K \u2286 V , we let WK(n,m) denote the set of w \u2208 W(n,m) that visit K (that is, forwhich there exists n \u2264 i \u2264 m such that w(i) \u2208 K), and let WK be the union WK =\u22c3{WK(n,m) :\u2212\u221e \u2264 n \u2264 m \u2264 \u221e}.Given w \u2208 W(n,m) and a \u2264 b \u2208 Z, we define w|[a,b] \u2208 W(n \u2228 a,m \u2227 b) to be the restriction ofw to the subgraph L(n \u2228 a,m \u2227 b) of L(n,m). Given w \u2208 WK(n,m), let H\u2212K(w) = inf{n \u2264 i \u2264 m :w(i) \u2208 K}, let H+K(w) = sup{n \u2264 i \u2264 m : w(i) \u2208 K}, and letwK = w|[H\u2212k (w),H+K(w)]be the restriction of w to between the first it visits K and last time it visits K. We equip W withthe topology generated by open sets of the form{w \u2208 W : w visits K and wK = w\u2032K},where K \u2282 V is finite and w\u2032 \u2208 WK . (Note that this topology is not the weakest topology makingthe evaluation maps w 7\u2192 w(i) and w 7\u2192 w(i, i+ 1) continuous. First and last hitting times of finitesets are not continuous with respect to that topology, but are continuous with respect to ours. TheBorel \u03c3-algebras generated by the two topologies are the same.) We also equip W with the Borel\u03c3-algebra generated by this topology.The time shift \u03b8k :W \u2192W is defined by \u03b8k :W(n,m) \u2212\u2192W(n\u2212 k,m\u2212 k),\u03b8k(w)(i) = w(i+ k), \u03b8k(w)(i, i+ 1) = w(i+ k, i+ k + 1).The space W\u2217 is defined to be the quotientW\u2217 =W\/ \u223c where w1 \u223c w2 if and only if w1 = \u03b8k(w2) for some k.Let pi : W \u2192 W\u2217 denote the quotient map. W\u2217 is equipped with the quotient topology andassociated quotient \u03c3-algebra. An element of W\u2217 is called a trajectory.5.3.3 The interlacement processGiven a network G = (G, c), the conductance c(v) of a vertex v is defined to be the sum of theconductances of the edges emanating from v, and, for each pair of vertices (u, v), the conductancec(u, v) is defined to be the sum of the conductances of the edges connecting u to v. Recall thatthe random walk X on the network G is the Markov chain on V with transition probabilitiesp(u, v) = c(u, v)\/c(u). In case G has multiple edges, we will also keep track of the edges crossed by915.3. Background and definitionsX, considering X to be a random element of W(0,\u221e). We write either Pu or PGu for the law of therandom walk started at u on the network G, depending on whether the network under considerationis clear from context. When X is a random walk on a network G and K is a set of vertices in G,we let \u03c4K denote the first time that X hits K and let \u03c4+K denote the first positive time that X hitsK.Let G = (V,E) be a transient network. Given w \u2208 W(n,m), let w\u2190 \u2208 W(\u2212n,\u2212m) be thereversal of w, defined by setting w\u2190(i) = w(\u2212i) for all \u2212m \u2264 i \u2264 \u2212n, and w\u2190(i, i + 1) =w(\u2212i\u2212 1,\u2212i) for all \u2212m \u2264 i \u2264 \u2212n\u2212 1. For each subset A \u2286 W, let A\u2190 denote the setA\u2190 := {w \u2208 W : w\u2190 \u2208 A }.For each finite set K \u2282 V , define a measure QK on WK by settingQK({w \u2208 W : w(0) \/\u2208 K}) = 0and, for each u \u2208 K and each two Borel subsets A ,B \u2208 W,QK({w \u2208 W : w|(\u2212\u221e,0] \u2208 A , w(0) = u and w|[0,\u221e) \u2208 B})= c(u)Pu(\u3008Xk\u3009k\u22650 \u2208 A\u2190 and \u03c4+K =\u221e)Pu(\u3008Xk\u3009k\u22650 \u2208 B).Let W\u2217K = pi(WK) be the set of trajectories that visit K.Theorem 5.3.1 (Sznitman [216] and Teixeira [218]: Existence and uniqueness of the interlacementintensity measure). Let G be a transient network. There exists a unique \u03c3-finite measure Q\u2217 onW\u2217 such that for every Borel set A \u2286 W\u2217 and every finite K \u2282 V ,Q\u2217(A \u2229W\u2217K) = QK(pi\u22121(A )). (5.3.1)The measure Q\u2217 is referred to as the interlacement intensity measure.Definition 5.3.2. Let \u039b denote the Lebesgue measure on R. The interlacement process I onG is defined to be a Poisson point process on W\u2217 \u00d7 R with intensity measure Q\u2217 \u2297 \u039b. For eacht \u2208 R, we denote by It the set of w \u2208 W\u2217 such that (w, t) \u2208 I . We also write I[a,b] for theintersection of I with W\u2217 \u00d7 [a, b].Let \u3008Vn\u3009n\u22650 be an exhaustion of an infinite transient network G. The interlacement process onG can be constructed as a limit of Poisson processes on random walk excursions from the boundaryvertices \u2202n to itself in the networks G\u2217n.Let N be a Poisson point process on R with intensity measure c(\u2202n)\u039b. Conditional on N , forevery t \u2208 N , let Wt be a random walk started at \u2202n and stopped when it first returns to \u2202n, wherewe consider each Wt to be an element of W\u2217. We define I n to be the point processI n :={(Wt, t) : t \u2208 N}.925.3. Background and definitionsProposition 5.3.3. Let G be an infinite transient network and let \u3008Vn\u3009n\u22650 be an exhaustion of G.Then the Poisson point processes I n converge in distribution to the interlacement process I asn\u2192\u221e.A similar construction of the random interlacement process is sketched in [217, \u00a74.5].Proof. Let K \u2282 V be finite, and let n be sufficiently large that K is contained in Vn. Define ameasure QnK on W by settingQnK({w \u2208 W : w(0) \/\u2208 K}) = 0and, for each u \u2208 K, each r,m \u2265 0 and each two Borel subsets A ,B \u2208 W,QnK({w \u2208 W(\u2212r,m) : w|[\u2212r,0] \u2208 A , w(0) = u and w|[0,m] \u2208 B})= c(u)PG\u2217nu(\u3008Xk\u3009rk=0 \u2208 A\u2190 and \u03c4+K > \u03c4\u2202n = r)PG\u2217nu (\u3008Xk\u3009mk=0 \u2208 B and \u03c4\u2202n = m). (5.3.2)By reversibility of the random walk, the right-hand side of (5.3.2) is equal toc(\u2202n)PG\u2217n\u2202n(\u3008Xk\u3009nk=0 \u2208 A and X\u03c4K = u and \u03c4K = r < \u03c4+\u2202n)\u00b7 PG\u2217nu(\u3008Xk\u3009mk=0 \u2208 B and \u03c4V \\Vn = m). (5.3.3)It follows that QnK(W) = c(\u2202n)PG\u2217n\u2202n(\u03c4K < \u03c4+\u2202n) and that the normalized measure QnK\/QnK(W)coincides with the law of a random walk excursion from \u2202n to itself in G\u2217n that has been conditionedto hit K and reparameterised so that it first hits K at time 0. Thus, by the splitting property ofPoisson processes, I n is a Poisson point process onW\u2217\u00d7R with intensity measure Qn\u2217\u2297\u039b, whereQn\u2217 satisfiesQn\u2217(A \u2229W\u2217K) = QnK(pi\u22121(A )).We conclude the proof by noting that QnK converges weakly to QK as n\u2192\u221e.5.3.4 Hitting probabilitiesRecall that the capacity (a.k.a. the conductance to infinity) of a finite set of vertices K in anetwork G is defined to beCap(K) =\u2211v\u2208Kc(v)Pv(\u03c4+K =\u221e),and observe that QK(W) = Cap(K) for every finite set of vertices K. If K is infinite, we defineCap(K) = limn\u2192\u221eCap(Kn), where Kn is any increasing sequence of finite sets of vertices with\u22c3Kn = K. We say that a set K of vertices is hit by I[a,b] if there exists (W, t) \u2208 I[a,b] such thatW hits K. By the definition of I , we have thatP(K hit by I[a,b]) = 1\u2212 exp(\u2212(b\u2212 a)QK(W)) = 1\u2212 exp (\u2212(b\u2212 a) Cap(K))935.3. Background and definitionsfor each finite set K. This formula extends to infinite sets by taking limits over exhaustions: IfK \u2286 V is infinite, let Kn be an exhaustion of K by finite sets. ThenP(K hit by I[a,b]) = limn\u2192\u221eP(Kn hit by I[a,b])= 1\u2212 limn\u2192\u221e exp(\u2212(b\u2212 a) Cap(Kn)) = 1\u2212 exp (\u2212(b\u2212 a) Cap(K)) .Similarly, the expected number of trajectories in I[a,b] that hit K is equal to (b\u2212 a) Cap(K). Weapply the above formulas to deduce the following simple 0-1 law.Lemma 5.3.4. Let G be a transient network, let I be the interlacement process on G. Then forall a < b \u2208 R and every set of vertices K \u2286 V , we haveP(K is hit by infinitely many trajectories in I[a,b])= 1(Cap(K) =\u221e) .Proof. If Cap(K) is finite then the expected number of trajectories in I[a,b] that hit K is finite, sothat the number of trajectories in I[a,b] that hit K is finite a.s. Conversely, if Cap(K) is infinite,then there is a trajectory in I[b\u22122\u2212n,b\u22122\u2212n\u22121] that hits K a.s. for every n \u2265 1 a.s. Since b\u2212 2\u2212n \u2265 afor all but finitely many n, it follows that I[a,b] hits K infinitely often a.s.We next prove that any set that has a positive probability to be hit infinitely often by anysingle trajectory will in fact be hit by infinitely many trajectories. Recall the method of randompaths [191, Theorem 10.1]: If G is an infinite network, A is a finite subset of G and \u0393 is a randominfinite simple path in G starting at A, thenCap(A)\u22121 \u2264\u2211e\u2208EP(e \u2208 \u0393)2.In particular, if the sum on the right hand side is finite for some random infinite simple path \u0393starting at A, then the capacity of A is positive and G is therefore transient. Moreover, for everyfinite set A in a transient network, there exists a random infinite simple path \u0393 starting in A suchthatCap(A)\u22121 =\u2211e\u2208EP(e \u2208 \u0393)2.The following lemma is presumably well-known, but we were unable to find a reference.Lemma 5.3.5. Let G be a transient network and let K \u2286 V . If the random walk on G hits Kinfinitely often with positive probability, then Cap(K) =\u221e.Proof. Let c be the conductance function of G. First suppose that K is hit infinitely often withprobability one. Let X = \u3008Xi\u3009i\u22650 be a random walk on G started at a vertex of K, and let Ni bethe ith time that X visits K. DefinecK(u, v) = c(u)Pu(XN1 = v)945.3. Background and definitionsfor all u, v \u2208 K, so that XNi is the random walk on the (non-locally finite) network H :=((K,K2), cK). Note that if A is a finite subset of K, then the capacity of A considered as aset of vertices in H is the same as the capacity of A considered as a set of vertices in G. That is,Cap(A) =\u2211v\u2208Ac(v)Pv(\u3008Xi\u3009i\u22650 returns to A) =\u2211v\u2208Ac(v)Pv(\u3008XNi\u3009i\u22650 returns to A).Let \u3008Kn\u3009n\u22651 be an increasing sequence of finite sets with\u22c3n\u22651Kn = K, and let \u0393 be a randominfinite simple path in H starting at K1 such that12\u2211u,v\u2208VP({u, v} \u2208 \u0393)2 = Cap(K1)\u22121 <\u221e.For each n \u2265 2, let \u0393n be the subpath of \u0393 beginning at the last time \u0393 visits Kn. Then, by themonotone convergence theorem,Cap(K) = limn\u2192\u221eCap(Kn) \u2265 limn\u2192\u221e(12\u2211u,v\u2208VP({u, v} \u2208 \u0393n)2)\u22121=\u221e.This concludes the proof in this case.Now suppose that K is hit infinitely often by the random walk on G with positive probability,and, for each vertex u of G, let h(u) be the probability that a random walk on G started at u hitsK infinitely often. We have that, for each two vertices u and v of G,Pu(X1 = v | X hits K infinitely often) = Pu(X1 = v)Pv(X hits K infinitely often)Pu(X hits K infinitely often)=h(v)c(u, v)h(u)c(u).It follows by an elementary calculation that the random walk on G conditioned to hit K infinitelyoften is reversible, and is equal to the random walk on the network (G, c\u02c6), where the conductancesc\u02c6 are defined byc\u02c6(u, v) = c(u, v)h(u)h(v), c\u02c6(u) = c(u)h(u)2.This is an example of Doob\u2019s h-transform. Since h \u2264 1, Rayleigh monotonicity implies that thecapacity of K with respect to the h-transformed conductances c\u02c6 is less than the capacity of K withrespect to the original conductances c. Thus, we may conclude the proof by applying the argumentof the previous paragraph to the h-transformed network.The following lemma is an immediate consequence of Lemma 5.3.4 and Lemma 5.3.5.Lemma 5.3.6. Let G be a transient network, let I be the interlacement process on G. Then for955.4. Interlacement Aldous-Broderall a < b \u2208 R and every set of vertices K \u2286 V , we haveP(infinitely many vertices of K are hit by I[a,b])=P(K is hit by infinitely many trajectories in I[a,b])= 1(Cap(K) =\u221e) .5.4 Interlacement Aldous-BroderIn this section we describe the Interlacement Aldous-Broder algorithm and investigate its basicproperties. Let G be an infinite transient network. For each set A \u2286 W\u2217 \u00d7 R, and each vertexv \u2208 V , define\u03c4t(A, v) := inf{s \u2265 t : \u2203(W, s) \u2208 A such that W hits v} .Let I be the interlacement process on G and write \u03c4t(v) = \u03c4t(I , v). Let A be the set of subsetsA of W\u2217 \u00d7R that satisfy the property that for every for every vertex v \u2208 V and every t \u2208 R thereexists a unique trajectory W\u03c4t(A,v) such that (W\u03c4t(A,v), \u03c4t(A, v)) \u2208 A and W\u03c4t(A,v) hits v. It is clearthat I \u2208 A a.s. Define et(v) = et(I , v) to be the oriented edge pointing into v that is traversedby the trajectory W\u03c4t(v) as it enters v for the first time. For each t \u2208 R and T \u2208 (t,\u221e], we definethe set ABTt (I ) \u2286 E byABTt (I ) :={\u2212et(v) : v \u2208 V, \u03c4t(v) \u2264 T} . (5.4.1)We write ABt(I ) = AB\u221et (I ). We define ABTt (A) similarly for all A \u2208 A. Let \u3008Vn\u3009n\u22650 be anexhaustion of G, and for each n \u2265 0 let I n be defined as in Section 5.3.3. Since the process I nis just a decomposition of a single random walk trajectory into excursions, Theorem 5.1.1 followsimmediately from the following lemma together with the correctness of the classical Aldous-Broderalgorithm.Lemma 5.4.1. Let G be a transient network with exhaustion \u3008Vn\u3009n\u22650 and let I be the interlacementprocess on G. Then ABTt (In) converges weakly to ABTt (I ) for each t \u2208 R and T \u2208 [t,\u221e].Proof. Let E\u2192 be the set of oriented edges of G, let S be a finite subset of E\u2192, and consider thesetCTt (S) :={A \u2286 W\u2217 \u00d7 R : S \u2286 ABTt (A)}.Let \u2202CTt (S) denote the topological boundary of CTt (S). Observe that\u2202CTt (S) \u2286{A \u2286 A : \u03c4t(e\u2212) \u2208 {t, T} for some e \u2208 S}\u222a (W\u2217 \u00d7 R \\ A).It follows that P(I \u2208 \u2202CTt (S)) = 0. The Portmanteau Theorem [155, Theorem 13.16] therefore965.4. Interlacement Aldous-Broderimplies thatP(S \u2286 ABTt (I ))= P(I \u2208 CTt (S))= limn\u2192\u221eP(I n \u2208 CTt (S))= limn\u2192\u221eP(S \u2286 ABTt (I n))for every finite set S \u2286 E\u2192.We next establish the basic properties of the process \u3008Ft\u3009t\u2208R = \u3008AB\u221et (I )\u3009t\u2208R. We first recallthat a process \u3008Xt\u3009t\u2208R is said to be ergodic if P(\u3008Xt\u3009t\u2208R \u2208 A ) \u2208 {0, 1} whenever A is an eventthat is shift invariant in the sense that \u3008Xt\u3009t\u2208R \u2208 A implies that \u3008Xt+s\u3009t\u2208R \u2208 A for every s \u2208 R.The process \u3008Xt\u3009t\u2208R is said to be mixing if for every two events A and B,P(\u3008Xt\u3009t\u2208R \u2208 A and \u3008Xt+s\u3009t\u2208R \u2208 B) \u2212\u2212\u2212\u2192s\u2192\u221e P(\u3008Xt\u3009t\u2208R \u2208 A )P (\u3008Xt\u3009t\u2208R \u2208 B) .Every mixing process is clearly ergodic, but the converse need not hold in general [155, \u00a720.5].Proposition 5.4.2. Let G be a transient network and let I be the interlacement process on G.Then \u3008Ft\u3009t\u2208R = \u3008AB\u221et (I )\u3009t\u2208R is an ergodic, mixing, Markov process.Proof. The fact that \u3008Ft\u3009t\u2208R is a Markov process follows from the following identity, which im-mediately implies that \u3008Fs\u3009s\u2264t and \u3008Fs\u3009s\u2265t are conditionally independent given Ft for each t \u2208 R:whenever t \u2208 R and s \u2265 0,ABt\u2212s(I ) = ABtt\u2212s(I ) \u222a {e \u2208 ABt(I ) : e\u2212 is not hit by I[t\u2212s,t)}. (5.4.2)We now prove that \u3008Ft\u3009t\u2208R is mixing. Let a1, a2, . . . , an and b1, b2, . . . , bm be two increasingsequences of real numbers, and let A1, A2, . . . , An and B1, B2, . . . Bm be finite subsets of E\u2192. Let Kdenote the set of endpoints of edges in the union\u22c3ni=1Ai. Let s be sufficiently large that b1+s \u2265 an.Let A = {Ai \u2286 AB\u221eai (I ) for all 1 \u2264 i \u2264 n}, Bs = {Bi \u2286 AB\u221ebi+s(I ) for all 1 \u2264 i \u2264 m}, andA \u2032s = {Ai \u2286 ABb1+sai (I ) for all 1 \u2264 i \u2264 n} \u2286 A . The events A \u2032s and Bs are independent andP(A \\A \u2032s ) \u2264 P(I[an,b1+s] does not hit some vertex in K) \u2264\u2211v\u2208Kexp(\u2212(b1 + s\u2212 an)Cap(v)) .We deduce that|P(A \u2229Bs)\u2212 P(A )P(Bs)| \u2264 |P(A \u2229Bs)\u2212 P(A \u2032s \u2229Bs)|+ |P(A \u2032s \u2229Bs)\u2212 P(A )P(Bs)|= |P(A \u2229Bs)\u2212 P(A \u2032s \u2229Bs)|+ |P(A \u2032s )P(Bs)\u2212 P(A )P(Bs)|\u2264 2\u2211v\u2208Kexp(\u2212(b1 + s\u2212 an)Cap(v)) \u2212\u2212\u2212\u2192s\u2192\u221e 0.Since events of the form {Ai \u2286 AB\u221eai (I ) for all 1 \u2264 i \u2264 n} generate the Borel \u03c3-algebra on thespace of (E\u2192){0,1}-valued processes, it follows that \u3008Ft\u3009t\u2208R is mixing and therefore ergodic.975.5. Proof of Theorems 5.2.1, 5.2.2 and 5.2.45.5 Proof of Theorems 5.2.1, 5.2.2 and 5.2.4Recall that an end of a tree is an equivalence class of infinite simple paths in the tree, where twoinfinite simple paths are equivalent if their traces have finite symmetric difference. Similarly, if Fis a spanning forest of a network G, we define an end of F to be an equivalence class of infinitesimple paths in G that eventually only use edges of F, where, again, two infinite simple paths areequivalent if their traces have finite symmetric difference. If an infinite tree T is oriented so thatevery vertex of the tree has exactly one oriented edge emanating from it, then there is exactly oneend \u03be of T for which the paths representing \u03be eventually follow the orientation of T . We call thisend the primary end of T . We call ends of T that are not the primary end excessive endsof T . Note that if \u03be is an excessive end of T and \u03b3 is a simple path representing \u03be, then all butfinitely many of the edges traversed by the path \u03b3 are traversed in the opposite direction to theirorientation in T .If I is the interlacement process on a transient network G, we write I[a,b] for the set of verticesof G hit by I[a,b].The proofs of Theorems 5.2.1 and 5.2.4 both rely on the following criterion. See Figure 5.1 foran illustration of the proof.Lemma 5.5.1. Let G be a transient network, let I be the interlacement process on G, and let\u3008Ft\u3009t\u2208R = \u3008ABt(I )\u3009t\u2208R. If the connected component containing v of pastF0(v) \\ I[\u2212\u03b5,0] is finite a.s.for every vertex v of G and every \u03b5 > 0, then every component of F0 is one-ended a.s.Proof. If u is in the past of v in F\u2212\u03b5, then \u03c4\u2212\u03b5(u) \u2265 \u03c4\u2212\u03b5(v). Thus, on the event that v is not hit byI[\u2212\u03b5,0], the past of v in F\u2212\u03b5 is equal to the component containing v in the subgraph of pastF0(v)induced by the complement of I[\u2212\u03b5,0] (see Figure 5.1). By assumption, the connected componentcontaining v in this subgraph is finite a.s., and so, by stationarity,P(pastF0(v) is infinite) = P(pastF\u2212\u03b5(v) is infinite) \u2264 P(v \u2208 I[\u2212\u03b5,0]) = 1 \u2212 e\u2212\u03b5Cap(v) \u2212\u2212\u2212\u2192\u03b5\u21920 0.Since v was arbitrary, we deduce that every component of F0 is one-ended a.s.The proof of Theorem 5.2.1 requires both of the following theorems.Theorem 5.5.2 (Lyons, Morris and Schramm [170]; Lyons and Peres [173, Theorem 6.41]). Let Gbe a network satisfying an anchored f(t)-isoperimetric inequality, where f is an increasing functionsuch that f(t) \u2264 t, f(2t) \u2264 \u03b1f(t) for some constant \u03b1, and \u222b\u221e1 f(t)\u22122dt < \u221e. Then for everyvertex v of G there exists a positive constant cv such that, for every connected set K containing v,Cap(K) \u2265 c2v4\u03b12(\u222b \u221e|K|1f(t)2)\u22121.In particular, G is transient.985.5. Proof of Theorems 5.2.1, 5.2.2 and 5.2.4vFigure 5.1: A schematic illustration of the proof of Lemma 5.5.1. Left: a tree in F0 with twoexcessive ends in the past of the vertex v. Centre: each of the excessive ends is hit by a trajectoryof I[\u2212\u03b5,0] (red), but v is not hit by such a trajectory. Right: the resulting forest F\u2212\u03b5.Theorem 5.5.3 (Morris [185, Theorem 9]: WUSF components are recurrent). Let G be an infi-nite network with edge conductances bounded above. Then every component of the wired uniformspanning forest of G is recurrent a.s.An equivalent statement of Theorem 5.5.3 is the following.Lemma 5.5.4. Let G be an infinite network with infe c(e) > 0. Then every component of the wireduniform spanning forest of G is recurrent when given unit conductances a.s.Proof. Form a network G\u02c6 by replacing each edge e of G with dc(e)e parallel edges each withconductance c(e)\/dc(e)e. It follows immediately from the definition of the UST of a network thatthe WUSF F\u02c6 of G\u02c6 may be coupled with the WUSF F of G so that an edge e of G is containedin F if and only if one of the edges corresponding to e in G\u02c6 is contained in F\u02c6. Since G\u02c6 has edgeconductances bounded above, Theorem 5.5.3 implies that every component of F\u02c6 is recurrent a.s.Since the edge conductances of G\u02c6 are bounded away from zero, it follows by Rayleigh monotonicitythat every component of F\u02c6 is recurrent a.s. when given unit conductances, and consequently thatthe same is true of F.Proof of Theorem 5.2.1. By Theorem 5.5.2, G is transient. Let I be the interlacement process onG. Let v be a fixed vertex of G. Then for each vertex u of G contained in the same componentof F = AB0(I ) as v, the conditional probability given F that u is connected to v in F \\ I[\u2212\u03b5,0]is equal to exp(\u2212\u03b5Cap(\u03b3u,v)), where \u03b3u,v is the trace of the path connecting u to v in F. WritedF(u, v) for the graph distance in F between two vertices u and v in G. Since the conductancesof G are bounded below by some positive constant \u03b4, we have that |\u03b3u,v| \u2265 \u03b4dF(u, v) and so, byTheorem 5.5.2,P(u connected to v in F \\ I[\u2212\u03b5,0] | F)\u2264 1(u connected to v in F) exp\uf8eb\uf8ed\u2212\u03b5 c2v4\u03b12(\u222b \u221e\u03b4dF(u,v)1f(t)2)\u22121\uf8f6\uf8f8 . (5.5.1)995.5. Proof of Theorems 5.2.1, 5.2.2 and 5.2.4Lemma 5.5.5. Let T be an infinite tree, let v be a vertex of T and let \u03c9 be a random subgraphof T . For each vertex u of G, let \u2016u\u2016 denote the distance between u and v in T , and suppose thatthere exists a function p : N\u2192 [0, 1] such thatP(u is connected to v in \u03c9) \u2264 p (\u2016u\u2016)for every vertex u in T and \u2211n\u22651p(n) <\u221e.ThenP(The component of v in \u03c9 is infinite) \u2264 Cap(v)\u2211n\u22651p(n).In particular, if T is recurrent then the component containing v in \u03c9 is finite a.s.Proof. Suppose that the component containing v in \u03c9 is infinite with positive probability; theinequality holds trivially otherwise. Denote this event A . Fix a drawing of T in the plane rootedat v. On the event A , let \u0393 = \u0393(\u03c9) be the leftmost simple path from v to infinity in \u03c9. ObservethatP(u \u2208 \u0393 | A ) \u2264 P(u is connected to v in \u03c9)P(A )\u2264 p(n)P(A )and\u2211\u2016u\u2016=nP(u \u2208 \u0393 | A ) = 1,so that \u2211\u2016u\u2016=nP(u \u2208 \u0393 | A )2 \u2264 p(n)P(A ).and hence\u2211e\u2208EP(e \u2208 \u0393 | A )2 =\u2211u\u2208V \\{v}P(u \u2208 \u0393 | A )2 =\u2211n\u22651\u2211\u2016u\u2016=nP(u \u2208 \u0393 | A )2 \u2264 1P(A )\u2211n\u22651p(n).Applying the method of random paths (taking our measure on random paths to be the conditionaldistribution of \u0393 given A ), we deduce thatCap(v) \u2265\uf8eb\uf8ed 1P(A )\u2211n\u22651p(n)\uf8f6\uf8f8\u22121 ,which rearranges to give the desired inequality.Comparing sums with integrals, hypothesis (2) of Theorem 5.2.1 implies that\u2211n\u22651exp(\u2212\u03b5 c2v4\u03b12(\u222b \u221e\u03b4n1f(t)2)\u22121)<\u221e1005.5. Proof of Theorems 5.2.1, 5.2.2 and 5.2.4for every \u03b5 > 0, and we deduce from eq. (5.5.1), Lemma 5.5.5 and Lemma 5.5.4 that the componentcontaining v in F \\ I[\u2212\u03b5,0] is finite a.s. Since v was arbitrary, every component of F \\ I[\u2212\u03b5,0] is finitea.s. We conclude by applying Lemma 5.5.1.5.5.1 Unimodular random rooted graphsProof of Theorem 5.2.4. Let (G, \u03c1) be a transient unimodular random rooted network, let I bethe interlacement process on G and let F = AB0(I ). It is known [7, Theorem 6.2, Proposition 7.1]that every component of F has at most two ends a.s. Suppose for contradiction that F containsa two-ended component with positive probability. The trunk of a two-ended component of F isdefined to be the unique doubly infinite simple path that is contained in the component. Definetrunk(F) to be the set of vertices of G that are contained in the trunk of some two-ended componentof F. For each vertex v of G, let e(v) be the unique oriented edge of G emanating from v that iscontained in F. For each vertex v \u2208 trunk(F), let s(v) be the unique vertex in trunk(F) that hase(s(v))+ = v, and let sn(v) be defined recursively for n \u2265 0 by s0(v) = v, sn+1(v) = s(sn(v)).Let \u03b5 > 0. We claim that sn(\u03c1) \u2208 I[\u2212\u03b5,0] for infinitely many n a.s. on the event that \u03c1 \u2208 trunk(F).Let k \u2265 1 and define the mass transportfk(G, u, v,F, I[\u2212\u03b5,0]) = 1\uf8eb\uf8ed u is in the trunk of its component in F,v = sk(u) and I[\u2212\u03b5,0] \u2229 {sn(v) : n \u2265 0} 6= \u2205\uf8f6\uf8f8 .Applying the mass-transport principle to fk, we deduce thatP(I[\u2212\u03b5,0] \u2229 {sn(\u03c1) : n \u2265 0} 6= \u2205 | \u03c1 \u2208 trunk)= P(I[\u2212\u03b5,0] \u2229 {sn(\u03c1) : n \u2265 k} 6= \u2205 | \u03c1 \u2208 trunk)for all k \u2265 0. (Here we are using the fact that (G, \u03c1,F, I[\u2212\u03b5,0]) is a unimodular rooted markedgraph, see [7] for appropriate definitions.) By taking the limit as k \u2192\u221e, we deduce thatP(sn(\u03c1) \u2208 I[\u2212\u03b5,0] for infinitely many n | \u03c1 \u2208 trunk)= P(I[\u2212\u03b5,0] \u2229 {sn(\u03c1) : n \u2265 0} 6= \u2205 | \u03c1 \u2208 trunk).It follows that sn(\u03c1) \u2208 I[\u2212\u03b5,0] for infinitely many n almost surely on the event that \u03c1 \u2208 trunk(F) \u2229I[\u2212\u03b5,0]. Since \u03c1 \u2208 trunk(F) \u2229 I[\u2212\u03b5,0] with positive probability conditional on F and the event that\u03c1 \u2208 trunk(F), it follows from Lemma 5.3.6 that infinitely many vertices of {sn(\u03c1) : n \u2265 1} are hitby I[\u2212\u03b5,0] a.s. on the event that \u03c1 \u2208 trunk(F).It follows from [7, Lemma 2.3] that for every vertex v \u2208 trunk(F), sn(v) \u2208 I[\u2212\u03b5,0] for infinitelymany n a.s., and consequently that the component containing v in pastF(v) \\ I[\u2212\u03b5,0] is finite forevery vertex v of G a.s. We conclude by applying Lemma 5.5.1.1015.5. Proof of Theorems 5.2.1, 5.2.2 and 5.2.45.5.2 Excessive endsProof of Theorem 5.2.2. We may assume that G is transient: if not, the WUSF of G is connected,the number of excessive ends of G is tail measurable, and the claim follows by tail-triviality of theWUSF [44]. Let I be the interlacement process on G and let \u3008Ft\u3009t\u2208R = \u3008ABt(I )\u3009t\u2208R. The eventthat F0 has uncountably many ends is tail measurable, and hence has probability either 0 or 1,again by tail-triviality of the WUSF. If the number of ends of F0 is uncountable a.s., then F0 mustalso have uncountably many excessive ends a.s., since the number of components of F0 is countable.Thus, it suffices to consider the case that F0 has countably many ends a.s.For each t \u2208 R, we call an excessive end \u03be of Ft indestructible if Cap({\u03b3i : i \u2265 0}) is finitefor some (and hence every) simple path \u3008\u03b3i\u3009i\u22650 in G representing \u03be, and destructible otherwise.Given a simple path \u03b3 = \u3008\u03b3i\u3009i\u22650, write \u03b3i,i+1 for the oriented edge that is traversed by \u03b3 as itmoves from \u03b3i to \u03b3i+1, and let \u03b3i,i\u22121 = \u2212\u03b3i\u22121,i. Observe, as we did at the beginning of Section 5.5,that a simple path \u3008\u03b3i\u3009i\u22650 in G represents an excessive end of Ft if and only if et(\u03b3i,I ) = \u03b3i,i\u22121for all sufficiently large values of i (equivalently, if and only if the reversed oriented edges \u2212\u03b3i,i+1are contained in Ft for all sufficiently large values of i). Since F0 has countably many ends a.s. byassumption, it follows from Lemma 5.3.4 that for every destructible end \u03be of F0 and every infinitesimple path \u3008\u03b3i\u3009i\u22650 in G representing \u03be, the trace {\u03b3i : i \u2265 0} of \u03b3 is hit by I[\u2212\u03b5,0] infinitely oftena.s. for every \u03b5 > 0. Recall from the proof of Lemma 5.5.1 that, on the event that v \/\u2208 I[\u2212\u03b5,0],the past of v in F\u2212\u03b5 is contained in subgraph of pastF0(v) induced by the complement of I[\u2212\u03b5,0].It follows that v a.s. does not have any destructible ends in its past in F\u2212\u03b5 a.s. on the event thatv \/\u2208 I[\u2212\u03b5,0], and so, by stationarity,P(v has a destructible end in its past in F0) \u2264 P(v \u2208 I[\u2212\u03b5,0]) \u2212\u2212\u2212\u2192\u03b5\u219200.Since the vertex v was arbitrary, we deduce that F0 does not contain any destructible excessiveends a.s.Since every excessive end of F0 is indestructible a.s., it follows from Lemma 5.3.6 that for everyexcessive end \u03be of F0 and every path \u3008vi\u3009 representing \u03be, only finitely many of the vertices vi arehit by I[t,0] a.s. for every t \u2264 0. Since F0 has at most countably many excessive ends a.s., wededuce that every path \u3008vi\u3009i\u22650 that represents an excessive end of F0 also represents an excessiveend of Ft for every t \u2264 0. In particular, the cardinality of the set of excessive ends of Ft is at leastthe cardinality of the set of excessive ends of F0 a.s. for every t \u2264 0. Since, by Proposition 5.4.2,\u3008Ft\u3009t\u2208R is stationary and ergodic, we deduce that the cardinality of the set of excessive ends of F0is a.s. equal to some constant.Proof of Corollary 5.2.3. Let G\u2032\u2032 be the network that has all the edges of both G and G\u2032. Bysymmetry, it suffices to show that the wired uniform spanning forests of G and G\u2032\u2032 have the samenumber of excessive ends a.s. Let F and F\u2032\u2032 be samples of the WUSFs of G and G\u2032\u2032 respectively,and let A be the set of edges of G\u2032\u2032 that are not edges of G. Since A is finite and G is connected,the event A = {A \u2229 F\u2032\u2032 = \u2205} has positive probability (this implication is easily proven in several1025.6. Ends and rough isometriesways, e.g. using either Wilson\u2019s algorithm, the Aldous-Broder algorithm, or the Transfer CurrentTheorem [62]). The spatial Markov property of the WUSF implies that the conditional distributionof F\u2032\u2032 given A is equal to the distribution of F, and in particular the conditional distribution of thenumber of excessive ends of F\u2032\u2032 given A has the same distribution as the number of excessive endsof F. The claim now follows from Theorem 5.2.2.5.6 Ends and rough isometriesRecall that a rough isometry from a graph G = (V,E) to a graph G\u2032 = (V \u2032, E\u2032) is a function\u03c6 : V \u2192 V \u2032 such that, letting dG and dG\u2032 denote the graph distances on V and V \u2032, there existpositive constants \u03b1 and \u03b2 such that the following conditions are satisfied:1. (\u03c6 roughly preserves distances.) For every pair of vertices u, v \u2208 V ,\u03b1\u22121dG(u, v)\u2212 \u03b2 \u2264 dG\u2032(\u03c6(u), \u03c6(v)) \u2264 \u03b1dG(u, v) + \u03b2.2. (\u03c6 is almost surjective.) For every vertex v\u2032 \u2208 V \u2032, there exists a vertex v \u2208 V such thatdG(\u03c6(v), v\u2032) \u2264 \u03b2.For background on rough isometries, see [173, \u00a72.6]. The final result of this paper answers negativelyQuestion 7.6 of Lyons, Morris and Schramm [170], which asked whether the property of having one-ended WUSF components is preserved under rough isometry of graphs.Theorem 5.6.1. There exist two rough-isometric, bounded degree graphs G and G\u2032 such that everycomponent of the wired uniform spanning forest of G has one-end a.s., but the wired uniformspanning forest of G\u2032 contains a component with uncountably many ends a.s.The proof of Theorem 5.6.1 uses Wilson\u2019s algorithm rooted at infinity. We refer the reader to[173, Proposition 10.1] for an exposition of this algorithm. The description as a branching processof the past of the WUSF of a regular trees with height-dependent exponential edge stretching isadapted from [44, \u00a711], and first appeared in the work of Ha\u00a8ggstro\u00a8m [111].Proof of Theorem 5.6.1. Let T = (V,E) be a 3-regular tree with root \u03c1. We write \u2016u\u2016 for thedistance between u \u2208 V and \u03c1. For each positive integer k, let Tk = (Vk, Ek) denote the treeobtained from T by replacing every edge connecting a vertex u of T to its parent by a path oflength k\u2016u\u2016. We identify the degree 3 vertices of Tk with the vertices of T . For each vertex u \u2208 V ,let S(u) be a binary tree with root \u03c1u and let Sk(u) be the tree obtained from S(u) by replacingevery edge with a path of length k\u2016u\u2016+1. Finally, for each pair of positive integers (k,m), let Gmk bethe graph obtained from Tk by, for each vertex u \u2208 V , adding a path of length k\u2016u\u2016+1 connectingu to \u03c1u and then replacing every edge in each of these added paths and every edge in each of thetrees Sk(u) by m parallel edges. The vertex degrees of Gmk are bounded by 3 +m, and the identity1035.6. Ends and rough isometriesFigure 5.2: An illustration of the graph G32. Red vertices correspond to vertices of the 3-regulartree T . Only three generations of each of the trees S(v) are pictured.map is an isometry (and hence a rough isometry) between Gmk and Gm\u2032k whenever k,m and m\u2032 arepositive integers. See Figure 1 for an illustration.Let k and m be positive integers. Observe that for every vertex v of T and every child u of vin T , the probability that simple random walk on Gmk started at u ever hits v does not depend onthe choice of v or u. Denote this probability p(m, k). We can bound p(m, k) as follows.kk + 2 +m\u2264 p(m, k) \u2264 k + 2k + 2 + 2m. (5.6.1)The lower bound of k\/(k+2+m) is exactly the probability that the random walk started at u visitsv before visiting any other vertex of T or visiting \u03c1u. The upper bound of (k + 2)\/(k + 2 + 2m) isexactly the probability that the random walk started at u ever visits a neighbour of u in T . This canbe computed by a straightforward network reduction (see [173] for background): The conductanceto infinity from the root of a binary tree is 1, so that, by the series and parallel laws, the effectiveconductance to infinity from u in the subgraph of Gmk spanned by the vertices of Sk(u) and thepath connecting u to Sk(u) is 2mk\u2212\u2016u\u2016\u22121. On the other hand, the effective conductance betweenu and its parent v is k\u2212\u2016u\u2016, while the effective conductance between u and each of its children isk\u2212\u2016u\u2016\u22121. It follows that the probability that a random walk started at u ever visits a neighbour ofu in T is exactlyk\u2212\u2016u\u2016 + 2k\u2212\u2016u\u2016\u22121k\u2212\u2016u\u2016 + 2k\u2212\u2016u\u2016\u22121 + 2mk\u2212\u2016u\u2016\u22121=k + 2k + 2 + 2mas claimed.Let Fmk be a sample of WUSFGmk generated using Wilson\u2019s algorithm on Gmk , starting with theroot \u03c1 of T . Let \u03be be the loop-erased random walk in Gmk beginning at \u03c1 that is used to start our1045.7. Closing discussion and open problemsforest. The path \u03be includes either one or none of the neighbours of \u03c1 in T and so, in either case,there are at least two neighbours v1 and v2 of \u03c1 in T that are not contained in this path. Continuingto run Wilson\u2019s algorithm from v1 and v2, we see that, conditional on \u03be, the events A1 = {v1 isin the past of \u03c1 in Fmk } and A2 = {v2 is in the past of \u03c1 in Fmk } are independent and each haveprobability p(m, k). Furthermore, on the event Ai, we add only the path connecting vi and \u03c1 inGmk to the forest during the corresponding step of Wilson\u2019s algorithm. Recursively, we see that therestriction to T of the past of \u03c1 in F contains a Galton-Watson branching process with Binomialoffspring distribution (2, p(m, k)). If k \u2265 m+ 3 this branching process is supercritical, so that Fmkcontains a component with uncountably many ends with positive probability. By tail triviality ofthe WUSF [173, Theorem 10.18], Fmk contains a component with uncountably many ends a.s. whenk \u2265 m+ 3.On the other hand, a similar analysis shows that the restriction to T of past of \u03c1 in Fmk isstochastically dominated by a binomial (3, p(m, k)) branching process. (The 3 here is to accountfor the possibility that every child of \u03c1 in T is in its past). If m \u2265 k + 2, this branching processis either critical or subcritical, and we conclude that the restriction to T of the past of \u03c1 in Fmk isfinite a.s. Condition on this restriction. Similarly again to the above, the restriction to Sk(v) ofthe past of v in Fkm is stochastically dominated by a critical binomial (2, 1\/2) branching process foreach vertex v of T , and is therefore finite a.s. We conclude that the past of \u03c1 in Fmk is finite a.s.whenever m \u2265 k + 2. A similar analysis shows that the past in Fmk of every vertex of Gmk is finitea.s., and consequently that every component of Fmk is one-ended a.s. whenever m \u2265 k + 2.Since 4 \u2265 1+3 and 6 \u2265 4+2, the wired uniform spanning forest F14 of G14 contains an infinitely-ended component a.s., and every component of the wired uniform spanning forest F64 of G64 isone-ended a.s.5.7 Closing discussion and open problems5.7.1 The FMSF of the interlacement orderingOne way to think about the Interlacement Aldous-Broder algorithm is as follows. Given the in-terlacement process I on a transient network G, we can define a total ordering of the edges ofG according to the order in which they are traversed by the trajectories of I[0,\u221e). That is, wedefine a strict total ordering \u227a of E by setting e1 \u227a e2 if and only if either e1 is first traversed bya trajectory of I[0,\u221e) at a smaller time than e2 is first traversed by a trajectory of I[0,\u221e), or if e1and e2 are both traversed for the first time by the same trajectory of I[0,\u221e), and this trajectorytraverses e1 before it traverses e2. We call \u227a the interlacement ordering of the edge set E.It is easily verified that AB0(I ) is the wired minimal spanning forest of G with respect tothe interlacement ordering. That is, an edge e \u2208 E is included in AB0(I ) if and only if there doesnot exist either a finite cycle or a bi-infinite path in G containing e for which e is the \u227a-maximalelement. See [173] for background on minimal spanning forests. In light of this, it is natural towonder what might be said about the free minimal spanning forest of the interlacement ordering,1055.7. Closing discussion and open problemsthat is, the spanning forest of G that includes an edge e \u2208 E if and only if there does not exist afinite cycle in G containing e for which e is the \u227a-maximal element. Indeed, if this forest were theFUSF of G, this could be used to solve the monotone coupling problem [173, Question 10.6] (seealso [57, 177, 181]) and the almost-connectivity problem [173, Question 10.12].Unfortunately there is little reason for this to be the case other than wishful thinking. Indeed,let I be the interlacement process on a transient network G, and definetc = inf{t \u2208 (0,\u221e) : I[0,t] is connected a.s.}.Teixeira and Tykesson [219] proved that if G is transitive, then tc is positive if and only if G isnonamenable. (The amenable case of their result generalises the corresponding result for Zd, dueto Sznitman [216].) We can apply this result to prove that the free minimal spanning forest of theinterlacement ordering is distinct from the WUSF on any nonamenable transitive graph: This issimilar to how the usual FMSF and WMSF (where the edge weights are i.i.d.) are distinct if andonly there is a nonempty nonuniqueness phase for Bernoulli bond percolation [175]. Since there aremany nonamenable transitive graphs where the WUSF and FUSF coincide (e.g. the product of a3-regular tree with Z, see [173, Chapter 10]), we deduce that there are transitive graphs (indeed,Cayley graphs) for which the FUSF does not coincide with the free minimal spanning forest of theinterlacement ordering.We now give a quick sketch of this argument. Suppose that G is a transitive nonamenablegraph. Observe that for every t < tc, there must exist a connected component of I[0,t) and a vertexu of G such that a random walk started at u has a positive probability not to hit the component:If not, we would have that I[0,s] was connected for every s > t, contradicting the assumption thatt < tc. Moreover, by finding a path from u to the component and considering the last vertex of thepath before we reach the component, the vertex u can be taken to be adjacent to the component.Let \u03c4 be the first time after tc\/2 that u is hit by a trajectory of I , and let etc\/2(u) be the orientededge that is traversed by this trajectory as it enters u for the first time. Denote this trajectory byW . No other trajectories of I appear at time \u03c4 a.s. In light of the above discussion, by makinglocal modifications to finitely many trajectories in I[0,\u03c4), we see that the following event occurswith positive probability: \u03c4 is strictly less than tc, the vertices u and etc\/2(u)\u2212 are both in differentcomponents of I[0,\u03c4) (and, in particular, are both in I[0,\u03c4)), and W hits the component of u in I[0,\u03c4)for the first time at u. On this event we must have that etc\/2(u) is included in the free minimalspanning forest of the interlacement ordering, but is not in AB0(I ), and hence the two forests donot coincide.5.7.2 Exceptional timesA natural question raised by the Interlacement Aldous-Broder algorithm concerns the existenceor non-existence of exceptional times for the process \u3008Ft\u3009t\u2208R = \u3008ABt(I )\u3009t\u2208R, that is, times atwhich Ft has properties markedly different from the a.s. properties of F0. For example, we might1065.7. Closing discussion and open problemsask whether, considering the process \u3008Ft\u3009t\u2208R on Zd (d \u2265 3), there are exceptional times when theforest has multiply ended components, is disconnected (if d = 3, 4), or is connected (if d \u2265 5).(Note that the proof of Theorem 5.2.2 implies that there do not exist exceptional times at whichFt contains indestructible excessive ends.)The answers to the first of these questions turn out to rather simple. Given a trajectory W in agraph G and a vertex u of G visited by the path, we define e(W,u) to be the oriented edge pointinginto u that is traversed by W as it enters u for the first time, and defineAB(W ) = {\u2212e(W,u) : u is visited by W}.Note that if the trace of W is infinite then AB(W ) is an infinite oriented tree. We define the treeof first entry edges AB(X) similarly when X = \u3008Xn\u3009n\u22650 is a path in G.Proposition 5.7.1 (Exceptional times for excessive ends). Let G be a transient network, let Ibe the interlacement process, and let \u3008Ft\u3009t\u2208R = \u3008ABt(I )\u3009t\u2208R. Let E be the set of times t \u2208 R suchthat Ft has a multiply ended component, and let E \u2032 be the set of times t \u2208 R for which there existsa trajectory Wt in It such that AB(W ) is multiply ended. If every component of F0 is one-endedalmost surely, then the following hold almost surely.1. E = E \u2032, and E = \u2205 if and only if F0 is connected almost surely.2. For every t \u2208 E , there is exactly one two-ended component of Ft, and all other componentsare one-ended. The unique two-ended component is the union of the tree AB(Wt) with somefinite bushes.Since Proposition 5.7.1 is tangential to the paper, we leave out some details from the proof.Proof. We first prove that E = E \u2032 almost surely. The containment E \u2032 \u2286 E is immediate, and holdsdeterministically. Let \u2126 be the almost sure event that every component of Ft is one-ended for everyrational t, and that no two trajectories of I have the same arrival time. We claim that E = E \u2032pointwise on the event \u2126. Suppose that \u2126 holds and that t \u2208 E , so that there exists a sequence ofvertices \u3008vi\u3009i\u22650 such that vi = et(vi+1)\u2212 for each i \u2265 0. In particular, the arrival times \u03c4t(vi) areincreasing. We claim that we must have \u03c4t(vi) = t for all i \u2265 0. Indeed, if \u03c4t(vi) \u2265 t + \u03b5 for some\u03b5 > 0 and all i larger than some i0, then we would have that et+\u03b4(vi) = et(vi) for all 0 < \u03b4 \u2264 \u03b5and all i \u2265 i0. In this situation, we would therefore have that Ft+\u03b4 contained a multiply endedcomponent for every 0 < \u03b4 \u2264 \u03b5, contradicting the assumption that the event \u2126 occured. Thus,we must have that there exists a trajectory Wt \u2208 It (which is unique by definition of \u2126), andthe sequence vi gives an excessive end in the tree AB(W ). Since the sequence vi represented anarbitrary excessive end of Ft, it follows that every excessive end of Ft arises from the tree AB(W )on the event \u2126.Now, if F0 is not connected a.s., then there is a vertex v of G such that two independent randomwalks from v do not intersect with positive probability, and it follows that there a.s. exist trajectories1075.7. Closing discussion and open problemsin I such that AB(W ) has at least two ends. It remains to prove that the trees AB(W ) have atmost two-ends for every trajectory W in I , and are all one-ended if F0 is a.s. connected. Sincethere are only countably many trajectories in I , it suffices to analyze a single bi-infinite randomwalk. To prove this, it is convenient to introduce a variant of the interlacement Aldous-Broderin which we first run a simple random walk started from a fixed vertex (considered to arrive attime zero), and then run the interlacement process I[0,\u221e), and form a forest from the first entryedges. It is not difficult to see, by a slight modification of the proof Theorem 5.1.1, that the forestproduced this was is the wired uniform spanning forest: In the finite exhaustion, this correspondsto first running a random walk from v until hitting the distinguished boundary vertex, and thendecomposing the rest of the walk into excursions from the boundary vertex. Using this algorithm,it follows that AB(X) is one-ended a.s. whenever X is a random walk on a transient graph G forwhich the wired uniform spanning forest is one-ended.Now suppose that W = \u3008Wn\u3009n\u2208Z is a bi-infinite random walk. If \u3008vi\u3009i\u22650 is a sequence of verticesin G corresponding to an excessive end of AB(W ), then we must have that vi = e(vi+1)\u2212 for alli sufficiently large, and it follows that this excessive end must be an end of the tree AB(\u3008Wi\u3009i\u22650),completing the proof that AB(W ) has at most two ends. On the other hand, we note that theunique path to infinity from W0 in AB(\u3008Wi\u3009i\u22650) is exactly the loop-erasure of \u3008Wi\u3009i\u22650, and if F0is connected a.s. then this path is hit infinitely often a.s. by \u3008Wi\u3009i<0. We deduce that in this casethis end is not present in the tree AB(W ), completing the proof.We do not know if there exist exceptional times for (dis)connectivity. We expect that suchtimes do not exist, but it would be very interesting if they do. kirchhoff1847ueberQuestion 5.7.2. Let d \u2265 3, let I be the interlacement process on Zd, and let \u3008Ft\u3009t\u2208R = \u3008ABt(I )\u3009t\u2208R.If d = 3, 4, do there exist times at which Ft is disconnected? If d \u2265 5, do there exist times at whichFt is connected?If the answer to Question 5.7.2 is positive, it would be interesting to further understand thestructure of the set of exceptional times and the geometry of the forest Ft at a typical exceptionaltime. It is easy to see that, unlike for excessive ends, the arrival times of trajectories are notexceptional times for connectivity, so if exceptional times do exist they are likely to have a moreinteresting structure. We note that there is a rich theory of exceptional times for other models suchas dynamical percolation, addressing many analogous questions. See e.g. [99, 115, 125, 213].A related question concerns the decorrelation of connectivity events under the dynamics.Question 5.7.3. Let d \u2265 5, let I be the interlacement process on Zd, and let \u3008Ft\u3009t\u2208R = \u3008ABt(I )\u3009t\u2208R.How doesP(x is connected to y in both F0 and Ft)behave as a function of the vertices x, y \u2208 Zd and the number t > 0? Does the behaviour as afunction of x, y undergo a phase transition as t is increased?1085.7. Closing discussion and open problemsRecall that for the USF of Zd, d \u2265 5, the probability that two vertices x and y are in the samecomponent of the USF decays like \u2016x \u2212 y\u2016\u2212(d\u22124) as \u2016x \u2212 y\u2016 \u2192 \u221e [41]. A successful approach toQuestions 5.7.2 and 5.7.3 might need to draw more deeply on the interlacement literature than wehave needed to in this paper.5.7.3 Excessive ends via update toleranceA key tool in the study of the USFs carried out in [127, 131, 223] is the update-tolerance ofthe USFs (referred to as weak insertion tolerance by Tima\u00b4r [223]). Given a sample F of eitherthe WUSF and the FUSF of a network G and an oriented edge e of G not in F, update-tolerancestates that there exists a forest U(F, e), obtained from F by adding e and deleting some otherappropriately chosen edge d, such that the law of U(F, e) is absolutely continuous with respect tothat of F. The forest U(F, e) is called the update of F at e. See [127, 131, 223] for further details.Since the number of excessive ends does not change when we perform an update, a positivesolution of the following conjecture would yield an alternative proof of Theorem 5.2.2. We say thata Borel set A \u2286 {0, 1}E is update-stable if for every oriented edge e of G, the updated forestU(F, e) is in A if and only if F is in A almost surely. The conjecture would also imply a positivesolution to [44, Question 15.7].Conjecture 5.7.4. Let G be an infinite network, and let F be either the wired or free spanningforest of G. Then for every update-stable Borel set A \u2286 {0, 1}E, the probability that F is in A iseither zero or one.5.7.4 Ends in uniformly transient networksThe following natural question remains open. If true, it would strengthen the results of Lyons,Morris and Schramm [170]. A network is said to be uniformly transient if the capacities of thevertices of the network are bounded below by a positive constant.Question 5.7.5. Let G be a uniformly transient network with infe c(e) > 0. Does it follow thatevery component of the wired uniform spanning forest of G is one-ended almost surely?The argument used in the proof of Theorem 5.2.2 can be adapted to show that, under thehypotheses of Question 5.7.5, every component of the WUSF is either one-ended or has uncountablymany ends, with no isolated excessive ends. To answer Question 5.7.5 positively, it remains to rulethis second case out.109Chapter 6Indistinguishability of trees inuniform spanning forestsSummary. We prove that in both the free and the wired uniform spanning forest (FUSF and WUSF)of any unimodular random rooted network (in particular, of any Cayley graph), it is impossible todistinguish the connected components of the forest from each other by invariantly defined graphproperties almost surely. This confirms a conjecture of Benjamini, Lyons, Peres and Schramm [44].We also answer positively two additional questions of [44] under the assumption of unimodu-larity. We prove that on any unimodular random rooted network, the FUSF is either connected orhas infinitely many connected components almost surely, and, if the FUSF and WUSF are distinct,then every component of the FUSF is transient and infinitely-ended almost surely. All of theseresults are new even for Cayley graphs.6.1 IntroductionThe Free Uniform Spanning Forest (FUSF) and the Wired Uniform Spanning Forest(WUSF) of an infinite graphG are defined as weak limits of the uniform spanning trees on large finitesubgraphs of G, taken with either free or wired boundary conditions respectively (see Section 6.1.2for details). First studied by Pemantle [190], the USFs are closely related many other areas ofprobability, including electrical networks [62, 154], Lawler\u2019s loop-erased random walk [44, 161, 228],sampling algorithms [197, 228], domino tiling [150], the Abelian sandpile model [137, 138, 178], therotor-router model [126], and the Fortuin-Kasteleyn random cluster model [105, 109]. The USFsare also of interest in group theory, where the FUSFs of Cayley graphs are related to the `2-Bettinumbers [94, 169] and to the fixed price problem of Gaboriau [95], and have also been used toapproach the Dixmier problem [87].Although both USFs are defined as limits of trees, they need not be connected. Indeed, aprincipal result of Pemantle [190] is that the FUSF and WUSF coincide on Zd for all d \u2265 1 andthat they are connected almost surely (a.s.) if and only if d \u2264 4. A complete characterisation ofthe connectivity of the WUSF was given by Benjamini, Lyons, Peres and Schramm (henceforthreferred to as BLPS) in their seminal work [44], who showed that the WUSF of a graph G isconnected a.s. if and only if the traces of two simple random walks started at arbitrary vertices ofG a.s. intersect. This recovers Pemantle\u2019s result on Zd, and shows more generally that the WUSFof a Cayley graph is connected a.s. if and only if the corresponding group has polynomial growth1106.1. Introductionof degree at most 4 [124, 173].Besides connectivity, several other basic features of the WUSF are also understood ratherfirmly. This understanding mostly stems from Wilson\u2019s algorithm rooted at infinity, which allowsthe WUSF to be sampled by joining together loop-erased random walks [173, 228]. For example,other than connectivity, the simplest property of a forest is the number of ends its componentshave. Here, an infinite graph G is said to be k-ended if, over all finite sets of vertices W , thesubgraph induced by V \\W has a maximum of k infinite connected components. In particular,an infinite tree is one-ended if and only if it does not contain a simple bi-infinite path. Followingearlier work by Pemantle [190], BLPS [44] proved that the number of components of the WUSFof any graph is non-random, that the WUSF of any unimodular transitive graph (e.g., any Cayleygraph) is either connected or has infinitely many components a.s., and that in both cases everycomponent of the WUSF is one-ended a.s. unless the underlying graph is itself two-ended. Morris[185] later proved that every component of the WUSF is recurrent a.s. on any graph, confirming aconjecture of BLPS [44, Conjecture 15.1], and several other classes of graphs have also been shownto have one-ended WUSF components [7, 127, 170].Much less is known about the FUSF. No characterisation of its connectivity is known, nor is itknown whether the number of components of the FUSF is non-random on an any graph. In [44] itis proved that if the FUSF and WUSF differ on a unimodular transitive graph, then a.s. the FUSFhas a transient tree with infinitely many ends, in contrast to the WUSF. However, it remainedan open problem [44, Question 15.8] to prove that, under the same hypotheses, every connectedcomponent of the FUSF is transient and infinitely ended a.s. In light of this, it is natural to askthe following more general question:Question. Let G be a unimodular transitive graph. Can the components of the free uniform span-ning forest of G be very different from each other?Questions of this form were first studied by Lyons and Schramm [176] in the context insertion-tolerant automorphism-invariant random subgraphs. Their remarkable theorem asserts that in anysuch random subgraph (e.g. Bernoulli bond percolation or the Fortuin-Kasteleyn random clustermodel) on a unimodular transitive graph, one cannot distinguish between the infinite connectedcomponents using automorphism-invariant graph properties. For example, all such componentsmust have the same volume growth, spectral dimension, value of pc and so forth (see Section 6.1.3for further examples). They also exhibited applications of indistinguishability to statements not ofthis form, including uniqueness monotonicity and connectivity decay. Here, a random subgraph \u03c9of a graph G is insertion-tolerance if for every edge e of G, the law of the subgraph \u03c9 \u222a {e} formedby inserting e into \u03c9 is absolutely continuous with respect to the law of \u03c9. The uniform spanningforests are clearly not insertion-tolerant, since the addition of an edge may close a cycle.BLPS conjectured [44, Conjecture 15.9] that the components of both the WUSF and FUSF alsoexhibit this form of indistinguishability. In this paper we confirm this conjecture.Theorem 6.1.1 (Indistinguishability of USF components). Let G be a unimodular transitive graph,and let F be a sample of either the free uniform spanning forest or the wired uniform spanning forest1116.1. Introductionof G. Then for each automorphism-invariant Borel-measurable set A of subgraphs of G, either everyconnected component of F is in A or every connected component of F is not in A almost surely.As indicated by the above discussion, Theorem 6.1.1 implies the following positive answer to[44, Question 15.8] under the assumption of unimodularity.Theorem 6.1.2 (Transient trees in the FUSF). Let G be a unimodular transitive graph and let Fbe a sample of FUSFG. If the measures FUSFG and WUSFG are distinct, then every component ofF is transient and has infinitely many ends almost surely.However, rather than deducing Theorem 6.1.2 from Theorem 6.1.1, we instead prove Theo-rem 6.1.2 directly and apply it in the proof of Theorem 6.1.1.We also apply Theorem 6.1.1 to answer another of the most basic open problems about theFUSF [44, Question 15.6] under the assumption of unimodularity.Theorem 6.1.3 (Number of trees in the FUSF). Let G be a unimodular transitive graph and let Fbe a sample of the free uniform spanning forest of G. Then F is either connected or has infinitelymany components almost surely.The derivation of Theorem 6.1.3 from Theorem 6.1.1 is inspired by the proof of [176, Theorem4.1], and also establishes the following result.Theorem 6.1.4 (Connectivity decay in the FUSF). Let G be a unimodular transitive graph andlet F be a sample of the free uniform spanning forest of G. If F is disconnected a.s., then for everyvertex v of G,inf{FUSFG(u \u2208 TF(v)) : u \u2208 V (G)}= 0 ,where u \u2208 TF(v) is the event that u belongs to the component of v in F.We prove all of our results in the much more general setting of unimodular random rootednetworks, which includes all Cayley graphs as well as a wide range of popular infinite randomgraphs and networks [7]. For example, our results hold when the underlying graph is an infinitesupercritical percolation cluster in a Cayley graph, a hyperbolic unimodular random triangulation[37, 75] (for which the FUSF and WUSF are shown to be distinct in the upcoming work [20]), asupercritical Galton-Watson tree, or even a component of the FUSF of another unimodular randomrooted network. See Section 6.1.3 for the strongest and most general statements.Organization. In Section 6.1.1 we describe our approach and the novel ingredients of our proof.The necessary background, including definitions of USFs and unimodular random rooted networksare presented in Section 6.1.2. In Section 6.1.3 we define the graph properties we will work with,state the most general and strongest versions of our theorems (most importantly, Theorem 6.1.9),and provide several illustrative examples. In Section 6.2 we develop the update-tolerance propertyof the FUSF, and prove, in the setting of Theorem 6.1.9, that if the FUSF and WUSF are distinctthen every component of the FUSF is transient and infinitely-ended (Theorem 6.1.12), and then1126.1. Introductionprove indistinguishability of the components in this case. In Section 6.3, still in the case wherethe FUSF and WUSF are distinct, we prove that the FUSF is either connected or has infinitelymany connected components (Theorem 6.1.10) and in the latter case we show that connectivitydecay is exhibited (Theorem 6.1.11). In Section 6.4 we show that the WUSF components areindistinguishable, completing the proof of Theorem 6.1.9.Remark. After this paper was posted on the arXiv, Adam Tima\u00b4r posted independent work [223]in which he proves Theorem 6.1.1 for the FUSF only in the case that FUSF 6= WUSF, and alsoproves Theorems 6.1.2 and 6.1.3; Indistinguishability of components in the WUSF is not treated.In the present paper, we prove [44, Conjecture 15.1] in its entirety for both the FUSF and WUSF.6.1.1 About the proofIn [176], Lyons and Schramm argue that the coexistence of clusters of different types in an invariantedge percolation implies the existence of infinitely many pivotal edges, that is, closed edges thatchange the type of an infinite cluster if they are inserted. When the percolation is insertion-tolerant,this heuristically contradicts the Borel-measurability of the property, as the existence of pivotaledges far away from the origin should imply that we cannot approximate the event that the clusterhas the property by a cylinder event. This argument was made precise in [176]. Unimodularityof the underlying graph was used heavily \u2013 indeed, indistinguishability can fail without it [176,Remark 3.16].A crucial ingredient of our proof is an update-tolerance property of USFs. This property wasintroduced for the WUSF by the first author [127] and is developed for the FUSF in Section 6.2.1.This property allows us to make a local modification to a sample of the FUSF or WUSF in such away that the law of the resulting modified forest is absolutely continuous with respect to the lawof the forest that we started with. In this local modification, we add an edge of our choice to theUSF and, in exchange, are required to remove an edge emanating from the same vertex. The edgethat we are required to remove is random and depends upon both the edge we wish to insert andon the entire sample of the USF.Update-tolerance replaces insertion-tolerance and allows us to perform a variant of the keyargument in [176]. However, several obstacles arise as we are required to erase an edge at the sametime as inserting one. In particular, we cannot simply open a closed edge connecting two clustersof different types in order to form a single cluster. These obstacles are particularly severe for theWUSF (and the FUSF in the case that the two coincide), where it is no longer the case that thecoexistance of components of different types implies the existence of pivotal edges. To proceed,we separate the component properties into two types, tail and non-tail, according to whether theproperty is sensitive to finite modifications of the component. Indistinguishability is then provenby a different argument in each case: non-tail properties are handled by a variant of the Lyons-Schramm method, while tail properties are handled by a completely separate argument utilisingWilson\u2019s algorithm [228] and the spatial Markov property. The proof that components of the WUSFcannot be distinguished by tail properties also applies to transitive graphs without the assumption1136.1. Introductionof unimodularity.6.1.2 Background and definitionsNotationA tree is a connected graph with no cycles. A spanning tree of a graph G = (V,E) is a connectedsubgraph of G that contains every vertex and no cycles. A forest is a graph with no cycles, anda spanning forest of a graph G = (V,E) is a subgraph of G that contains every vertex and nocycles. Given a forest F and a vertex v we write TF(v) for the connected component of F containingv. An essential spanning forest is a spanning forest such that every component is infinite. Abranch of an infinite tree T is an infinite component of T \\ v for some vertex v. The core of aninfinite tree T , denoted core(T ), is the set of vertices of T such that T \\ v at least two infiniteconnected components.Recall that an infinite graph G is said to be k-ended if removing a finite set of vertices W fromG results in a maximum of k distinct infinite connected components. In particular, an infinite treeis one-ended if and only if it does not contain any simple bi-infinite paths. We say that a forest Fis one-ended if all of its components are one-ended. The past of a vertex v in a one-ended forest,denoted pastF(v), is the union of v and the finite components of F \\ v. The future of the vertex vis the set of u such that v \u2208 pastF(u).We write BG(v, r) for the graph-distance ball of radius r about a vertex v in a graph G.Uniform Spanning Trees and ForestsWe now briefly provide the necessary definitions, notation and background concerning USFs. Werefer the reader to [173, \u00a74 and \u00a710] for a comprehensive review of this theory. Given a graphG = (V,E) we will refer to an edge e \u2208 E both as an oriented and unoriented edge and it willalways be clear which one from the context. Most frequently we will deal with oriented edges andin this case we orient them from their tail e\u2212 to their head e+.A network (G, c) is a locally finite, connected multi-graph G = (V,E) together with a functionc : E \u2192 (0,\u221e) assigning a positive conductance to each edge of G. Graphs are considered to benetworks by setting c \u2261 1. The distinction between graphs and networks does not play much of arole for us, and we will mostly suppress the notation of conductances, writing G to mean either agraph or a network. Write c(u) for the sum of the conductances of the edges e\u2212 = u emanatingfrom u and c(u, v) for the conductance of the sum of the conductances of the (possibly many) edgeswith endpoints u and v. The random walk \u3008Xn\u3009n\u22650 on a network G is the Markov chain on Vwith transition probabilities p(u, v) = c(u, v)\/c(u).The uniform spanning tree measure USTG of a finite connected graph G is the uniformmeasure on spanning trees ofG (considered for measure-theoretic purposes as functions E \u2192 {0, 1}).When G is a network, USTG is the probability measure on spanning trees of G such that theprobability of a tree is proportional to the product of the conductances of its edges.1146.1. IntroductionLet G be an infinite network. An exhaustion \u3008Vn\u3009n\u22650 of G is an increasing sequence of finitesets of vertices Vn \u2282 V such that\u22c3n\u22650 Vn = V . Given such an exhaustion, we define Gn to be thesubgraph of G induced by Vn together with the conductances inherited from G, and define G\u2217n to bethe network obtained from G by identifying (or \u201cwiring\u201d) V \\ Vn into a single vertex and deletingall the self-loops that are created. The weak limits of the measures USTGn and USTG\u2217n exist forany network and do not depend on the choice of exhaustion [109, 190]. The limit of the USTGn iscalled the free uniform spanning forest measure FUSFG while the limit of the USTG\u2217n is calledthe wired uniform spanning forest measure WUSFG. Both limits are clearly concentrated onthe set of essential spanning forests of G.The measures FUSFG and WUSFG coincide if and only if G does not support any non-constantharmonic functions of finite Dirichlet energy [44], and in particular the two measures coincide whenG = Zd. The two measures also coincide on every amenable transitive graph [44, Corollary 10.9],and an analogous statement holds for unimodular random rooted networks once an appropriatenotion of amenability is adopted [7, \u00a78]. When G is a Cayley graph, the two measures FUSFGand WUSFG coincide if and only if the first `2-Betti number of the corresponding group is zero[169]. By taking various free or direct products of groups and estimating their Betti numbers, thischaracterization allows to construct an abundance of Cayley graphs in which the two measureseither coincide or differ [173, \u00a710.2].A very useful property of the UST and the USFs is the spatial Markov property. Let G be anetwork and let H and F be finite subsets of G. We write G\u02c6 = (G\u2212H)\/F for the network formedfrom G by deleting each edge h \u2208 H and contracting (i.e., identifying the two endpoints of) eachedge f \u2208 F . If G is finite and T is a sample of USTG, then the law of T conditioned on the event{F \u2286 T,H \u2229 T = \u2205} (assuming this event has positive probability) is equal to the law of the unionof F with an independent copy of USTG\u02c6, considered as a subgraph of G [173, \u00a74]. Now suppose thatG is an infinite network with exhaustion \u3008Vn\u3009n\u22650 and let F be a sample of either FUSFG or WUSFG.Applying the Markov property to the finite networks Gn and G\u2217n and taking the limit as n \u2192 \u221e,we see similarly that the conditional distribution of F conditioned on the event {F \u2286 F, H \u2229F = \u2205}is equal to the law of the union of F with an independent copy of FUSFG\u02c6 or WUSFG\u02c6 as appropriate.It is important here that H and F are finite.Lastly, throughout Section 6.4 we will use a recent result of the first author regarding ends ofthe WUSF\u2019s components. Components of the WUSF are known to be one-ended a.s. in severallarge classes of graphs and networks. The following is proven by the first author in [127], andfollows earlier works [7, 44, 170, 190].Theorem 6.1.5 ([127]). Let (G, \u03c1) be transient unimodular random rooted network with E[c(\u03c1)] <\u221e. Then every component of the wired uniform spanning forest of G is one-ended almost surely.1156.1. IntroductionUnimodular random networksWe present here the necessary definition of unimodular random networks and refer the reader to thecomprehensive monograph of Aldous and Lyons [7] for more details and many examples. A rootedgraph (G, \u03c1) is a locally finite, connected graph G together with a distinguished vertex \u03c1, the root.An isomorphism of graphs is an isomorphism of rooted graphs if it preserves the root. The ball ofradius r around a vertex v of G, denoted BG(v, r) is the graph induced on the set of vertices whichare at graph distance at most r from v. The local topology on the set of isomorphism classes ofrooted graphs is defined so that two (isomorphism classes of) rooted graphs (G, \u03c1) and (G\u2032, \u03c1\u2032) areclose to each other if and only if the rooted balls (BG(\u03c1, r), \u03c1) and (BG\u2032(\u03c1\u2032, r), \u03c1\u2032) are isomorphic toeach other for large r. We denote the space of isomorphism classes of rooted graphs endowed withthe local topology by G\u2022. We define an edge-marked graph to be a locally finite connected graphtogether with a function m : E(G) \u2192 X for some separable metric space X, the mark space (inthis paper, X will be a product of intervals and some copies of {0, 1}). For example, if G = (G, c)is a network and F is a sample of FUSFG, then (G, c,F) is a graph with marks in (0,\u221e) \u00d7 {0, 1}.The local topology on rooted marked graphs is defined so that two marked rooted graphs are closeif for large r there is an isomorphism (of rooted graphs) \u03c6 : (BG(\u03c1, r),m, \u03c1)\u2192 (BG\u2032(\u03c1\u2032, r), \u03c1\u2032) suchthat dX(m\u2032(\u03c6(e)),m(e)) is small for every edge e in BG(\u03c1, r). We denote the space of edge-markedgraphs with marks in X by GX\u2022 .Similarly, we define a doubly-rooted graph (G, u, v) to be a graph together with an orderedpair of distinguished vertices. The space G\u2022\u2022 of doubly-rooted graphs is defined similarly to G\u2022. Arandom rooted graph (G, \u03c1) is unimodular if it obeys the mass-transport principle. That is,for every non-negative Borel function f : G\u2022\u2022 \u2192 [0,\u221e] \u2013 which we call a mass transport \u2013 wehave thatE\u2211v\u2208Vf(G, \u03c1, v) = E\u2211u\u2208Vf(G, u, \u03c1).In other words, (G, \u03c1) is unimodular if for every mass transport f , the expected mass received bythe root equals the expected mass sent by the root. Every Cayley graph (rooted at any vertex)is a unimodular random rooted graph (whose law is concentrated on a singleton), as is everyunimodular transitive graph [173, \u00a78]. For many examples of a more genuinely random nature, see[7]. Unimodular random rooted networks and other edge-marked graphs are defined similarly.When (G, \u03c1) is a unimodular random rooted network and F is a sample of either FUSFG orWUSFG, (G, \u03c1,F) is also unimodular: Since the definitions of FUSFG and WUSFG do not dependon the choice of exhaustion, for each mass transport f : G(0,\u221e)\u00d7{0,1}\u2022\u2022 \u2192 [0,\u221e], the expectationsfF (G, u, v) = FUSFG[f(G, u, v,F)]and fW (G, u, v) = WUSFG[f(G, u, v,F)]are also mass transports. This allows us to deduce the mass-transport principle for (G, \u03c1,F) fromthat of (G, \u03c1).1166.1. IntroductionReversibility and stationarityLet (G, \u03c1) be a random rooted network and let \u3008Xn\u3009n\u22650 be a random walk on G started at \u03c1. Therandom rooted graph (G, \u03c1) is said to be stationary if(G, \u03c1)d= (G,X1)and reversible if(G, \u03c1,X1)d= (G,X1, \u03c1).While every reversible random rooted graph is trivially stationary, the converse need not hold ingeneral. Indeed, every transitive graph (rooted arbitrarily) is stationary, while it is reversible if andonly if it is unimodular. For example, the grandfather graph [173] is transitive but not reversible.The following correspondence between unimodular and reversible random rooted networks isimplicit in [7, \u00a74] and is proven explicitly in [38].Proposition 6.1.6. If (G, \u03c1) is a unimodular random rooted network with E[c(\u03c1)] < \u221e, thenbiasing the law of (G, \u03c1) by c(\u03c1) (that is, reweighting the law of (G, \u03c1) by the Radon-Nikodymderivative c(\u03c1)\/E[c(\u03c1)]) yields the law of a reversible random rooted network. Conversely, if (G, \u03c1)is a reversible random rooted network with E[c(\u03c1)\u22121] <\u221e then biasing the law of (G, \u03c1) by c(\u03c1)\u22121yields the law of a unimodular random rooted network.\uf8f1\uf8f2\uf8f3 (G, \u03c1) unimodularwith E[c(\u03c1)] <\u221e\uf8fc\uf8fd\uf8febias by c(\u03c1)\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2192\u2190\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212bias by c(\u03c1)\u22121\uf8f1\uf8f2\uf8f3 (G, \u03c1) reversiblewith E[c(\u03c1)\u22121] <\u221e\uf8fc\uf8fd\uf8fe .For example, a finite rooted network is unimodular if and only if, conditioned on G, its root isuniformly distributed on the network, and is reversible if and only if, conditioned on G, the root isdistributed according to the stationary distribution of the random walk on the network.Thus, to prove an almost sure statement about unimodular random rooted networks withE[c(\u03c1)] < \u221e we can bias by the conductance at the root and work in the reversible setting, andvice versa.A useful equivalent characterisation of reversibility is as follows. Let G\u2194 denote the space ofisomorphism classes of graphs equipped with a bi-infinite path (G, \u3008xn\u3009n\u2208Z), which is endowed witha natural variant of the local topology. Let (G, \u03c1) be a random rooted graph and let \u3008Xn\u3009n\u22650 and\u3008X\u2212n\u3009n\u22650 be two independent simple random walks started from X0 = \u03c1, so that (G, \u3008Xn\u3009n\u2208Z) isa random variable taking values in G\u2194. Then (G, \u03c1) is reversible if and only if(G, \u3008Xn\u3009n\u2208Z) d= (G, \u3008Xn+k\u3009n\u2208Z) \u2200 k \u2208 Z. (6.1.1)Indeed, (\u03c1,X\u22121, . . .) is a simple random walk started from \u03c1 independent of X1 and, conditional on(G,X1), reversibility implies that \u03c1 is uniformly distributed among the neighbours of X1, so that(X1, \u03c1,X\u22121, X\u22122, . . .) has the law of a simple random walk from X1 and (6.1.1) follows. Conversely,1176.1. Introduction(6.1.1) implies that (G, \u03c1) is reversible by taking k = 1 and restricting to the 0th and 1st coordinatesof the walk.A useful variant of Proposition 6.1.6 is the following. Suppose that (G, \u03c1) is a unimodularrandom rooted network with E[c(\u03c1)] <\u221e, F is a sample of either FUSFG or WUSFG, and let cF(v)denote the sum of the conductances of the edges of G emanating from v that are included in F.Then, if we sample (G, \u03c1,F) biased by cF(\u03c1) and let \u3008Xn\u3009n\u22650 and \u3008X\u2212n\u3009n\u22650 be independent randomwalks on F starting at \u03c1, then, by [7, Theorem 4.1],(G, \u3008Xn\u3009n\u2208Z,F) d= (G, \u3008Xn+k\u3009n\u2208Z,F) \u2200 k \u2208 Z. (6.1.2)ErgodicityWe say that a unimodular random rooted network with E[c(\u03c1)] <\u221e is ergodic if any (and henceall) of the below hold.Theorem 6.1.7 (Characterisation of ergodicity [7, \u00a74]). Let (G, \u03c1) be a unimodular random rootednetwork with E[c(\u03c1)] <\u221e. The following are equivalent.1. When the law of (G, \u03c1) is biased by c(\u03c1) to give an equivalent reversible random rooted network,the stationary sequence \u3008(G,Xn)\u3009n\u22650 is ergodic.2. Every event A \u2282 G(0,\u221e)\u2022 invariant to changing the root has probability in {0, 1}.3. The law of (G, \u03c1) is an extreme point of the weakly closed convex set of laws of unimodularrandom rooted networks.A similar statement holds for edge-marked networks. Tail triviality of the USFs [44, Theorem8.3] implies that if (G, \u03c1) is an ergodic unimodular random rooted network and F is a sample ofeither FUSFG or WUSFG, then (G, \u03c1,F) is also ergodic.The extremal characterisation (3) implies (by Choquet theory) that every unimodular randomrooted network with E[c(\u03c1)] <\u221e can be written as a mixture of ergodic unimodular random rootednetworks. Thus, to prove a.s. statements about general unimodular random rooted networks itsuffices for us to consider ergodic unimodular random rooted networks.6.1.3 Component properties and indistinguishability on unimodular randomrooted networksGeneral unimodular random rooted graphs and networks have few automorphisms, so that it isnot appropriate at this level of generality to phrase indistinguishability in terms of automorphism-invariant properties. Instead, we consider properties that are invariant under rerooting within acomponent as follows. Consider the space G{0,1}\u2022 of rooted graphs with edges marked by \u03c9(e) \u2208{0, 1}, which we think of as a rooted graph together with a distinguished subgraph spanned by theedges \u03c9 = {e : \u03c9(e) = 1}. Given such a (G, v, \u03c9) we define K\u03c9(v) to be the connected componentof v in \u03c9.1186.1. IntroductionDefinition 6.1.8. A Borel-measurable set A \u2282 G\u2022 is called a component property if and onlyif it is invariant to rerooting within the component of the root, i.e.,(G, v, \u03c9) \u2208 A =\u21d2 (G, u, \u03c9) \u2208 A \u2200u \u2208 K\u03c9(v) .Again, this may be formulated for networks with the obvious modifications. This definition isequivalent to the one given in [7, Definition 6.14]. We say that a connected component K of \u03c9 hasproperty A (and abuse notation by writing K \u2208 A ) if (G, u, \u03c9) \u2208 A for some (and hence every)vertex u \u2208 K. We are now ready to state our main theorem in its full generality and strength.Theorem 6.1.9 (Indistinguishability of USF components). Let (G, \u03c1) be a unimodular randomnetwork with E[c(\u03c1)] < \u221e, and let F be a sample of either FUSFG or WUSFG. Then for everycomponent property A , either every connected component of F has property A or none of theconnected components of F have property A almost surely.And we may now restate Theorems 6.1.3, 6.1.4 and 6.1.2 in their full generality.Theorem 6.1.10. Let (G, \u03c1) be a unimodular random rooted network with E[c(\u03c1)] < \u221e and letF be a sample of FUSFG. Then F is either connected or has infinitely many components almostsurely.Theorem 6.1.11. Let (G, \u03c1) be a unimodular random rooted network with E[c(\u03c1)] <\u221e and let Fbe a sample of FUSFG. If F is disconnected a.s., then a.s. for every vertex v of G,inf{FUSFG(u \u2208 TF(v)) : u \u2208 V (G)} = 0.Theorem 6.1.12. Let (G, \u03c1) be a unimodular random rooted network and let F be a sample of theFUSFG. On the event that the measures FUSFG and WUSFG are distinct, every component of F istransient and has infinitely many ends almost surely. This holds both when the edges of F are giventhe conductances inherited from G and when they are given unit conductances.It follows that, under the assumptions of Theorem 6.1.12, every component of the FUSF of G haspositive speed and critical percolation probability pc < 1 [7].We remark that, by [96, Proposition 5], Theorem 6.1.9 is equivalent to the following ergodicitystatement.Corollary 6.1.13. Let (G, \u03c1) be an ergodic unimodular random rooted network with E[c(\u03c1)] < \u221eand let F be a sample of either FUSFG or WUSFG. Then (TF(\u03c1), \u03c1) is an ergodic unimodu-lar random rooted network. Moreover, if we bias the distribution of (G, \u03c1,F) by cF(\u03c1) and let\u3008Xn\u3009n\u22650 and \u3008X\u2212n\u3009n\u22650 be independent random walks on F started at \u03c1, then the stationary se-quence\u2329(G, \u3008Xn+k\u3009n\u2208Z,F)\u232ak\u2208Z is ergodic.1196.1. IntroductionExamples of component propertiesExample 6.1.14 (Automorphism-invariant properties). Let G0 be a transitive graph, and let Abe an automorphism-invariant set of subgraphs of G0, that is, \u03b3A = A for any automorphism \u03b3of G0. Fix an arbitrary vertex v0 of G0 and letA \u2032 ={(G, v, \u03c9) :\u2203 an isomorphism \u03c6 : (G, v)\u2192 (G0, v0)such that \u03c6(K\u03c9(v)) \u2208 A}Then A \u2032 is a component property such that (G0, v0, \u03c9) \u2208 A \u2032 if and only if K\u03c9(v0) \u2208 A . Thus,Theorem 6.1.1 follows from Theorem 6.1.1 and similarly Theorems 6.1.3, 6.1.4 and 6.1.2 follow fromTheorems 6.1.10, 6.1.11 and 6.1.12, respectively.Example 6.1.15 (Intrinsic properties). A graph H is said to have volume-growth dimensiond if|BH(v, r)| = rd+o(1)for any (and hence all) vertices v of H. Let pn(\u00b7, \u00b7) denote the n-step transition probabilities ofsimple random walk on H. We say that H has spectral dimension d ifpn(v, v) = n\u2212d\/2+o(1)for any (and hence all) vertices v of H. Lastly, recall that the critical percolation probabilitypc(H) of an infinite connected graph H is the supremum over p \u2208 [0, 1] such that independentpercolation with edge probability p a.s. does not exhibit an infinite cluster. Then, under thehypotheses of Corollary 6.1.13, the volume-growth, spectral dimension of TF(\u03c1) (if they exist) andvalue of pc are non-random, and consequently are a.s. the same for every tree in F.Component properties can be \u2018extrinsic\u2019 and depend upon how the component sits inside of thebase graph G.Example 6.1.16 (Extrinsic properties). A subgraph H of G is said to have discrete Hausdorffdimension \u03b1 if|BG(v, n) \u2229H| = n\u03b1+o(1),for any (and hence all) vertices v of G. The event that a component has a particular discreteHausdorff dimension is a component property, and consequently Theorem 6.1.9 implies that allcomponents of the USF in a unimodular random rooted network have the same discrete Hausdorffdimension (if this dimension exists). In fact, the discrete Hausdorff dimension of every componentof the USF in Zd was proven to be 4 for all d \u2265 4 by Benjamini, Kesten, Peres and Schramm [41].Even for unimodular transitive graphs, the conclusion of Theorem 6.1.9 is strictly strongerthan that of Theorem 6.1.1. This is because a component property can also depend on the wholeconfiguration, as the following example demonstrates.1206.1. IntroductionExample 6.1.17. Define N(v, \u03c9, r) to be the number of distinct components of \u03c9 that are adjacentto the ball B\u03c9(v, r) of radius r about v in the intrinsic distance on K\u03c9(\u03c1). Asymptotic statementsabout the growth of N(v, \u03c9, r), can be used to define component properties that depend on theentire configuration \u03c9, e.g.A = {(G, v, \u03c9) : N(v, \u03c9, r) = r\u03b2+o(1)}.All of the examples above are what we call tail properties. That is, these properties can beverified by looking at all but finitely many edges of both the component and of the configuration(see the next section for the precise definition). Let us give now an interesting example of a non-tailproperty.Example 6.1.18 (Non-tail property). The component propertyA (n) =\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3(G, v, \u03c9) :for each connected component K of \u03c9, there exists a path \u3008ei\u3009in G connecting K\u03c9(v) to K such that at most n of the ei\u2019s arenot contained in \u03c9.\uf8fc\uf8f4\uf8fd\uf8f4\uf8feis not a tail property. Benjamini, Kesten, Peres and Schramm [41] proved the remarkable resultthat the property A (n) \\A (n\u2212 1) holds a.s. for every component of the USF of Zd if and only if4(n\u2212 1) < d \u2264 4n.Tail propertiesDefinition 6.1.19. We say that a component property A is a tail component property if(G, v, \u03c9) \u2208 A =\u21d2 (G, v, \u03c9\u2032) \u2208 A \u2200\u03c9\u2032 \u2286 E(G) such that \u03c94\u03c9\u2032 andK\u03c9(v)4K\u03c9\u2032(v) are both finite.Indistinguishability of USF components by properties that are not tail can fail without theassumption of unimodularity \u2013 see [176, Remark 3.16] and [36, Example 3.1]. However, our nexttheorem shows that unimodularity is not necessary for indistinguishability of WUSF componentsby tail properties when the WUSF components are a.s. one-ended. In [170] it is shown that the lastcondition holds in every transient transitive graph. The following theorem, which is used in theproof of Theorem 6.1.9, implies that WUSF components are indistinguishable by tail properties inany transient transitive graph (not necessarily unimodular).Theorem 6.1.20. Let (G, \u03c1) be a stationary random network and let F be a sample of WUSFG.Suppose that every component of F is one-ended almost surely. Then for every tail componentproperty A , either every connected component of F has property A or none of the connectedcomponents of F have property A almost surely.1216.2. Indistinguishability of FUSF componentsSharpnessWe present a construction showing that the condition E[c(\u03c1)] < \u221e in Theorem 6.1.9 is indeednecessary. For integers n and k > 2, denote by Tn(k) the finite network on a binary tree of heightn such that edges at distance h from the leaves have conductance kh. Choose a uniform randomroot in Tn(k) and take n to \u221e while keeping k fixed. The limit of this process can be seen to bethe transient unimodular random rooted network T (k) in which the underlying graph is the canopytree [5] and edges of distance h from the leaves have conductance kh.Consider the finite network Gn obtained by gluing a copy of Tn(3) and a copy of Tn(4) at theirleaves in such a way that the resulting network Gn is planar, and let \u03c1n a uniformly chosen rootvertex of Gn. Then the randomly rooted graphs (Gn, \u03c1n) converge to a unimodular random rootednetwork which is formed by gluing a copy of T (3) and a copy of T (4) at their leaves. It can easily beseen via Wilson\u2019s algorithm (see Section 6.4) that the WUSF will contain precisely two one-endedcomponents corresponding to the two infinite rays of T (3) and T (4). These two trees are clearlydistinguishable from each other by measuring the frequency of edges with conductances 3 or 4 ontheir infinite ray. This example can be made into a graph rather than a network simply by replacingan edge with conductance kh by kh parallel edges.It is also possible to construct a unimodular random rooted network on which there are infinitelymany WUSF components almost surely and every cluster is distinguishable from every other cluster.Let G be a 3-regular tree, and let F1 be a sample of WUSFT . For every component T of F1, let U(T )be i.i.d. uniform [0, 1]. For each edge e of G that is contained in F1, let T (e) be the component ofF1 containing e. Define conductances on G by, for each edge e of G, setting c(e) = 1 if e \/\u2208 F1 andotherwise settingc(e) = exp((1 + U(T ))\u00d7 |The finite component of T (e) \\ e|) .These strong drifts ensure that a random walk on the network (G, c) will eventually remain in asingle component of F1. Running Wilson\u2019s algorithm on the network (G, c) to sample a copy F2of WUSF(G,c), we see that the components of F2 correspond to the components of F1. We candistinguish these components from each other by observing the rate of growth of the conductancesalong a ray in each component.6.2 Indistinguishability of FUSF componentsOur goal in this section is to prove Theorem 6.1.9 for the FUSF on a unimodular random rootednetwork (G, \u03c1) when the measures FUSFG and WUSFG are distinct.6.2.1 Cycle breaking in the FUSFLet G be a finite network. For each spanning tree t of G and oriented edge e of G that is not aself-loop, we define the direction D(e) = D(t, e) to be the first edge in the unique simple pathfrom e\u2212 to e+ in t.1226.2. Indistinguishability of FUSF componentsLemma 6.2.1. Let G be an infinite network with exhaustion \u3008Vn\u3009n\u22651 and let Tn be a sampleof USTGn for each n. Then for every oriented edge e of G, the random variables (Tn, D(Tn, e))converge in distribution to some limit (F, D(e)), where D(e) is an edge adjacent to e\u2212 and themarginal distribution of F is given by FUSFG.Proof. Since the distribution of Tn converges to FUSFG, it suffices to show that the conditionalprobabilitiesP(D(Tn, e) = d | f1, . . . , fk \u2208 Tn, h1, . . . hl \/\u2208 Tn) (6.2.1)converge, where d = (d\u2212, d+) is any oriented edge with d\u2212 = e\u2212 and F = {f1, . . . , fk} andH = {h1, . . . , hl} are any two finite collections of edges in G for which the event A = {f1, . . . , fk \u2208F, g1, . . . gl \/\u2208 F} has non-zero probability. Fix such d, F and H. If F includes a path from e\u2212 toe+ then D(Tn, e) is determined and there is convergence in (6.2.1). Also, if d \u2208 H then (6.2.1) iszero. So let us assume now that neither is the case.Let us first explain why convergence holds in (6.2.1) when F = H = \u2205. In that case, byKirchhoff\u2019s effective resistance formula (see [44, Theorem 4.1]), the probability that the uniquepath between e\u2212 to e+ in Tn goes through d equals the amount of current on the edge d when aunit current flows from e\u2212 to e+ in the network Gn. It is well known that this quantity convergesas n\u2192\u221e to the current passing through d in the free unit current flow from e\u2212 to e+ in G [173,Proposition 9.1].When F and H are non-empty, we take n to be sufficiently large such that the edges e, d andall the edges of F and H are contained in Gn, and that the event An = {F \u2282 Tn, H \u2229 Tn = \u2205}has non-zero probability (i.e, that Gn \\ H is connected and there are no cycles in F ). We write(Gn \u2212 H)\/F for the network formed from Gn by deleting each edge h \u2208 H and contracting eachedge f \u2208 F . By the Markov property (see Section 6.1.2), the laws of Tn conditioned on An and ofF conditioned on A can be sampled from by taking the union of F with a sample of the UST of(Gn \u2212 H)\/F or the FUSF of (G \u2212 H)\/F respectively. Thus, the same argument as above workswhen d 6\u2208 F and shows that the limit of (6.2.1) is equal to the current passing through d in the freeunit current flow from e\u2212 to e+ in (G\u2212H)\/F .Finally suppose that d \u2208 F . Let Vd be the set of vertices connected to e\u2212 by a simple path in Fpassing through d, and let Ed be the set of oriented edges with tail in Vd. By our previous discussion,(6.2.1) equals the sum of the currents flowing through the edges of Ed in the unit current flow frome\u2212 to e+ in (Gn \u2212H)\/F . As before, [173, Proposition 9.1] shows that this quantity converges tothe corresponding sum of currents in the free unit current flow from e\u2212 to e+ in (G\u2212H)\/F .For each oriented edge e of G and FUSFG-a.e. spanning forest f of G, we define the updateU(f, e) as follows. If e is either a self-loop or already contained in f , set U(f, e) = f . Otherwise,sample D(e) from its conditional distribution given F = f , and set U(f, e) = F \u222a {e} \\ D(e). Itseems likely that this conditional distribution is concentrated on a point.Question 6.2.2. Let G be a network and let e be an edge of G. Does U(f, e) coincide FUSFG-a.e.with some measurable function of f?1236.2. Indistinguishability of FUSF componentsIf any additional randomness is required to perform an update, it will always be taken to beindependent of any other random variables considered.Lemma 6.2.3. Let G be a network and F be a sample of FUSFG. Let v be a vertex of G and let Ebe an element of the set {e : e\u2212 = v} chosen independently of F and with probability proportionalto its conductance. Then U(F, E) and F have the same distribution.Proof. Let \u3008Vn\u3009n\u22650 be an exhaustion of G and let Tn be a sample of the UST on Gn for each n. Wemay assume that Gn contains v and every edge adjacent to v for all n \u2265 1. We define the updateU(t, e) of a spanning tree t of Gn at the oriented edge e to beU(t, e) = t \u222a {e} \\D(t, e).Since U(Tn, E) converges to U(F, E) in distribution, and so it suffices to verify that U(Tn, E)d= Tnfor each n \u2265 0: this may be done by checking that USTGn satisfies the detailed balance equationsfor the Markov chain on the set of spanning trees of G with transition probabilitiesp(t1, t2) =1c(v)c({e : e\u2212 = v and U(t1, e) = t2}).This simple calculation is carried out in [127, Lemma 6].This has the following immediate consequence.Corollary 6.2.4 (Update Tolerance for the FUSF). Let G be a network. Fix an edge e, and letFUSFeG denote the joint distribution of a sample F of FUSFG and of the update U(F, e). ThenFUSFG(F \u2208 A ) \u2265 c(e)c(e\u2212)FUSFeG(U(F, e) \u2208 A )Proof. Lemma 6.2.3 implies thatFUSFG(F \u2208 A ) = 1c(e\u2212)\u2211e\u02c6\u2212=e\u2212c(e\u02c6)FUSFe\u02c6G(U(F, e\u02c6) \u2208 A )\u2265 c(e)c(e\u2212)FUSFeG(U(F, e) \u2208 A ).6.2.2 All FUSF components are transient and infinitely-endedA weighted tree is a network whose underlying graph is a tree. Recall that a branch of an infinitetree T is an infinite connected component of T \\ v for some vertex v.Lemma 6.2.5. Let (T, \u03c1) be a unimodular random rooted weighted tree that is transient with positiveprobability. On the event that T is transient, T a.s. does not have any recurrent branches.Proof. For each vertex u of T , let V (u) be the set of vertices v 6= u such that the componentcontaining u in T \\ v is recurrent. We first claim that if T is a transient weighted tree and u is a1246.2. Indistinguishability of FUSF componentsvertex of T , then V (u) is either empty, or a finite simple path starting from a neighbour of u inT , or an infinite transient ray (i.e. a ray such that the sum of the edge resistances along the rayis finite) starting from a neighbour of u in T . First, Rayleigh\u2019s monotonicity principle implies thatif v \u2208 V (u) then every other vertex on the unique path from u to v in T is also in V (u). Second,if there exist v1, v2 \u2208 V (u) which do not lie on a simple path from u in T , then the component ofu in T \\ v1 and the component of u in T \\ v2 are both recurrent and have all of T as their unionimplying that T is recurrent.Thus, if V (u) is not empty, it must be a finite path or ray in T starting at u. Let us rule outthe case that V (u) is a recurrent infinite ray. Assume that V (u) is infinite and denote this rayby (v0, v1, v2, . . .) with v0 = u. For each integer n \u2265 0 the component of T \\ vn+1 containing vnis recurrent and so, by Rayleigh\u2019s monotonicity principle, the components of T \\ vn that do notcontain vn+1 are all recurrent. Thus T is decomposed to the union of the ray (v0, v1, . . .) and acollection of recurrent branches hanging on this ray. If the ray V (u) is a recurrent, we concludethat T is recurrent as well.Suppose for contradiction that with positive probability T is transient but there exists an edgee such that the component of \u03c1 in T \\ e is infinite and recurrent. Take \u03b5 > 0 sufficiently small sothat this edge e may be taken to have c(e) \u2265 \u03b5 with positive probability. Denote the event thatsuch an edge exists by B\u03b5. We will show that this contradicts the Mass-Transport Principle byexhibiting a mass transport such that every vertex sends a mass of at most one but some verticesreceive infinite mass on the event B\u03b5.From each vertex u such that V (u) is finite and non-empty, send mass one to the vertex v inV (u) that is farthest from u in T . From each vertex u such that V (u) is a transient ray, sendmass one to the end-point v = e\u2212 of the last edge e in the path from u spanned by V (u) such thatc(e) \u2265 \u03b5. If V (u) is empty, u sends no mass. Clearly every vertex sends a total mass of at most one.However, on the event B\u03b5, the vertex v that \u03c1 sends mass to receives infinite mass. Indeed, everyvertex in the infinite recurrent component of T \\ v containing \u03c1 sends mass one to v, contradictingthe Mass-Transport Principle.Lemma 6.2.6. Let G be an infinite network and let F be a sample of FUSFG. If with positiveprobability F has a recurrent component and a transient component with a non-empty core, thenwith positive probability F has a component that is a transient tree with a recurrent branch. Thisholds both when edges of the trees are given the conductances inherited from G and when they aregiven unit conductances.Proof. We consider the case that the edges of the trees are given the conductances inherited fromG, the other case is similar. If two such components exist, then one can find a finite path startingat a vertex of a recurrent component T and ending in a vertex of the core of a transient componentT \u2032. Moreover, by taking the shortest such path, the starting vertex is the only vertex in T and theend vertex is the only vertex in core(T \u2032).Thus, there exists a non-random finite simple path \u03b3 = \u3008\u03b3i\u3009ni=0 in G such that the followingevent, denoted B(\u03b3), holds with positive probability:1256.2. Indistinguishability of FUSF componentsTransient component,no recurrent branchesRecurrentComponentTransient componentwith a recurrent branche1 e2 e3Figure 6.1: When recurrent components and transient components with non-empty cores and norecurrent branches coexist, a finite sequence of updates can create a transient component with arecurrent branch.\u2022 TF(\u03b30) is recurrent,\u2022 TF(\u03b3i) 6= TF(\u03b30) for 0 < i \u2264 n,\u2022 TF(\u03b3n) is transient and\u2022 \u03b3n \u2208 core(TF(\u03b3n)) and it is the only such vertex in \u03b3.For each 1 \u2264 i \u2264 n, let ei be an oriented edge of G with e\u2212i = \u03b3i and e+i = \u03b3i\u22121. Define the forests\u3008Fi\u3009ni=0 by setting F0 = F and recursively,Fi = U(Fi\u22121, ei) , i = 1, . . . , n .We claim that on the event B(\u03b3), at least one of the two forests F0 or Fn contains a transienttree with a recurrent branch. If F0 contains such a tree we are done, so suppose not. We claimthat in this case TFn(\u03b3n) is a transient tree with a recurrent branch. Indeed, at each step of theprocess we are add the edge ei and remove some other edge adjacent to \u03b3i in TFi\u22121(\u03b3i), so thatTFn(\u03b3n) contains the tree TF0(\u03b30) (since \u03b3i 6\u2208 TF0(\u03b30) for i \u2265 1) and the path e1, . . . , en. Moreover,since \u03b3i 6\u2208 core(TF0(\u03b3n)) for all 0 \u2264 i < n, the tree TFn(\u03b3n) contains a branch of TF0(\u03b3n) and istherefore transient by our assumption. Thus, removing \u03b31 from the transient tree TFn(\u03b3n) yieldsthe recurrent branch TF0(\u03b30) as required.Denote by E the set of subgraphs of G that are transient trees with a recurrent branch. Wehave shown that P(F0 \u2208 E ) + P(Fn \u2208 E ) > 0, while by update-tolerance (Corollary 6.2.4)P(F0 \u2208 E ) \u2265( n\u220fi=1c(ei)c(e\u2212i ))P(Fn \u2208 E )so that P(F0 \u2208 E ) > 0 as claimed.Proof of Theorem 6.1.12. We may assume that (G, \u03c1) is ergodic, see Section 6.1.2. We apply [7,Proposition 4.9 and Theorem 6.2] to deduce that whenever (T, \u03c1) is an infinite unimodular randomrooted (unweighed) tree with E[c(\u03c1)] <\u221e, the event that T is infinitely-ended and the event that1266.2. Indistinguishability of FUSF componentsT is transient coincide up to a null set and, moreover, T has positive probability to be transientand infinitely-ended if and only if E[degT (\u03c1)] > 2. The expected degree of the WUSF is 2 inany unimodular random rooted network, and since the FUSFG stochastically dominates WUSFG,the assumption that FUSFG 6= WUSFG implies that E[degF(\u03c1)] > 2. Let M > 0 and let F\u2032 bethe forest obtained by deleting from F every edge e such that max(degF(e\u2212), degF(e+)) \u2265 M . IfM is sufficiently large then E[degF\u2032(\u03c1)] > 2 by the monotone convergence theorem. It follows bythe above that TF\u2032(\u03c1) is infinitely-ended and and transient (when given unit conductances) withpositive probability, and consequently that the same holds for TF(\u03c1) by Rayleigh monotonicity.Ergodicity of (G, \u03c1,F) then implies that the forest F contains a component that is infinitely-endedand transient (when given unit conductances) a.s.Assume for contradiction that with positive probability F has a component that is finitely-ended, or equivalently a component that is recurrent when given unit conductances. Lemma 6.2.6then implies that with positive probability F has a transient component with a recurrent branch(when all components are given unit conductances), contradicting Lemma 6.2.5.Thus, we have that all components of F are a.s. infinitely-ended and are transient when givenunit conductances. It follows from [7, Proposition 4.10] that every component is also a.s. transientwhen given the conductances inherited from G.6.2.3 Pivotal edges for the FUSFLet G be a network, let F be a sample of FUSFG and let A be a component property. We say thatan oriented edge e of G is a \u03b4-additive pivotal for a vertex v if1. e+ \u2208 TF(v) and e\u2212 6\u2208 TF(v) and,2. given F, the components TU(F,e)(v) and TF(v) have different types with probability at least \u03b4.We say that an oriented edge e is a \u03b4-subtractive pivotal for v if1. e\u2212 \u2208 TF(v) and e+ 6\u2208 TF(v) and,2. given F, the components TU(F,e)(v) and TF(v) have different types with probability at least \u03b4.We emphasize that when we say \u201cwith probability at least \u03b4\u201d above, this is over the randomnessof U(F, e), rather than of F.Lemma 6.2.7. Let G be a network and let F be a sample of FUSFG. Assume that a.s. all thecomponents of F are transient trees with non-empty cores and that with positive probability F hascomponents of both types A and \u00acA . Then for some small \u03b4 > 0, with positive probability thereexists a vertex v and an edge e such that v \u2208 core(F ) and e is a \u03b4-pivotal for v.Proof. We argue similarly to Lemma 6.2.6. Due to the assumptions of this lemma, there must bea component T of type A and a component T \u2032 of type \u00acA and an edge e 6\u2208 F connecting them.1276.2. Indistinguishability of FUSF componentsSo we may form a path starting with e that ends in a core vertex of T \u2032 such that all edges of thepath except for e are in T \u2032, and the last vertex of the path is the only vertex in core(T \u2032).Hence, there exists a non-random simple path \u03b3 = \u3008\u03b3i\u3009ni=0 in G such that the following event,denoted B(\u03b3), holds with positive probability:\u2022 TF(\u03b30) has type A ,\u2022 TF(\u03b3i) = TF(\u03b3j) 6= TF(\u03b30) for all 0 < i \u2264 j \u2264 n,\u2022 TF(\u03b3n) has type \u00acA and,\u2022 \u03b3n \u2208 core(TF(\u03b3n)) and it is the only such vertex in \u03b3.For each 1 \u2264 i \u2264 n, let ei be an oriented edge of G with e\u2212i = \u03b3i and e+i = \u03b3i\u22121. Define the forests\u3008Fi\u3009ni=0 by setting F0 = F and, recursively,Fi = U(Fi\u22121, ei) , i = 1, . . . , n.We claim that given B(\u03b3) there exists some small \u03b4 > 0 such that either one of the edges ei isa \u03b4-additive pivotal for \u03b30 in the forest Fi\u22121, or one of the edges ei is a \u03b4-subtractive pivotal for \u03b3nin the forest Fi\u22121. Indeed, if there is 1 \u2264 i \u2264 n such thatP(TFi(\u03b30) \u2208 \u00acA | Fi\u22121)> 0 ,(i.e., the component of \u03b30 changes type with positive probability in the transition from Fi\u22121 to Fi),then for the first such i, the edge ei is a \u03b4-additive pivotal for \u03b30 in Fi\u22121 (since the cluster of \u03b30only grows) where \u03b4 > 0 is the conditional probability above.If this does not occur, then a.s. TFn(\u03b30) is of type A . However, TFn(\u03b30) = TFn(\u03b3n) and TF0(\u03b3n)is of type \u00acA , so there is the first 1 \u2264 i \u2264 n in which TFi(\u03b3n) \u2208 A (i.e., the component of \u03b3nchanges type in the transition from Fi\u22121 to Fi). For this i we have ei is a \u03b4\u2032-subtractive pivotal for\u03b3n in Fi\u22121 with\u03b4\u2032 = P(TFi(\u03b3n) \u2208 A | Fi\u22121)> 0 .Let E\u03b4 be the event that there exists a vertex v such that v \u2208 core(F) and there exists an edgee that is a \u03b4-additive pivotal or \u03b4-subtractive pivotal for v in F. We proved that for some small\u03b4 > 0 we get\u2211ni=0 P(Fi \u2208 E\u03b4) > 0. However, by update tolerance (Corollary 6.2.4) it follows thatP(F0 \u2208 E ) \u2265( i\u220fj=1c(ej)c(e\u2212j ))P(Fi \u2208 E ) .Hence P(F0 \u2208 E\u03b4) > 0 for some small \u03b4 > 0 as claimed.6.2.4 Proof of Theorem 6.1.9 for the FUSFOur goal in this section is to prove the following theorem.1286.2. Indistinguishability of FUSF componentsTheorem 6.2.8. Let (G, \u03c1) be a unimodular random network with E[c(\u03c1)] < \u221e, and let F be asample of FUSFG. On the event that FUSFG 6= WUSFG, we have that for every component propertyA , either every connected component of F has property A or none of the connected components ofF have property A almost surely.We follow the strategy of Lyons and Schramm [176] while making the changes necessary to useupdate-tolerance.Proof of Theorem 6.2.8. Let (G, \u03c1) be a unimodular random rooted network with E[c(\u03c1)] <\u221e, letF be a sample of FUSFG and let A be a component property. Let \u3008Xn\u3009n\u2208Z be a bi-infinite randomwalk on F started at \u03c1 (that is, the concatenation of two independent random walks starting at \u03c1,as in Proposition 6.1.6). Conditioned on the random walk \u3008Xn\u3009n\u2208Z, let en be oriented edges chosenuniformly and independently from the set of edges at distance at most r from Xn in G. Finally, let{U(F, e) : e \u2208 E} be updates of F at each edge e of G, sampled independently of each other and of\u3008Xn\u3009n\u2208Z and \u3008en\u3009n\u2208Z conditional on (G, \u03c1,F). We bias by cF(\u03c1) so that, by [7, Theorem 4.1],(G, \u3008Xn\u3009n\u2208Z,F) d= (G, \u3008Xn+k\u3009n\u2208Z,F) \u2200 k \u2208 Z. (6.2.2)Let P\u0302 denote joint distribution of the random variables (G, \u03c1), F, \u3008Xn\u3009n\u2208Z, \u3008en\u3009n\u2208Z, and {U(F, e) :e \u2208 E} under this biasing, and let P\u0302(G,\u03c1) denote the conditional distribution given (G, \u03c1) of F,\u3008Xn\u3009n\u2208Z, \u3008en\u3009n\u2208Z and {U(F, e) : e \u2208 E} under the same biasing.By Theorem 6.1.12, every component of F is a.s. transient and infinitely-ended. By Lemma 6.2.7,there exists \u03b4 > 0 such that with positive probability \u03c1 \u2208 core(F) and there exists a either a \u03b4-additive or \u03b4-subtractive pivotal edge e for \u03c1 in F. By decreasing \u03b4 if necessary, it follows that thereexists an integer r such that with positive probability \u03c1 \u2208 core(F) and there exists a \u03b4-pivotal edgefor \u03c1 in F such that \u03c1 and e are at graph distance at most r in G and c(e)\/c(e\u2212) \u2265 \u03b4.Conditional on (G, \u03c1), for each edge e in G and n \u2208 Z, denote by E ne the event that en = e andthat the trace {Xn+k}k\u2208Z is disjoint from the components of F \\ Xn containing e\u2212 and e+. Forevery essential spanning forest f of G and n \u2208 Z, we haveP\u0302(G,\u03c1)(E ne | F = f) = P\u0302(G,\u03c1)(E ne | U(F, e) = f) .Thus, for every event B \u2286 {0, 1}E(G) such that P\u0302(G,\u03c1)(F \u2208 B) > 0, we have thatP\u0302(G,\u03c1)(E ne \u2229 {F \u2208 B}) = P\u0302(G,\u03c1)(E ne | F \u2208 B)P\u0302(G,\u03c1)(F \u2208 B)= P\u0302(G,\u03c1)(E ne | {U(F, e) \u2208 B})P\u0302(G,\u03c1)(F \u2208 B)=P\u0302(G,\u03c1)(F \u2208 B)P\u0302(G,\u03c1)(U(F, e) \u2208 B)P\u0302(G,\u03c1)(E ne \u2229 {U(F, e) \u2208 B})=FUSFG[cF(\u03c1)1(F \u2208 B)]FUSFeG[cF(\u03c1)1(U(F, e) \u2208 B)]P\u0302(G,\u03c1)(E ne \u2229 {U(F, e) \u2208 B}).1296.2. Indistinguishability of FUSF componentsObserve that if e does not have \u03c1 as an endpoint then, by Lemma 6.2.3,FUSFG[cF(\u03c1)1(F \u2208 B)] = 1c(e\u2212)\u2211e\u2032\u2212=e\u2212c(e\u2032)FUSFe\u2032G[cU(F,e\u2032)(\u03c1)1(U(F, e\u2032) \u2208 B)]\u2265 c(e)c(e\u2212)FUSFeG[cU(F,e)(\u03c1)1(U(F, e) \u2208 B)]=c(e)c(e\u2212)FUSFeG[cF(\u03c1)1(U(F, e) \u2208 B)],and so, for every edge e of G not having \u03c1 as an endpoint,P\u0302(G,\u03c1)(E ne \u2229 {F \u2208 B}) \u2265c(e)c(e\u2212)P\u0302(G,\u03c1)(E ne \u2229 {U(F, e) \u2208 B}) . (6.2.3)Update-tolerance also implies that (6.2.3) holds trivially when P\u0302(G,\u03c1)(F \u2208 B) = 0.Fix \u03b5 > 0, and let R be sufficiently large such that there exists an event A \u2032 that is measurablewith respect to (G, \u03c1) and F \u2229 BG(\u03c1,R) and satisfies P\u0302(A 4A \u2032) \u2264 \u03b5. Such an A \u2032 exists by theassumption that A is measurable. Define the disjoint unionsE n :=\u22c3c(e)\/(e\u2212)\u2265\u03b4E ne and EnR :=\u22c3e\u2212 \/\u2208BG(\u03c1,R) ,c(e)\/c(e\u2212)\u2265\u03b4E ne .Condition on (G, \u03c1), and letB = {\u03c9 \u2208 {0, 1}E : (G, \u03c1, \u03c9) \u2208 A \u2032 \\A }.Summing over (6.2.3) with this B yields that, for every R \u2265 1,P\u0302(G,\u03c1)(F \u2208 B) \u2265 P\u0302(G,\u03c1)(E nR \u2229 {F \u2208 B})\u2265 \u03b4P\u0302(G,\u03c1)(E nR \u2229 {U(F, en) \u2208 B})and hence, taking expectations,P\u0302((G, \u03c1,F) \u2208 A \u2032 \\A ) \u2265 \u03b4P\u0302(E nR \u2229 {(G, \u03c1,F) \u2208 A \u2032 \\A )}).By the definition of A \u2032 we have thatE nR \u2229 {(G, \u03c1, U(F, en)) \u2208 A \u2032} = E nR \u2229 {(G, \u03c1,F) \u2208 A \u2032},and soP\u0302((G, \u03c1,F) \u2208 A \u2032 \\A ) \u2265 \u03b4P\u0302(E nR \u2229{(G, \u03c1,F) \u2208 A \u2032} \u2229 {(G, \u03c1, U(F, en)) \u2208 \u00acA }).Let Pn denote the event that en is either a \u03b4-additive or \u03b4-subtractive pivotal edge for Xn. On1306.3. The FUSF is either connected or has infinitely many componentsthe event E nR , the vertices \u03c1 and Xn are in the same component of U(F, en), so that, on the eventPn \u2229 E nR \u2229 {(G, \u03c1,F) \u2208 A }, we have that (G, \u03c1, U(F, en)) \u2208 \u00acA with probability at least \u03b4. Thus,P\u0302((G, \u03c1,F) \u2208 A \u2032 \\A ) \u2265 \u03b42P\u0302(E nR \u2229 {(G, \u03c1,F) \u2208 A \u2032} \u2229Pn)\u2265 \u03b42P\u0302(E nR \u2229 {(G, \u03c1,F) \u2208 A } \u2229Pn)\u2212 \u03b42\u03b5 ,by definition of A \u2032. Since \u3008Xn\u3009n\u22650 is transient, we can take n to be sufficiently large thatP\u0302({(G, \u03c1,F) \u2208 A } \u2229 E n \\ E nR ) \u2264 \u03b5. Thus, for such n,P\u0302((G, \u03c1,F) \u2208 A \u2032 \\A ) \u2265 \u03b42P\u0302(E n \u2229 {(G, \u03c1,F) \u2208 A } \u2229 Pn)\u2212 2\u03b42\u03b5since A is a component property. Stationarity of \u3008(G,F, \u3008Xn+k\u3009k\u2208Z)\u3009n\u2208Z implies that P\u0302(E n \u2229{(G, \u03c1,F) \u2208 A } \u2229Pn) does not depend on n, so that it suffices to show it is positive to obtain acontradiction by choosing \u03b5 > 0 sufficiently small.As mentioned earlier, with positive probability \u03c1 \u2208 core(F) and there exists a \u03b4-pivotal edge ein F at distance at most r from \u03c1 such that c(e)\/(e\u2212) \u2265 \u03b4. Hence, eitherP\u0302({\u03c1 \u2208 core(F)} \u2229 {(G, \u03c1,F) \u2208 A } \u2229P0) > 0orP\u0302({\u03c1 \u2208 core(F)} \u2229 {(G, \u03c1,F) \u2208 \u00acA } \u2229P0) > 0.Since \u00acA is also a component property, we may assume without loss of generality that the formeroccurs. Conditioned on the events {\u03c1 \u2208 core(F)}, {(G, \u03c1,F) \u2208 A } and P0, the event E n is theevent that two independent random walks from \u03c1 stay within the components of F \\ \u03c1 that do notcontain e\u22120 or e+0 . This occurs with positive probability since every infinite component of TF(\u03c1) \\ \u03c1is transient by Theorem 6.1.12 and Lemma 6.2.5, concluding the proof.6.3 The FUSF is either connected or has infinitely manycomponentsOur goal in this section is to prove Theorems 6.1.10 and 6.1.11. Let (G, \u03c1) be a unimodular randomrooted network with E[c(\u03c1)] <\u221e and let F be a sample of FUSFG. We may assume that (G, \u03c1) isergodic, otherwise we take an ergodic decomposition. We may also assume that FUSFG 6= WUSFGa.s., since otherwise the result follows from [44].The following is an adaptation of [176, Lemma 4.2] from the unimodular transitive graph settingto our setting. We omit the proof, which is very similar to that of [176].Lemma 6.3.1 (Component Frequencies). Let (G, \u03c1) be an ergodic unimodular random rooted net-work with E[c(\u03c1)] < \u221e and let F be a sample of FUSFG. Conditional on (G, \u03c1), let Pv denote thelaw of a random walk \u3008Xn\u3009n\u22650 on G started at v for each vertex v of G, independent of F. Then1316.3. The FUSF is either connected or has infinitely many componentsthere exists a measurable function Freq : G{0,1}\u2022 \u2192 [0, 1] such that for every vertex v of G and everycomponent T of F,limN1NN\u2211n=11(Xn \u2208 T ) = Freq(G, \u03c1, T ) Pv-a.s.For each subset W of V , we refer to Freq(W ) = Freq(G, \u03c1,W ) as the frequency of W .Lemma 6.3.2. Let (G, \u03c1) be an ergodic unimodular random rooted network with E[c(\u03c1)] <\u221e suchthat FUSFG 6= WUSFG a.s. and let F be a sample of FUSFG. Conditioned on F let Px denote thelaw of a random walk \u3008Xn\u3009n\u22650 on G started at x, independent of F. Assume that with positiveprobability there exist a component of F with positive frequency. Then on this event, we a.s. havethat for every vertex u of G such that Freq(TF(u)) > 0 and every edge e \u2208 TF(u) such that F \\ econsists of two infinite components K1 and K2,Px(lim supN\u2192\u221e1NN\u2211n=11(Xn \u2208 Ki) > 0\u2223\u2223\u2223\u2223\u2223 (G, \u03c1,F))> 0.for both i = 1, 2 and every vertex x \u2208 G.Proof. We argue similarly to Lemma 6.2.5. For each vertex u of G and each edge e in TF(u), letKu(e) denote the component of TF(u)\\e containing u. For every vertex u such that Freq(TF(u)) > 0,let E(u) be the set of edges e in TF(u) such thatPx(lim supN\u2192\u221e1NN\u2211n=11(Xn \u2208 Ku(e)) > 0\u2223\u2223\u2223\u2223\u2223 (G, \u03c1,F))= 0.for some vertex x (and hence every vertex).Similarly to the proof of Lemma 6.2.5, we have that for every vertex u, if E(u) is non-emptythen it is either a finite simple path or a ray in TF(u) starting at u. Indeed, if e \u2208 E(u) then everyedge on the path between u and e is also in E(u), while if e1 and e2 do not lie on a simple pathfrom u in TF(u) then the union of Ku(e1) and Ku(e2) is all of TF(u) hencelim supN\u2192\u221e1NN\u2211n=11(Xn \u2208 Ku(e1)) + lim supN\u2192\u221e1NN\u2211n=11(Xn \u2208 Ku(e2)) \u2265 Freq(TF(u)) > 0a.s. for every starting point x of the random walk Xn.First suppose that with positive probability there exists a vertex u \u2208 core(F ) such that E(u) isa finite path. Define a mass transport by sending mass one from each vertex u such that E(u) isfinite, to the endpoint of the last edge in E(u); from all other vertices send no mass. Every vertexsends a mass of at most one while, if E(u) is a finite path for some u \u2208 core(F ) and e = (e\u2212, e+)is the last edge of this path, then e+ receives mass one from every vertex in the infinite set Ku(e),contradicting the Mass-Transport Principle.1326.3. The FUSF is either connected or has infinitely many componentsNext suppose that with positive probability there exists a vertex u such that E(u) is an infiniteray emanating from u. In this case, for any other vertex u\u2032 \u2208 TF(u) all but finitely many edges eof E(u) satisfy that u\u2032 \u2208 Ku(e) and it follows that E(u\u2032) is also an infinite ray and E(u\u2032)4E(u)is finite. By Theorem 6.2.8 and ergodicity of (G, \u03c1,F), the set E(u) is therefore an infinite ray forevery vertex u in G a.s. Transport unit mass from each vertex u to the first vertex following uin the ray E(u). Then every vertex u sends unit mass, and receives degF(u) \u2212 1 mass. By theMass-Transport Principle E[degF(\u03c1)] = 2, contradicting [7, Proposition 7.1] and the assumptionthat FUSFG 6= WUSFG a.s.By Theorem 6.1.12 each component of F has a non-empty core a.s. and by the above argumentE(u) = \u2205 for every core vertex u for which Freq(TF(u)) > 0, concluding our proof.Proof of Theorem 6.1.10. Suppose that F has some finite number k \u2265 2 of components a.s., whichwe denote T1, . . . Tk. Then for every N and vk\u2211i=11NN\u2211n=11(Xn \u2208 Ti) = 1and so\u2211ki=1 Freq(Ti) = 1. The frequency of a component is a component property and so, byTheorem 6.2.8 we must have that Freq(Ti) = 1\/k for all i = 1, . . . , k.As in Lemma 6.2.5, conditional on (G, \u03c1,F) there exists a simple path \u3008\u03b3i\u3009mj\u22650 in G that doesnot depend on F such that, with positive probability\u2022 TF(\u03b3j) 6= TF(\u03b30) for j = 1, . . . ,m and\u2022 All the edges (\u03b3j , \u03b3j+1) for j = 1, . . . ,m\u2212 1 belong to the same component, and\u2022 \u03b3m \u2208 core(TF(\u03b3m)) and it is the only such vertex in \u03b3.Denote this event by B(\u03b3). For each 1 \u2264 j \u2264 m, let ej be an oriented edge of G with e\u2212j = \u03b3j ande+j = \u03b3j\u22121. Define the forests \u3008Fj\u3009mj=0 recursively by setting F0 = F andFj = U(Fj\u22121, ej).Condition on this event B(\u03b3). The choice of the edges \u3008ej\u3009j\u22650 is such thatFm = F \u222a {e1} \\ {d}for some edge d, which we orient so that d\u2212 = \u03b3m, that disconnects TF(\u03b3m) and whose removal dis-connects TF(\u03b3m) into two infinite connected components. We denote the component in F containing\u03b3m by K1 and the component containing d+ by K2. As sets of vertices, we have thatTFm(\u03b30) = TF(\u03b30) \u222aK1 and TFm(d+) = K2.1336.4. Indistinguishability of WUSF componentsThus, by update-tolerance (Corollary 6.2.4), we have a.s. thatFreq(TFm(\u03b30)) = Freq(TF(\u03b30)) + Freq(K1) =1k, (6.3.1)and so Freq(K1) = 0 a.s., contradicting Lemma 6.3.2.Proof of Theorem 6.1.11. By Theorem 6.1.10 and the assumption that F is disconnected a.s., Fhas infinitely many components. For every N and v, letting \u3008Xn\u3009n\u22650 be a simple random walkindependent of F started at v,\u2211Ti is a component of F1NN\u2211n=11(Xn \u2208 Ti) = 1and so, taking the limit \u2211Ti a component of FFreq(Ti) \u2264 1.By Theorem 6.2.8 all the component frequencies are equal and so must all equal zero. Thus, forevery component T of F,lim1NN\u2211n=11(Xn \u2208 T )\u2192 0 a.s. ,which in particular implies that for any \u03b5 > 0 there exists some n for which P(Xn \u2208 TF(\u03c1)) \u2264 \u03b5,hence there exists a vertex v with P(v \u2208 TF(\u03c1)) \u2264 \u03b5, as required.Remark 6.3.3. It is possible to remove the application of indistinguishability in the above proofs.If there is a unique component with non-zero frequency a.s., Lemma 6.3.2 allows us to performupdates to create two components of non-zero frequency, contradicting update-tolerance and er-godicity. Otherwise, consider the component of maximal frequency. The maximal componentfrequency is non-random in an ergodic unimodular random rooted network. However, updating al-lows us to attach an infinite part of another positive-frequency component to the maximal frequencycomponent, increasing its frequency by Lemma 6.3.2, contradicting update-tolerance.6.4 Indistinguishability of WUSF components6.4.1 Indistinguishability of WUSF components by tail properties.Let G be a transient network. Recall that Wilson\u2019s algorithm rooted at infinity [44, 197, 228]allows us to generate a sample of WUSFG by joining together loop-erased random walks on G. Let\u03b3 be a path in G that is either finite or transient, i.e. visits each vertex of G at most finitely manytimes. The loop-erasure LE(\u03b3) of the path \u03b3 (introduced by Lawler [161]) is formed by erasingcycles from \u03b3 chronologically as they are created. More formally, we define LE(\u03b3)i = \u03b3ti where thetimes ti are defined recursively by t0 = 0 and ti = 1 + max{t \u2265 ti\u22121 : \u03b3t = \u03b3ti\u22121}.1346.4. Indistinguishability of WUSF componentsWilson\u2019s algorithm rooted at infinity generates a sample F of WUSFG as follows. Let {vj : j \u2208 N}be an enumeration of the vertices of G and define a sequence of forests \u3008Fi\u3009i\u22650 in G as follows:1. Let F0 = \u2205.2. Given Fj , start an independent random walk from vj+1 stopped if and when it hits the set ofvertices already included in Fj .3. Form the loop-erasure of this random walk path and let Fj+1 be the union of Fj with thisloop-erased path.4. Let F =\u22c3j\u22650 Fj .It is a fact that the choice of enumeration does not affect the law of F. We will require the followingslight variation on Wilson\u2019s algorithm rooted at infinity.Lemma 6.4.1. Let W be a finite set of vertices in G, and let {vj : j \u2208 N} and {wj : 1 \u2264 j \u2264 |W |}be enumerations of V (G) \\W and W respectively. Let F be the spanning forest of G generated asfollows.1. Let F\u20320 = \u2205.2. Given F\u2032j, start an independent random walk from vj+1 stopped if and when it hits the set ofvertices already included in F\u2032j.3. Form the loop-erasure of this random walk path and let F\u2032j+1 be the union of F\u2032j with thisloop-erased path.4. Let F0 =\u22c3j\u22650 F\u2032j.5. Given Fj, start an independent random walk from wj+1 stopped if and when it hits the set ofvertices already included in Fj.6. Let F =\u22c3|W |j=0 Fj.Then F is distributed according to WUSFG.Proof. Let j0 = max{j : vj is adjacent to W} and consider the enumerationv1, . . . , vj0 , w1, . . . , w|W |, vj0+1, . . .of V (G). Let {Xv : v \u2208 V (G)} be a collection of independent random walks, one from each vertexv of G. Let F be generated using the random walks {Xv : v \u2208 V (G)} as above and let F\u2032 be asample of WUSFG generated using Wilson\u2019s algorithm rooted at infinity, using the enumerationv1, . . . , vj0 , w1, . . . , w|W |, vj0+1, . . . and the same collection of random walks {Xv}. Then F\u2032 = F,and so F\u2032 has distribution WUSFG as claimed.1356.4. Indistinguishability of WUSF componentsnR\u03c1Figure 6.2: Illustration of the forest FR and the event CR,n. The forest FR is defined to be theunion of the futures in F of each of the vertices of G outside the ball of radius R about \u03c1. CR,n isthe event that none of these futures intersect the ball of radius n about \u03c1.Proof of Theorem 6.1.20. If G is recurrent, then its WUSF is a.s. connected and the statementholds trivially, so let us assume that G is a.s. transient.For each r > 0 let Gr be the \u03c3-algebra generated by the random variables (G, \u03c1) and F\u2229BG(\u03c1, r).Let G\u0302 be the network obtained from G by contracting every edge of F\u2229BG(\u03c1, r) and deleting everyedge of BG(\u03c1, r) \\F. Conditional on Gr, the forest F is distributed as the union of F\u2229BG(\u03c1, r) anda sample of WUSFG\u0302by the WUSF\u2019s spatial Markov property (see Section 6.1.2).For each integer R \u2265 0, let FR be the subgraph of F defined to be the union of the futures inF of every vertex in G \\ BG(\u03c1,R). For each R, we can sample FR conditioned on Gr by runningWilson\u2019s algorithm on G\u0302 starting from every vertex G\u0302 \\ BG(\u03c1,R) (in arbitrary order). A vertexv is contained in FR if and only if its past pastF(v) intersects G \\ BG(\u03c1,R), and so\u22c2R\u22650 FR = \u2205a.s. by the assumption that F has one-ended components a.s. Conditional on Gr and FR, the restof F may be sampled by finishing the run of Wilson\u2019s algorithm on G\u0302 as described in Lemma 6.4.1.For each R and n such that R \u2265 n, let CR,n be the event that FR \u2229 BG(\u03c1, n) = \u2205, so thatlimR\u2192\u221e P(CR,n) = 1 for each fixed n (see Figure 6.2). Let \u3008Xi\u3009i\u22650 be a random walk on G startedat \u03c1 and let \u3008X\u0302i\u3009i\u22650 be a random walk on G\u0302 started at (the equivalence class in G\u0302 of) \u03c1. Fixr \u2264 n \u2264 R and condition on Gr,FR and on the event CR,n. The definition of FR ensures thatF \\ FR is finite and that TF(v) \\ TFR(v) is finite for every v \u2208 FR. Thus, since A is a tail property,the event that TF(v) has property A is already determined by (G, \u03c1) and FR for every for vertexv \u2208 FR. Thus, by Lemma 6.4.1, the conditional probability that \u03c1 is in an A cluster equals theconditional probability that the random walk \u3008X\u0302i\u3009i\u22650 first hits FR at a vertex that belongs to an1366.4. Indistinguishability of WUSF componentsA cluster, so thatP(TF(\u03c1) \u2208 A | Gr,FR,CR,n) = P(\u3008X\u0302i\u3009i\u22650 first hits FR at an A cluster | Gr,FR,CR,n).On the event CR,n the walk \u3008X\u0302i\u3009i\u22650 must hit FR at time n\u2212 r or greater, and so for all j < n\u2212 rwe haveP(TF(\u03c1) \u2208 A | Gr,FR,CR,n) = P(\u3008X\u0302i\u3009i\u2265j first hits FR at an A cluster | Gr,FR,CR,n)= P(TF(X\u0302j) \u2208 A | Gr,FR,CR,n)where the last equality is due to Lemma 6.4.1. That is,P({TF(\u03c1) \u2208 A } \u2229 CR,n | Gr,FR) = P({TF(X\u0302j) \u2208 A } \u2229 CR,n | Gr,FR)a.s. for all j < n\u2212 r. Taking conditional expectations with respect to Gr givesP({TF(\u03c1) \u2208 A } \u2229 CR,n | Gr) = P({TF(X\u0302j) \u2208 A } \u2229 CR,n | Gr)a.s. for all j < n\u2212 r. Since the events CR,n are increasing in R and P(CR,n)\u2192 1 as R\u2192\u221e, takingthe limit R\u2192\u221e in the above equality gives thatP(TF(\u03c1) \u2208 A | Gr) = P(TF(X\u0302j) \u2208 A | Gr) (6.4.1)for all j < n\u2212 r. This equality holds for all j by taking n to infinity.Let \u03c4 and \u03c4\u0302 be the last times that \u3008Xn\u3009n\u22650 and \u3008X\u0302n\u3009n\u22650 visit BG(\u03c1, r + 1) respectively. Thenfor each vertex v \u2208 BG(\u03c1, r + 1), the conditional distributions\u3008X\u03c4+n\u3009n\u22650 conditioned on X\u03c4 = v and (G, \u03c1) and\u3008X\u0302\u03c4\u0302+n\u3009n\u22650 conditioned on X\u0302\u03c4\u0302 = v and Gr (6.4.2)are equal.Let I denote the invariant \u03c3-algebra of the stationary sequence \u3008(G,Xn,F)\u3009n\u22650. The ErgodicTheorem implies that1NN\u2211i=11(TF(Xi) \u2208 A ) a.s.\u2212\u2212\u2212\u2212\u2192N\u2192\u221eY := P(TF(\u03c1) \u2208 A | I).Moreover, the random variable Y is measurable with respect to the completion of the \u03c3-algebra1376.4. Indistinguishability of WUSF componentsgenerated by (G, \u03c1,F): to see this, note that for every a < b \u2208 [0, 1], Levy\u2019s 0-1 law implies thatP(limN\u2192\u221e1NN\u2211i=11(TF(Xi) \u2208 A ) \u2208 [a, b]\u2223\u2223\u2223\u2223 (G, \u03c1),F, \u3008Xn\u3009kn=0)a.s.\u2212\u2212\u2212\u2192k\u2192\u221e1(limN\u2192\u221e1NN\u2211i=11(TF(Xi) \u2208 A ) \u2208 [a, b]).ButP(limN\u2192\u221e1NN\u2211i=11(TF(Xi) \u2208 A ) \u2208 [a, b]\u2223\u2223\u2223\u2223 (G, \u03c1),F, \u3008Xn\u3009kn=0)= P(limN\u2192\u221e1NN\u2211i=11(TF(Xk+i) \u2208 A ) \u2208 [a, b]\u2223\u2223\u2223\u2223 (G,Xk),F)and so, by stationarity, P(Y \u2208 [a, b] \u2223\u2223 (G, \u03c1,F)) \u2208 {0, 1} a.s. In particular, Y is independent of X\u03c4given (G, \u03c1).Since G is transient, \u03c4 is finite a.s. and so1NN\u2211i=11(TF(X\u03c4+i) \u2208 A ) a.s.\u2212\u2212\u2212\u2212\u2192N\u2192\u221eY.In particular, this a.s. convergence holds conditioned on X\u03c4 = v for each v such that P(X\u03c4 = v) > 0.Since the support of X\u0302\u03c4\u0302 is contained in the support of X\u03c4 and the conditioned measures in (6.4.2)are equal, we have by the above that there exists a random variable Y\u0302 such that1NN\u2211i=11(TF(X\u0302\u03c4\u0302+i) \u2208 A ) a.s.\u2212\u2212\u2212\u2212\u2192N\u2192\u221eY\u0302 ,and the distribution of Y\u0302 given Gr and X\u0302\u03c4\u0302 = v is equal to the distribution of Y given (G, \u03c1) andX\u03c4 = v, so that Y\u0302 is in fact independent of Gr and X\u0302\u03c4\u0302 given (G, \u03c1). That is, for every a < b \u2208 [0, 1],P(Y\u0302 \u2208 [a, b] | Gr, X\u0302\u03c4\u0302 = v) = P(Y \u2208 [a, b] | (G, \u03c1), X\u03c4 = v) = P(Y \u2208 [a, b] | (G, \u03c1))so that, taking conditional expectations with respect to (G, \u03c1),P(Y\u0302 \u2208 [a, b] | (G, \u03c1)) = P(Y \u2208 [a, b] | (G, \u03c1)) = P(Y\u0302 \u2208 [a, b] | Gr, X\u0302\u03c4\u0302 = v) (6.4.3)establishing the independence of Y\u0302 from Gr and X\u0302\u03c4\u0302 conditional on (G, \u03c1).1386.4. Indistinguishability of WUSF componentsSince \u03c4\u0302 is finite a.s. we also have that1NN\u2211i=11(TF(X\u0302i) \u2208 A ) a.s.\u2212\u2212\u2212\u2212\u2192N\u2192\u221eY\u0302 .Hence, by (6.4.1) and the conditional Dominated Convergence Theorem,P(TF(\u03c1) \u2208 A | Gr) = E[1(TF(X\u0302j) \u2208 A ) | Gr]= E[1NN\u2211j=11(TF(X\u0302j) \u2208 A )\u2223\u2223\u2223\u2223\u2223Gr]a.s.\u2212\u2212\u2212\u2212\u2192N\u2192\u221eE[Y\u0302\u2223\u2223\u2223Gr] = E [Y\u0302 \u2223\u2223\u2223 (G, \u03c1)] .It follows by similar reasoning to (6.4.3) that the event {TF(\u03c1) \u2208 A } is independent of F\u2229BG(\u03c1, r)conditional on (G, \u03c1) for every r. It follows that WUSFG(TF(\u03c1) \u2208 A ) \u2208 {0, 1} a.s., and henceWUSFG(TF(v) \u2208 A ) \u2208 {0, 1} for every vertex v of G a.s. by stationarity. But, given (G, \u03c1), everyvertex v of G has positive probability of being in the same component of F as \u03c1, and so we musthave that the probabilitiesWUSFG(TF(v) \u2208 A ) = WUSFG(TF(\u03c1) \u2208 A ) \u2208 {0, 1}agree for every vertex v of G a.s., so that either every component of F has type A a.s. conditionalon (G, \u03c1) or every component of F does not have type A a.s. conditional on (G, \u03c1), completing theproof.6.4.2 Indistinguishability of WUSF components by non-tail properties.Wired cycle-breaking for the WUSFLet G be an infinite network and let f be a spanning forest of G such that every component of fis infinite and one-ended. For every oriented edge e of G, we define the update U(f, e) of f at eas follows:If e is a self-loop, or is already contained in f , let U(f, e) = f . Otherwise:1. If e\u2212 and e+ are in the same component of f , so that f \u222a {e} contains a cycle, letd be the unique edge of f that is both contained in this cycle and adjacent to e\u2212and let U(f, e) = f \u222a {e} \\ {d}.2. Otherwise, let d be the unique edge of f such that d is adjacent to e\u2212 and thecomponent containing e\u2212 in f \\ {d} is finite, and let U(f, e) = f \u222a {e} \\ {d}.The following are proved in [127] (in which an appropriate update rule is also developed for thecase that the WUSF has multiply-ended components) and can also be proved similarly to the proofin Section 6.2.1.1396.4. Indistinguishability of WUSF componentsProposition 6.4.2. Let G be a network, let F be a sample of WUSFG and suppose that everycomponent of F is one-ended almost surely. Let v be a fixed vertex of G and let E be an edge chosenfrom the set {e : e\u2212 = v} independently of F and with probability proportional to its conductance.Then U(F, E) and F have the same distribution.Corollary 6.4.3 (Update-tolerance for the WUSF). Let G be a network and let F be a sample ofWUSFG. If every component of F is one-ended almost surely, then for every event A \u2282 {0, 1}E(G)and every oriented edge e in G,WUSFG(F \u2208 A ) \u2265 c(e)c(e\u2212)WUSFG(U(F, e) \u2208 A ).Proof. By Proposition 6.4.2,WUSFG(F \u2208 A ) = 1c(e\u2212)\u2211e\u02c6\u2212=e\u2212c(e\u02c6)WUSFG(U(F, e\u02c6) \u2208 A )\u2265 c(e)c(e\u2212)WUSFG(U(F, e) \u2208 A ).Pivotal edges for the WUSFLet G be a network, and let f be a spanning forest of G such that every component of f is infiniteand one-ended and let A be a component property. We call an oriented edge e of G a good pivotaledge for a vertex v of G if either1. e+ \u2208 Tf (v), e\u2212 \u2208 Tf (v), and the type of TU(f,e)(v) is different from the type of Tf (v) (inwhich case we say e is a good internal pivotal edge for v),2. e+ \/\u2208 Tf (v), e\u2212 \/\u2208 Tf (v), and the type of TU(f,e)(v) is different from the type of Tf (v) (inwhich case we say e is a good external pivotal edge for v),3. e+ \u2208 Tf (v), e\u2212 \/\u2208 Tf (v), and the type of TU(f,e)(v) is different from the type of Tf (v) (inwhich case we say e is a good additive pivotal edge for v), or4. e\u2212 \u2208 Tf (v), e+ \/\u2208 Tf (v), the component of v in Tf (v) \\ {e\u2212} is infinite and the type ofTU(f,e)(v) is different from the type of Tf (v) (in which case we say e is a good subtractivepivotal edge for v).In particular, e is a good pivotal for some vertex v only if infinitely many vertices change the typeof their component when we update from f to U(f, e).Lemma 6.4.4. Let G be a network, let F be a sample of WUSFG and let A be a componentproperty. If every component of F is one-ended a.s., then eitherthere exists a good pivotal edge for some vertex v with positive probabilityor1406.4. Indistinguishability of WUSF componentsA is WUSFG-equivalent to a tail component property. That is, there exists a tail com-ponent property A \u2032 such thatWUSFG((G, v,F) \u2208 A 4A \u2032) = 0.for every vertex v of G.Proof. Suppose that no good pivotal edges exist a.s. and let A \u2032 \u2286 G(0,\u221e)\u00d7{0,1}\u2022 be the componentpropertyA \u2032 =\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3(G, v, \u03c9) :There exists a vertex u \u2208 K\u03c9(v) and a one-ended essentialspanning forest f of G such that (G, u, f) \u2208 A and thesymmetric differences \u03c94f and K\u03c9(u)4Kf (u) are finite.\uf8fc\uf8f4\uf8fd\uf8f4\uf8fe .Note that by definition A \u2032 is a tail component property. Our goal is to show that A and A \u2032 haveWUSFG((G, v,F) \u2208 A 4A \u2032) = 0 for every vertex v of G. One part of this assertion is easy, indeed,let \u21260 \u2282 G(0,\u221e)\u00d7{0,1}\u2022 be the event \u21260 = {(G, v, \u03c9) : \u03c9 is a one-ended essential spanning forest} sothat WUSFG(\u21260) = 1 by assumption. Then A \u2229 \u21260 \u2282 A \u2032 since one can take f = \u03c9 and u = v inthe definition of A \u2032.The second part of this assertion is slightly more difficult and requires the use of update-tolerance. Given a one-ended essential spanning forest F and a finite sequence of oriented edges\u3008ei\u3009ni=1 of G we define U(F; e1, . . . , en) recursively by U(F; e1) = U(F, e1) andU(F; e1, . . . , en) = U(U(F; e1, . . . , en\u22121), en).Let F be a sample of WUSFG and let \u21261 be the event that for any finite sequence of edges \u3008ei\u3009ni=1the forest U(F; e1, . . . , en) has no good pivotal edges. By update-tolerance and the assumption thatF has no good pivotal edges a.s., WUSFG(\u21261) = 1. Thus, it suffices to show that A\u2032\u2229\u21260\u2229\u21261 \u2282 A .Let (G, v,F) \u2208 A \u2032 \u2229 \u21260 \u2229 \u21261 and let f be a one-ended essential spanning forest such thatF4f and TF(u)4Tf (u) are finite and (G, u, f) \u2208 A for some vertex u \u2208 TF(v). We will prove byinduction on |f \\ F| that there exists a vertex u\u2032 \u2208 TF(v) with (G, u\u2032, f) \u2208 A and a finite sequenceof oriented edges \u3008ei\u3009ni=1 of G such that U(F; e1, . . . , en) = f and TF(u\u2032) and TU(F;e1,...,ei)(u\u2032) havethe same type for every 1 \u2264 i \u2264 n. Since (G, u\u2032, f) \u2208 A by assumption, this will imply that(G, v,F) \u2208 A as desired.To initialize the induction assume that |f \\ F| = 0. Then f \u2282 F and, since both F and f areone-ended essential spanning forests, we must have that F \u2282 f since any addition of an edge to fcreates either a cycle or a two-ended component, so that F = f and the claim is trivial.Next, assume that |f \\ F| > 0 and let h \u2208 f \\ F. Since F is a one-ended essential spanningforest, F\u222a{h} contains either a cycle or a two-ended component and we can therefore find an edgeg \u2208 F \\ f such that F\u2032 = F \u222a {h} \\ {g} is a one-ended essential spanning forest. The choice of g isnot unique, and will be important in the final case below.First suppose F \u222a {h} contains a cycle. In this case the choice of an edge g as above is not1416.4. Indistinguishability of WUSF componentsimportant. The edge g must be contained in this cycle since otherwise the cycle would be containedin F\u2032. Let e1, . . . , ek be an oriented simple path on this cycle so that e1 = g and ek = h. We havethat F\u2032 = U(F; ek, ek\u22121, . . . , e2) by definition of the update operation. Since none of the forestsU(F; ek, . . . , ei) have any good internal or external pivotal edges, we have that TF\u2032(u) has thesame type as TF(u). Lastly, (G, u,F) \u2208 \u21260 \u2229 \u21261 and |F\u2032 \u2229 f | < |F \u2229 f |, so that our inductionhypothesis provides us with a vertex u\u2032 \u2208 TF\u2032(u) and a sequence of edges e\u20321, . . . , e\u2032m such thatU(F; e\u20321, . . . , e\u2032m) = f and TF(u\u2032) and TU(F;e1,...,ei)(u\u2032) have the same type for every 1 \u2264 i \u2264 n. Sincethis also holds when u\u2032 is replaced by any vertex u\u2032\u2032 in the future of u\u2032 in F\u2032, we may take u\u2032\u2032 suchthat the above hold and u\u2032\u2032 \u2208 TF(v). We conclude the induction step by concatenating the twosequences e\u20321, . . . , e\u2032m, ek, . . . , e2.Now suppose that F\u222a{h} contains a two-ended component. Let us first consider the easier casein which u is not contained in this two-ended component, which is the case if and only if neither ofthe endpoints of h are in TF(u). In this case the choice of an edge g as above is not important. Theedge g must be such that the removal of g disconnects the component of F \u222a {h} containing g intotwo infinite connected components. We orient h so that its tail is in the component of g in F andorient g so that its head is in the component of F \\ {g} containing h\u2212. We then take an orientedsimple path in F from g+ to h\u2212 and append to it the edge h. As above, performing the updatesfrom the last edge of the path (that is, h) to the first (the edge in the path touching g+) yields F\u2032.Since none of the forests U(F; ek, . . . , ei) have any good external pivotal edges, we have that TF\u2032(u)has the same type as TF(u). We may now apply our induction hypothesis to (G, u,F\u2032) as before tocomplete the induction step in this case.Finally, if F \u222a {h} contains a two-ended component and one of the endpoints of h is in TF(u).The choice of g is important in this case. Orient h so that h+ \u2208 TF(u) and consider the uniqueinfinite rays from h+ and h\u2212 in F, denoted e1, e2, . . . and e\u22121, e\u22122, . . . respectively. Orient the ray\u3008ei\u3009i\u22651 towards infinity and the ray \u3008e\u2212i\u3009i\u22650 towards h\u2212 so that, writing e0 = h, \u3008ei\u3009i\u2208Z is anoriented bi-infinite path in F \u222a {h}.Next consider the unique infinite ray from u in F. Since the symmetric difference TF(u)4Tf (u)is finite, all but finitely many of the edges in the infinite ray from u in F must also be containedin the component of u in f . Let u\u2032 be the first vertex in the infinite ray from u in F such thatu\u2032 is contained in the ray from h+ in F and all of the ray from u\u2032 in F is contained in f , so thatu\u2032 = e+k = e\u2212k+1 for some k \u2265 0.Since f is a one-ended essential spanning forest and contains the ray \u3008ei\u3009i\u2265k+1, there existsan edge el with l < k such that el \/\u2208 f . By the definition of the update operation, we have thatF\u2032 = U(F;\u2212e0,\u2212e1, . . . ,\u2212el\u22121) if l > 0 and F\u2032 = U(F, e0, e\u22121, . . . , el+1) if l < 0. Let Fj denoteeither U(F;\u2212e0,\u2212e1, . . . ,\u2212ej) or U(F, e0, e\u22121, . . . , e\u2212j) for each j \u2264 l \u2212 1 as appropriate. In eithercase, u\u2032 is in an infinite connected component of Fj \\ ej+1 for each j and so, since good pivotaledges do not exist for any of the Fj , the type of TF\u2032(u\u2032) is the same as the type of TF(u\u2032). Lastly,we also have that (G, u\u2032,F) \u2208 \u21260\u2229\u21261 and |F\u2032\u2229f | < |F\u2229f |, and so we may use apply our inductionhypothesis to (G, u\u2032,F\u2032) as before, completing the proof.1426.4. Indistinguishability of WUSF componentsIndistinguishability of WUSF components by non-tail propertiesOur goal in this section is to prove the following, theorem, which in conjunction with Theorem 6.2.8completes the proof of Theorem 6.1.9.Theorem 6.4.5. Let (G, \u03c1) be a unimodular random network with E[c(\u03c1)] < \u221e, and let F be asample of WUSFG. Then for every component property A , either every connected component of Fhas property A or none of the connected components of F have property A almost surely.Proof of Theorem 6.4.5. We may assume that G is transient, since otherwise F is connected a.s. andthe claim is trivial. Since (G, \u03c1) becomes reversible when biased by c(\u03c1), Theorem 6.1.20 impliesthat the components of F are indistinguishable by tail properties (and therefore also by propertiesequivalent to tail properties), so that we may assume from now on that A is not equivalent to atail property. In this case, Lemma 6.4.4 implies that good pivotal edges exist for \u03c1 with positiveprobability. Without loss of generality, we may assume further that, with positive probability,TF(\u03c1) has property A and there exists a good pivotal edge for \u03c1: if not, replace A with \u00acA . Inthis case, there exist a natural numbers r such that, with positive probability TF(\u03c1) has propertyA and there exists a good pivotal edge e for \u03c1 at distance at most r from \u03c1 in G.Let {\u03b8(e) : e \u2208 E} be i.i.d. uniform [0, 1] random variables indexed by the edges of G, and let\u3008\u03c9n\u3009n\u22651 be Bernoulli (1\u22121\/(n+1))-bond percolations on G defined by setting \u03c9n(e) = 1 if and onlyif \u03b8(e) \u2265 1\u2212 1\/(n+ 1). By Theorem 6.1.5, every connected component of F is one-ended a.s. andso every component of F \u2229 \u03c9n is finite for every n a.s. Given (G, \u03c1,F, \u03b8), for each vertex u of G letvn(u) be a vertex chosen uniformly at random from the cluster of u in F \u2229 \u03c9n and let en(u) be anoriented edge chosen uniformly from the ball of radius r about vn(u) in G, where (vn(u), en(u)) and(vn\u2032(u\u2032), em\u2032(u\u2032)) are taken to be independent conditional on (G, \u03c1,F, \u03b8) if n\u2032 6= n or u\u2032 6= u. We writevn = vn(\u03c1), en = en(\u03c1) and let P\u0302 denote the joint law of (G, \u03c1,F, \u03b8, \u3008(vn(u), en(u)) : u \u2208 V \u3009n\u22651).The following is a special case of a standard fact about unimodular random rooted networks.Lemma 6.4.6. (G, \u03c1, vn,F, \u03b8) and (G, vn, \u03c1,F, \u03b8) have the same distribution.Proof. LetB \u2286 G(0,\u221e)\u00d7{0,1}\u00d7[0,1]\u2022\u2022 be an event, and for each vertex u of G let Kn(u) by the connectedcomponent of \u03c9n \u2229 F containing u. Define a mass transport by sending mass 1\/|Kn(u)| from eachvertex u to every vertex v \u2208 Kn(u) such that (G, u, v,F, \u03b8) \u2208 B (it may be that no such verticesexist, in which case u sends no mass). Then the expected mass sent by the root isE\u0302\uf8ee\uf8f0 1|Kn(\u03c1)|\u2211v\u2208Kn(\u03c1)1((G, \u03c1, v,F, \u03b8) \u2208 B)\uf8f9\uf8fb = P\u0302((G, \u03c1, vn,F, \u03b8) \u2208 B)while the expected mass received by the root isE\u0302\uf8ee\uf8f0 1|Kn(\u03c1)|\u2211v\u2208Kn(\u03c1)1((G, v, \u03c1,F, \u03b8) \u2208 B)\uf8f9\uf8fb = P\u0302((G, vn, \u03c1,F, \u03b8) \u2208 B).1436.4. Indistinguishability of WUSF componentsWe conclude by applying the Mass-Transport Principle.We will also require the following simple lemma.Lemma 6.4.7. Let f be an essential spanning forest of G such that every component of f isone-ended.1. For every edge e such that e \/\u2208 f but e+ and e\u2212 are in the same component of f , let C(f, e)denote the unique cycle contained in f \u222a {e}. Then for every vertex u in G,P\u0302(en(u) = e and C(f, e) \u2286 \u03c91 | (G, \u03c1), F = f)= P\u0302(en(u) = e and C(f, e) \u2286 \u03c91 | (G, \u03c1), F = U(f, e))for all n \u2265 0.2. For every edge e of G, there exists \u03ba(f, e) > 0 such that for every vertex u of G for whichat least one endpoint of e is not contained in Tf (u) and the component of u in f \\ {e\u2212} isinfinite,P\u0302(en(u) = e | (G, \u03c1), F = f) \u2265 \u03ba(f, e)P\u0302(en(u) = e | (G, \u03c1), F = U(f, e))for all n \u2265 0.Proof. Item (1) follows immediately from the observation that, under these assumptions, the setof vertices connected to u in \u03c9n \u2229 f and \u03c9n \u2229U(f, e) are equal on the event that C(f, e) \u2286 \u03c91. Wenow prove item (2). If e+ and e\u2212 are in the same component of f or if e+, e\u2212 \/\u2208 Tf (u) then theclaim holds trivially by setting \u03ba(f, e) = 1, so suppose not. Recall that K\u03c9n\u2229f (u) is defined to bethe connected component of u in \u03c9n \u2229 f . Define\u03ba1(u, f ;\u03c9n) =1|K\u03c9n\u2229f (u)|and\u03ba2(u, f, e;\u03c9n) =\u2211{v\u2208K\u03c9n\u2229f (u) : d(v,e)\u2264r}1|{e\u2032 \u2208 E : d(v, e\u2032) \u2264 r}| .Then conditional on (G, \u03c1), F = f , and \u03c9n, the probability that en(u) = e for each oriented edge eof G equals\u03ba1(u, f ;\u03c9n)\u03ba2(u, f, e;\u03c9n).Let W denote the union of the finite components of f \\ {e+, e\u2212}. Our assumptions on e, u and fimply that TU(f,e)(u)4 Tf (u) is contained in W , so that\u03ba1(u, f ;\u03c9n)\u22121 = |K\u03c9n\u2229f (u)| \u2264 |K\u03c9n\u2229U(f,e)(u)|+ |W | = \u03ba1(u, U(f, e);\u03c9n)\u22121+|W | ,1446.4. Indistinguishability of WUSF componentsand so\u03ba1(u, f ;\u03c9n) \u2265 11 + |W |\u03ba1(u, U(f, e);\u03c9n),since \u03ba1(u, U(f, e);\u03c9n) \u2264 1. Let\u03ba\u22122 (e) = min{|{e\u2032 \u2208 E : d(v, e\u2032) \u2264 r}|\u22121 : v \u2208 V (G), d(v, e) \u2264 r}> 0.Suppose that \u03ba2(u, U(f, e), e;\u03c9n) > 0. Then there is a vertex x in the tree K\u03c9n\u2229U(f,e)(u) such thatd(x, e) \u2264 r and x is still connected to u in K\u03c9n\u2229U(f,e)(u) \\ e. This x is therefore also be connectedto u in \u03c9n \u2229 f , and so\u03ba2(u, f, e;\u03c9n) \u2265 |{e\u2032 \u2208 E : d(x, e\u2032) \u2264 r}|\u22121 \u2265 \u03ba\u22122 (e) ,and thus,\u03ba2(u, f, e;\u03c9n) \u2265 \u03ba\u22122 (e)1(\u03ba2(u, U(f, e), e;\u03c9n) > 0).But \u03ba2(u, U(f, e), e;\u03c9n) is bounded above by\u03ba2(u, U(f, e), e;\u03c9n) \u2264 \u03ba+2 (e) :=\u2211{v: d(v,e)\u2264r}1|{e\u2032 \u2208 E : d(v, e\u2032) \u2264 r}|and so\u03ba2(u, f, e;\u03c9n) \u2265 \u03ba\u22122 (e)\u03ba+2 (e)\u03ba2(u, U(f, e), e;\u03c9n).We obtain that\u03ba1(u, f, e;\u03c9n)\u03ba2(u, f, e;\u03c9n) \u2265 \u03ba\u22122 (e)(1 + |W |)\u03ba+2 (e)\u03ba1(u, U(f, e);\u03c9n)\u03ba2(u, U(f, e), e;\u03c9n). (6.4.4)The claim follows by setting\u03ba(f, e) =\u03ba\u22122 (e)(1 + |W |)\u03ba+2 (e)and taking expectations over \u03c9n in (6.4.4).Given (G, \u03c1,F, \u03b8) and a positive \u03b4 > 0, we say that an oriented edge e of G is \u03b4 -update-friendly if1. c(e)\/c(e\u2212) \u2265 \u03b4, and2. \u03ba(F, e) \u2265 \u03b4, and3. if e \/\u2208 F but e+ and e\u2212 are in the same component of F, then C(F, e) \u2286 \u03c91.Note that if e is \u03b4-update-friendly for (G, \u03c1,F, \u03b8) then it is also \u03b4-update-friendly for (G, \u03c1, U(F, e), \u03b8).By assumption, there exists \u03b4 > 0 such that with positive probability TF(\u03c1) has property A and1456.4. Indistinguishability of WUSF componentsthere exists a good pivotal edge e for \u03c1 at distance at most r from \u03c1 in G such that e is \u03b4-update-friendly.Conditional on (G, \u03c1), for each edge e of G and n \u2208 Z, let E ne denote the event that e is\u03b4-update-friendly and en = e. Write P\u0302(G,\u03c1) for P\u0302 conditioned on (G, \u03c1). Applying part (2) ofLemma 6.4.7 if e+, e\u2212 are both in TF(\u03c1) and part (1) otherwise, we deduce from the definition of\u03b4-update-friendliness that for every event B \u2208 {0, 1}E(G) such that WUSFG(F \u2208 B) > 0,P\u0302(G,\u03c1)(E ne \u2229 {F \u2208 B}) = P\u0302(G,\u03c1)(E ne | F \u2208 B)WUSFG(F \u2208 B)\u2265 \u03b4P\u0302(G,\u03c1)(E ne | {U(F, e) \u2208 B})WUSFG(F \u2208 B)= \u03b41(c(e)c(e\u2212)\u2265 \u03b4)WUSFG(F \u2208 B)WUSFG(U(F, e) \u2208 B)\u00b7 P\u0302(G,\u03c1)(E ne \u2229 {U(F, e) \u2208 B})\u2265 \u03b42P\u0302(G,\u03c1)(E ne \u2229 {U(F, e) \u2208 B}) , (6.4.5)where the last inequality is by update-tolerance (Corollary 6.4.3). Update-tolerance also impliesthat this inequality holds trivially when WUSFG(F \u2208 B) = 0.Fix \u03b5 > 0, and let R be sufficiently large that there exists an event A \u2032 that is measurable withrespect to the \u03c3-algebra generated by (G, \u03c1) and F \u2229BG(\u03c1,R) and has P\u0302((G, \u03c1,F) \u2208 A 4A \u2032) \u2264 \u03b5.Define the disjoint unionsE n :=\u22c3c(e)\/(e\u2212)\u2265\u03b4E ne and EnR :=\u22c3e\u2212 \/\u2208BG(\u03c1,R) ,c(e)\/c(e\u2212)\u2265\u03b4E ne .Condition on (G, \u03c1), and letB = {\u03c9 \u2208 {0, 1}E : (G, \u03c1, \u03c9) \u2208 A \u2032 \\A }.Summing over (6.4.5) with this B yields thatP\u0302(G,\u03c1)(F \u2208 B) \u2265 P\u0302(G,\u03c1)(E nR \u2229 {F \u2208 B})\u2265 \u03b42P\u0302(G,\u03c1)(E nR \u2229 {U(F, en) \u2208 B})and hence, taking expectations,P\u0302((G, \u03c1,F) \u2208 A \u2032 \\A ) \u2265 \u03b42P\u0302(E nR \u2229 {(G, \u03c1,F) \u2208 A \u2032 \\A }).By the definition of A \u2032 we have thatE nR \u2229 {(G, \u03c1, U(F, en)) \u2208 A \u2032} = E nR \u2229 {(G, \u03c1,F) \u2208 A \u2032},1466.4. Indistinguishability of WUSF componentsand soP\u0302((G, \u03c1,F) \u2208 A \u2032 \\A )\u2265 \u03b42P\u0302(E nR \u2229{(G, \u03c1,F) \u2208 A \u2032} \u2229 {(G, \u03c1, U(F, en)) \u2208 \u00acA }). (6.4.6)Let Pn denote the event that en is a good pivotal edge for vn. We claim that if Pn occursand \u03c1 is not in the past of vn, then TU(F,en)(\u03c1) = TU(F,en)(vn) and (G, \u03c1, U(F, en)) \u2208 \u00acA . If en is agood internal, external or additive pivotal for vn, then clearly \u03c1 and vn are in the same componentof U(F, en), and, since en is pivotal for vn we deduce that (G, \u03c1, U(F, en)) \u2208 \u00acA . If en is a goodsubtractive pivotal edge for vn then the component of vn in F \\ {e\u2212n , e+n } is infinite and, since \u03c1 isnot in the past of vn, \u03c1 and vn must be in the same component of F \\ {e\u2212n , e+n }. It follows that \u03c1and vn are in the same component of U(F, en), and so U(F, en) \u2208 \u00acA\u03c1 as before. Combining thiswith (6.4.6), we haveP\u0302((G, \u03c1,F) \u2208 A \u2032 \\A ) \u2265 \u03b42P\u0302(E nR \u2229 {(G, \u03c1,F) \u2208 A \u2032} \u2229Pn \u2229 {\u03c1 6\u2208 pastF(vn)}) .Lemma 6.4.8. P\u0302(\u03c1 \u2208 past(vn))\u2192 0 as n\u2192\u221e.Proof. By Lemma 6.4.6,P\u0302(\u03c1 \u2208 pastF(vn))= P\u0302(vn \u2208 pastF(\u03c1)).Observe that past(\u03c1) is finite, while the size of the component of \u03c1 in TF(\u03c1) \u2229 \u03c9n tends to infinityas n\u2192\u221e. Since vn is defined to be a uniform vertex of the this component, it follows thatP\u0302(vn \u2208 pastF(\u03c1) | (G, \u03c1,F, \u03b8)) a.s.\u2212\u2212\u2212\u2192n\u2192\u221e 0and the claim follows by taking expectations.Thus, taking n sufficiently large that P\u0302(\u03c1 \u2208 pastF(vn)) < \u03b5, we have thatP((G, \u03c1,F) \u2208 A \u2032 \\A ) \u2265 \u03b42P\u0302(E nR \u2229 {(G, \u03c1,F) \u2208 A \u2032} \u2229Pn)\u2212 \u03b42\u03b5.By definition of A \u2032, we then have thatP((G, \u03c1,F) \u2208 A \u2032 \\A ) \u2265 \u03b42P\u0302(E nR \u2229 {(G, \u03c1,F) \u2208 A } \u2229Pn)\u2212 2\u03b42\u03b5 .We can further choose n to be sufficiently large that P\u0302(E n \\ E nR ) \u2264 \u03b5, so thatP((G, \u03c1,F) \u2208 A \u2032 \\A ) \u2265 \u03b42P\u0302(E n \u2229 {(G, \u03c1,F) \u2208 A } \u2229Pn)\u2212 3\u03b42\u03b5= \u03b42P\u0302(E n \u2229 {(G, vn,F) \u2208 A } \u2229Pn)\u2212 3\u03b42\u03b5 (6.4.7)1476.4. Indistinguishability of WUSF componentswhere in the second equality we have used the fact that A is a component property. Observe that,by Lemma 6.4.6, the probability P\u0302(E n \u2229 {(G, vn,F) \u2208 A } \u2229Pn) > 0 does not depend on n. Itdoes not depend on \u03b5 either, and so (6.4.7) contradicts the definition of A \u2032 when \u03b5 is taken to besufficiently small.148Part IIICircle Packing and Planar Graphs149Chapter 7Unimodular hyperbolic triangulations:circle packing and random walkSummary. We show that the circle packing type of a unimodular random plane triangulation isparabolic if and only if the expected degree of the root is six, if and only if the triangulation isamenable in the sense of Aldous and Lyons [7]. As a part of this, we obtain an alternative proof ofthe Benjamini-Schramm Recurrence Theorem [50].Secondly, in the hyperbolic case, we prove that the random walk almost surely converges to apoint in the unit circle, that the law of this limiting point has full support and no atoms, and thatthe unit circle is a realisation of the Poisson boundary. Finally, we show that the simple randomwalk has positive speed in the hyperbolic metric.Figure 7.1: A circle packing of a random hyperbolic triangulation.1507.1. Introduction7.1 IntroductionA circle packing of a planar graph G is a set of circles with disjoint interiors in the plane, onefor each vertex of G, such that two circles are tangent if and only if their corresponding verticesare adjacent in G. The Koebe-Andreev-Thurston Circle Packing Theorem [156, 221] states thatevery finite simple planar graph has a circle packing; if the graph is a triangulation (i.e. everyface has three sides), the packing is unique up to Mo\u00a8bius transformations and reflections. He andSchramm [121, 122] extended this theorem to infinite, one-ended, simple triangulations, showingthat each such triangulation admits a locally finite circle packing either in the Euclidean plane or inthe hyperbolic plane (identified with the interior of the unit disc), but not both. See Section 7.3.4for precise details. This result is a discrete analogue of the Uniformization Theorem, which statesthat every simply connected, non-compact Riemann surface is conformally equivalent to either theplane or the disc (indeed, there are deep connections between circle packing and conformal maps,see [202, 215] and references therein). Accordingly, a triangulation is called CP parabolic if itcan be circle packed in the plane and CP hyperbolic otherwise.Circle packing has proven instrumental in the study of random walks on planar graphs [47, 50,107, 121]. For graphs with bounded degrees, a rich theory has been established connecting thegeometry of the circle packing and the behaviour of the random walk. Most notably, a one-ended,bounded degree triangulation is CP hyperbolic if and only if random walk on it is transient [121]and in this case it is also non-Liouville, i.e. admits non-constant bounded harmonic functions [47].The goal of this work is to develop a similar, parallel theory for random triangulations. Partic-ular motivations come from the Markovian hyperbolic triangulations constructed recently in [23]and [75]. These are hyperbolic variants of the UIPT [24] and are conjectured to be the local limitsof uniform triangulations in high genus. Another example is the Poisson-Delaunay triangulationin the hyperbolic plane, studied in [49] and [37]. All these triangulations have unbounded degrees,rendering existing methods ineffective (for example methods used in [17, 47, 121]).Indeed, in the absence of bounded degree the existing theory fails in many ways. For example,in a circle packing of a triangulation with bounded degrees, radii of adjacent circles have uniformlybounded ratios (a fact known as the Ring Lemma [200]). The absence of such a uniform boundinvalidates important resistance estimates. This is not a mere technicality: one can add extracircles in the interstices of the circle packing of the triangular lattice to give the random walk driftin arbitrary directions. This does not change the circle packing type, but allows construction ofa graph that is CP parabolic but transient or even non-Liouville. Indeed, the main effort in [107]was to overcome this sole obstacle in order to prove that the UIPT is recurrent.The hyperbolic random triangulations of [75] and [37] make up for having unbounded degrees bya different useful property: unimodularity (essentially equivalent to reversibility, see Sections 7.3.1and 7.3.2). This allows us to apply probabilistic and ergodic arguments in place of the analyticarguments appropriate to the bounded degree case. Our first main theorem establishes a proba-bilistic characterisation of the CP type for unimodular random rooted triangulations, and connectsit to the geometric property of invariant (non-)amenability, which we define in Section 7.3.3.1517.1. IntroductionTheorem 7.1.1. Let (G, \u03c1) be an infinite, simple, one-ended, ergodic unimodular random rootedplanar triangulation. Then eitherE[deg(\u03c1)] = 6, in which case (G, \u03c1) is invariantly amenable and almost surely CPparabolic,or elseE[deg(\u03c1)] > 6, in which case (G, \u03c1) is invariantly non-amenable and almost surely CPhyperbolic.This theorem can be viewed as a local-to-global principle for unimodular triangulations. Thatis, it allows us to identify the circle packing type and invariant amenability, both global properties,by calculating the expected degree, a very local quantity. For example, if (G, \u03c1) is a simple, one-ended triangulation that is obtained as a local limit of planar graphs, then by Euler\u2019s formula andFatou\u2019s lemma its average degree is at most 6, so that Theorem 7.1.1 implies it is almost surely CPparabolic. If in addition (G, \u03c1) has bounded degrees, then it is recurrent by He-Schramm [121]. Inparticular, this gives an alternative proof of the Benjamini-Schramm Recurrence Theorem [50] inthe primary case of a one-ended limit. We handle the remaining cases in Section 7.4.1. Unlike theproof of [50], whose main ingredient is a quantitative estimate for finite circle packings [50, Lemma2.3], our method works with infinite triangulations directly and implies the following generalisation:Proposition 7.1.2. Any unimodular, simple, one-ended random rooted planar triangulation (G, \u03c1)with bounded degrees and E[deg(\u03c1)] = 6 is almost surely recurrent.This trivially extends the Benjamini-Schramm result, since any local limit of finite planargraphs is unimodular. An important open question is whether every unimodular random graphis a Benjamini-Schramm limit of finite graphs. In a forthcoming paper [20], we show that anyunimodular planar graph G is a limit of some sequence of finite graphs Gn, and that if G is atriangulation with E[deg(\u03c1)] = 6 then Gn can also be taken to be planar. In particular, any graphto which Proposition 7.1.2 applies is also a local limit of finite planar graphs with bounded degrees.Consequently there are no graphs to which this result applies and the Benjamini-Schramm Theoremdoes not. Note however, that for a given unimodular planar triangulation, it may not be obvioushow to find this sequence of graphs. We remark that the dichotomy of Theorem 7.1.1 has manyextensions, applying to more general maps and holding further properties equivalent. We addressthese in [20]. See [34] and [121, Theorem 10.2] for earlier connections between the CP type anddegree distributions in the deterministic setting.Our method of proof relies on the deep theorem of Schramm [206] that the circle packing of atriangulation in the disc or the plane is unique up to Mo\u00a8bius transformations fixing the disc or theplane as appropriate. We use this fact throughout the paper in an essential way: it implies thatany quantity derived from the circle packing in the disc or the plane that is invariant to Mo\u00a8biustransformations is determined by the graph G and not by our choice of circle packing. Key examplesof such quantities are angles between adjacent edges in the associated drawings with hyperbolic or1527.1. IntroductionEuclidean geodesics (see Section 7.4), hyperbolic radii of circles in the hyperbolic case, and ratiosof Euclidean radii in the parabolic case.Boundary Theory. Throughout, we realize the hyperbolic plane as the Poincare\u00b4 disc {|z| < 1}with metric dhyp. The unit circle {|z| = 1} is the boundary of the hyperbolic plane in severalgeometric and probabilistic senses. For a general graph embedded in the hyperbolic plane, the unitcircle may or may not coincide with probabilistic notions of the graph\u2019s boundary.When a bounded degree triangulation is circle packed in the disc, Benjamini and Schramm [47]showed that the random walk converges to a point in the circle almost surely and that the law ofthe limit point has full support and no atoms. More recently, it was shown by the first and thirdauthors together with Barlow and Gurel-Gurevich [17] that the unit circle is a realisation of boththe Poisson and Martin boundaries of the triangulation. Similar results regarding square tilingwere obtained in [48] and [100].Again, these theorems fail for some triangulations with unbounded degrees. Starting with anyCP hyperbolic triangulation, one can add circles in the interstices of the packing so as to createdrifts along arbitrary paths. In this way, one can force the random walk to spiral in the unitdisc and not converge to any point in the boundary. One can also create a graph for which thewalk can converge to a single boundary point from two or more different angles each with positiveprobability, so that the exit measure is atomic and the unit circle is no longer a realisation of thePoisson boundary. Our next result recovers the boundary theory in the unimodular setting.When C is a circle packing of a graph G in the disc D, we write C = (z, r) where z(v) is the(Euclidean) centre of the circle corresponding to v, and r(v) is its Euclidean radius. Recall thatthe hyperbolic metric on the unit disc is defined by|dhyp(z)| = 2|dz|1\u2212 |z|2 ,and that circles in the Euclidean metric are also hyperbolic circles (with different centres and radii).We write zh(v) and rh(v) for the hyperbolic centre and radius of the circle corresponding to v. Weuse PGv and EGv to denote the probability and expectation (conditioned on G) with respect torandom walk (Xn)n\u22650 on G started from a vertex v.Theorem 7.1.3. Let (G, \u03c1) be a simple, one-ended, CP hyperbolic unimodular random planartriangulation with E[deg2(\u03c1)] <\u221e. Let C be a circle packing of G in the unit disc, and let (Xn) bea simple random walk on G. The following hold conditional on (G, \u03c1) almost surely:1. z(Xn) and zh(Xn) both converge to a (random) point denoted \u039e \u2208 \u2202D,2. The law of \u039e has full support \u2202D and no atoms.3. \u2202D is a realisation of the Poisson boundary of G. That is, for every bounded harmonic1537.2. Examplesfunction h on G there exists a bounded measurable function g : \u2202D\u2192 R such thath(v) = EGv [g(\u039e)].We refer to the law of \u039e conditional on (G, \u03c1) as the exit measure from v. In Section 7.7we extend this result to weighted and non-simple triangulations, with the obvious changes. Oneingredient in the proof of the absence of atoms is a more general observation, Lemma 7.5.2, whichstates roughly that exit measures on boundaries of stationary graphs are either non-atomic or trivialalmost surely.Our final result relates exponential decay of the Euclidean radii along the random walk to speedin the hyperbolic metric.Theorem 7.1.4. Let (G, \u03c1) be a simple, one-ended, CP hyperbolic unimodular random rootedplanar triangulation with E[deg2(\u03c1)] <\u221e and let C be a circle packing of G in the unit disc. Thenalmost surelylimn\u2192\u221edhyp(zh(\u03c1), zh(Xn))n= limn\u2192\u221e\u2212 log r(Xn)n> 0.In particular, both limits exist. Moreover, the limits do not depend on the choice of packing, and if(G, \u03c1) is ergodic then this limit is an almost sure constant.Thus the random walk (Xn) has positive asymptotic speed in the hyperbolic metric, the Eu-clidean radii along the walk decay exponentially, and the two rates agree.Organization of the paper. In Section 7.2 we review the motivating examples of unimodularhyperbolic random triangulations to which our results apply. In Section 7.3.1 and Section 7.3.2 wegive background on unimodularity, reversibility and related topics. In Section 7.3.3 we recall Al-dous and Lyons\u2019s notion of invariant amenability [7] and prove one of its important consequences.In Section 7.3.4 we recall the required results on circle packing and discuss measurability. Sec-tion 7.4 contains the proof of Theorem 7.1.1 as well as a discussion of how to handle the remaining(easier) cases of the Benjamini-Schramm Theorem. Theorem 7.1.3 is proved in Section 7.5 andTheorem 7.1.4 is proved in Section 7.6. Background on the Poisson boundary is provided beforethe proof of Theorem 7.1.3(3) in Section 7.5.3. In Section 7.7 we discuss extensions of our resultsto non-simple and weighted triangulations. We end with some open problems in Section 7.8.7.2 ExamplesBenjamini-Schramm limits of random maps have been objects of great interest in recent years,serving as discrete models of 2-dimensional quantum gravity. Roughly, the idea is to consider auniformly random map from some class of rooted maps (e.g. all triangulations or quadrangulationsof the sphere of size n) and take a local limit as the size of the maps tends to infinity. The firstsuch construction was the UIPT [24]; see also [15, 18, 19, 39, 76].1547.2. ExamplesCurien\u2019s PSHT. Recently, hyperbolic versions of the UIPT and related maps have been con-structed: half-plane versions in [23] and full-plane versions in [75]. These are constructed directly,and are believed but not yet known to be the limits of finite maps (see below). The full planetriangulations form a one (continuous) parameter family {T\u03ba}\u03ba\u2208(0,2\/27) (known as the PSHT, forPlanar Stochastic Hyperbolic Triangulation). They are reversible and ergodic, have anchored ex-pansion and are therefore invariantly non-amenable. The degree of the root in T\u03ba is known to havean exponential tail, so that all of its moments are finite. These triangulations are not simple, soour main results do not apply to them directly, but by considering their simple cores we are stillable to obtain a geometric representation of their Poisson boundary (see Section 7.7).Benjamini-Schramm limits of maps in high genus. It is conjectured that the PSHT T\u03ba isthe Benjamini-Schramm limit of the uniform triangulation with n vertices of a surface of genusb\u03b8nc, for some \u03b8 = \u03b8(\u03ba) (see e.g. [198] for precise definitions of maps on general surfaces). In ourupcoming paper [20] we prove that all one-ended unimodular random rooted planar triangulationsare also Benjamini-Schramm limits of finite triangulations. If the triangulation has expected degreegreater than 6, then the finite approximating triangulations necessarily have genus linear in theirsize.In the context of circle packing, it may be particularly interesting to take the Benjamini-Schramm limit (T, \u03c1) of the uniform simple triangulation with n vertices of the b\u03b8nc-holed torusTn. This limit (which we conjecture exists) should be a simple variant of the PSHT. Letting \u03c1nbe a uniformly chosen root of Tn, it should also be the case that E[deg(\u03c1n)] \u2192 E[deg(\u03c1)] > 6 andE[deg(\u03c1)2] <\u221e, so that our results would be applicable to the circle packing of (T, \u03c1).Delaunay triangulations of the hyperbolic plane. Start with a Poisson point process in thehyperbolic plane with intensity \u03bb times the hyperbolic area measure, and add a root point at theorigin. Consider now the Delaunay triangulation with this point process as its vertex set, wherethree vertices u, v, w form a triangle if the circle through u, v, w contains no other points of theprocess. This triangulation, known as the Poisson-Delaunay triangulation, is naturally embeddedin the hyperbolic plane with hyperbolic geodesic edges. These triangulations, studied in [37, 49],are unimodular when rooted at the point at the origin. They are known to have anchored expansion[37] and are therefore invariantly non-amenable. (We also get a new proof of non-amenability fromTheorem 7.1.1, as one can show the expected degree to be greater than six by transporting anglesas in the proof of Theorem 7.1.1.) The Poisson-Delaunay triangulations are also simple and one-ended, and the degree of the root has finite second moment, so that our results apply directly totheir circle packings.1557.3. Background and definitions7.3 Background and definitions7.3.1 Unimodular random graphs and mapsUnimodularity of graphs (both fixed and random) has proven to be a useful and natural propertyin a number of settings. We give here the required definitions and some of their consequences, andrefer the reader to [7, 173] for further background.A rooted graph (G, \u03c1) is a graph G = (V,E) with a distinguished vertex \u03c1 called the root.We will allow our graphs to contain self-loops and multiple edges, and refer to graphs withouteither as simple. A graph is said to be one-ended if the removal of any finite set of vertices leavesprecisely one infinite connected component. A graph isomorphism between two rooted graphs is arooted graph isomorphism if it preserves the root.A map is a proper (that is, with non-intersecting edges) embedding of a connected graph into asurface, viewed up to orientation preserving homeomorphisms of the surface, so that all connectedcomponents of the complement (called faces) are topological discs.8 The map is planar if thesurface is homeomorphic to an open subset of the sphere, and is simply connected if the surfaceis homeomorphic to the sphere or the plane. A map is a triangulation if every face is incident toexactly three edges. Note that an infinite planar triangulation is simply connected if and only if itis one-ended.Every connected graph G can be made into a metric space by endowing it with the shortestpath metric dG. By abuse of notation, we use the ball Bn(G, u) to refer both to the set of vertices{v \u2208 V : dG(u, v) \u2264 n} and the induced subgraph on this set, rooted at u. The balls in a mapinherit a map structure from the full map.The local topology on the space of rooted connected graphs (introduced in [50]) is the topologyinduced by the metricdloc((G, \u03c1), (G\u2032, \u03c1\u2032))= e\u2212R where R = sup{n \u2265 0 : Bn(G, \u03c1) \u223c= Bn(G\u2032, \u03c1\u2032)}.The local topology on rooted maps is defined similarly by requiring the isomorphism of the ballsto be an isomorphism of rooted maps. We denote by G\u2022 andM\u2022 the spaces of isomorphism classesof rooted connected graphs and of maps with their respective local topologies. Random rootedgraphs and maps are Borel random variables taking values in these spaces.Several variants of these spaces will also be of use. A (countably) marked graph is a graphtogether with a mark function m : V \u222a E \u2192 M which gives every edge and vertex a mark insome countable set M . A graph isomorphism between marked graphs is an isomorphism of markedgraphs if it preserves the marks. The local topologies on rooted marked graphs and maps is definedin the obvious way. These spaces are denoted GM\u2022 andMM\u2022 . Sometimes we will consider maps withmarks only on vertices or only on edges; these fit easily into our framework. Marked graphs are8There is an additional constraint regarding boundaries of faces of infinite degree. However, this condition isautomatically satisfied for triangulations and for simply connected maps, so that we need not worry about it in thispaper.1567.3. Background and definitionsspecial cases of what Aldous and Lyons [7] call networks, for which the marks may take values inany separable complete metric space.Similarly, we define G\u2022\u2022 (resp. M\u2022\u2022) to be the spaces of doubly rooted (that is, with a distin-guished ordered pair of vertices) connected graphs (resp. maps) (G, u, v). These spaces, along withtheir marked versions, are equipped with natural variants of the local topology. All such spaces weconsider are Polish.A mass transport is a non-negative Borel function f : G\u2022\u2022 \u2192 R+. A random rooted graph(G, \u03c1) is said to be unimodular if it satisfies the mass transport principle: for any masstransport f ,E[ \u2211v\u2208V (G)f(G, \u03c1, v)]= E[ \u2211v\u2208V (G)f(G, v, \u03c1)].In other words,\u2018Expected mass out equals expected mass in.\u2019This definition generalises naturally to define unimodular marked graphs and maps. Importantly,any finite graph G with a uniformly chosen root vertex \u03c1 satisfies the mass transport principle.The laws of unimodular random rooted graphs form a weakly closed, convex subset of the spaceof probability measures on G\u2022, so that weak limits of unimodular random graphs are unimodular.In particular, a weak limit of finite graphs with uniformly chosen roots is unimodular: such a limitof finite graphs is referred to as a Benjamini-Schramm limit. It is a major open problem todetermine whether all unimodular random rooted graphs arise as Benjamini-Schramm limits offinite graphs [7, \u00a710]. As mentioned in Section 7.2, we provide a positive solution to this problemin the planar case in the upcoming work [20], proving that every simply connected unimodularrandom rooted planar map is a Benjamini-Schramm limit of finite maps.A common use of the mass transport principle to obtain proofs by contradiction is the following.If (G, \u03c1) is a unimodular random rooted graph and f is a mass transport such that the mass sentout from each vertex\u2211v f(G, u, v) \u2264 M is uniformly bounded almost surely, then almost surelythere are no vertices that receive infinite mass: if vertices receiving infinite mass were to existwith positive probability, the root would be such a vertex with positive probability [7, Lemma 2.3],contradicting the mass transport principle.7.3.2 Random walk, reversibility and ergodicityRecall that the simple random walk on a graph is the Markov chain that chooses Xn+1 fromamong the neighbours of Xn weighted by the number of shared edges. Define G\u2194 (resp.M\u2194) to bespaces of isomorphism classes of graphs (resp. maps) equipped with a bi-infinite path (G, (xn)n\u2208Z),which we endow with a natural variant of the local topology. When (G, \u03c1) is a random graph ormap, we let (Xn)n\u22650 and (X\u2212n)n\u22650 be two independent simple random walks started from \u03c1 andconsider (G, (Xn)n\u2208Z) to be a random element of G\u2194 or M\u2194 as appropriate.1577.3. Background and definitionsA random rooted graph (G, \u03c1) is stationary if (G, \u03c1)d= (G,X1) and reversible if (G, \u03c1,X1)d=(G,X1, \u03c1) as doubly rooted graphs. Equivalently, (G, \u03c1) is reversible if and only if (G, (Xn)n\u2208Z) isstationary with respect to the shift:(G, (Xn)n\u2208Z)d= (G, (Xn+k)n\u2208Z) for every k \u2208 Z.To see this, it suffices to prove that if (G, \u03c1) is a reversible random graph then (X1, \u03c1,X\u22121, X\u22122, . . .)has the law of a simple random walk started from X1. But (\u03c1,X\u22121, . . .) is a simple randomwalk started from \u03c1 independent of X1 and, conditional on (G,X1), reversibility implies that \u03c1 isuniformly distributed among the neighbours of X1, so (X1, \u03c1,X\u22121, X\u22122, . . .) has the law of a simplerandom walk as desired.We remark that if (G, \u03c1) is stationary but not necessarily reversible, it is still possible to extendthe walk to a doubly infinite path (Xn)n\u2208Z so that G is stationary along the path. The differenceis that in the reversible case the past (Xn)n\u22640 is itself a simple random walk with the same law asthe future.Reversibility is related to unimodularity via the following bijection, which is implicit in [7] andproven explicitly in [38]: if (G, \u03c1) is reversible, then biasing by deg(\u03c1)\u22121 (i.e. reweighing the lawof (G, \u03c1) by the Radon-Nikodym derivative deg(\u03c1)\u22121\/E[deg(\u03c1)\u22121]) gives an equivalent unimodularrandom rooted graph, and conversely if (G, \u03c1) is a unimodular random rooted graph with finiteexpected degree, then biasing by deg(\u03c1) gives an equivalent reversible random rooted graph. Thus,the laws of reversible random rooted graphs are in bijection with the laws of unimodular randomrooted graphs for which the root degree has finite expectation.An event A \u2282 G\u2194 is said to be invariant if (G, (Xn)n\u2208Z) \u2208 A implies (G, (Xn+k)n\u2208Z) \u2208A for each k \u2208 Z. A reversible or unimodular random graph is said to be ergodic if the lawof (G, (Xn)n\u2208Z) gives each invariant event probability either zero or one. An event A \u2286 G\u2022 isrerooting-invariant if (G, \u03c1) \u2208 A implies (G, v) \u2208 A for every vertex v of G.Theorem 7.3.1 (Characterisation of ergodicity [7, \u00a74]). Let (G, \u03c1) be a unimodular random rootedgraph with E[deg(\u03c1)] <\u221e (resp. a reversible random rooted graph). The following are equivalent.1. (G, \u03c1) is ergodic.2. Every rerooting-invariant event A \u2286 G\u2022 has probability in {0, 1}.3. The law of (G, \u03c1) is an extreme point of the weakly closed convex set of laws of unimodular(resp. reversible) random rooted graphs.(The equivalence of items 2 and 3 holds for unimodular random rooted graphs without theassumption of finite expected degree.) A consequence of the extremal characterisation is thatevery unimodular random rooted graph is a mixture of ergodic unimodular random rooted graphs,meaning that it may be sampled by first sampling a random law of an ergodic unimodular randomrooted graph, and then sampling from this randomly chosen law - this is known as an ergodicdecomposition and its existence is a consequence of Choquet\u2019s Theorem. In particular, whenever1587.3. Background and definitionswe want to prove that a unimodular random rooted graph with some almost sure property alsohas some other almost sure property, it suffices to consider the ergodic case. The same commentapplies for reversible random rooted graphs.7.3.3 Invariant amenabilityWe begin with a brief review of general amenability, before combining it with unimodularity for thenotion of invariant amenability. We refer the reader to [173, \u00a76] for further details on amenabilityin general, and [7, \u00a78] for invariant amenability.A weighted graph is a graph together with a weight function w : E \u2192 R+. Unweightedmultigraphs may always be considered as weighted graphs by setting w \u2261 1. The weight function isextended to vertices by w(x) =\u2211e3xw(e), and (with a slight abuse of notation) to sets of edges orvertices by additivity. The simple random walk X = (Xn)n\u22650 on a weighted graph is the Markovchain on V with transition probabilities p(x, y) = w(x, y)\/w(x). Here, our graphs are allowed tohave infinite degree provided w(v) is finite for every vertex.The (edge) Cheeger constant of an infinite weighted graph is defined to beiE(G) = inf{w(\u2202EW )w(W ): \u2205 6= W \u2282 V finite}where \u2202EW denotes the set of edges with exactly one end in W . A graph is said to be amenableif its Cheeger constant is zero and non-amenable if it is positive.The Markov operator associated to simple random walk on G is the bounded, self-adjointoperator from L2(V,w) to itself defined by (Pf)(u) =\u2211p(u, v)f(v). The norm of this operator iscommonly known as the spectral radius of the graph. If u, v \u2208 V then the transition probabilitiesare given by pn(u, v) =\u2329Pn1v,1u\/w(u)\u232aw, so that, by Cauchy-Schwarz,pn(u, v) \u2264\u221aw(v)w(u)\u2016P\u2016nw (7.3.1)and in fact \u2016P\u2016w = lim supn\u2192\u221e pn(u, v)1\/n. A fundamental result, originally proved for Cayleygraphs by Kesten [152], is that the spectral radius of a weighted graph is less than one if and onlyif the graph is non-amenable (see [173, Theorem 6.7] for a modern account). As an immediateconsequence, non-amenable graphs are transient for simple random walk.Invariant amenabilityThere are natural notions of amenability and expansion for unimodular random networks due toAldous and Lyons [7]. A percolation on a unimodular random rooted graph (G, \u03c1) is a randomassignment of \u03c9 : E \u222a V \u2192 {0, 1} such that the marked graph (G, \u03c1, \u03c9) is unimodular. We think of\u03c9 as a random subgraph of G consisting of the \u2018open\u2019 edges and vertices \u03c9(e) = 1, \u03c9(v) = 1, andmay assume without loss of generality that if an edge is open then so are both of its endpoints.1597.3. Background and definitionsThe cluster K\u03c9(v) at a vertex v is the connected component of v in \u03c9, i.e. the set of verticesfor which there is a path of open edges to v (by convention, if \u03c9(v) = 0 or if there are no openedges touching v we put K\u03c9(v) = {v}). A percolation is said to be finitary if all of its clusters arefinite almost surely. The invariant Cheeger constant of an ergodic unimodular random rootedgraph (G, \u03c1) is defined to beiinv((G, \u03c1)) = inf{E[ |\u2202EK\u03c9(\u03c1)||K\u03c9(\u03c1)|]: \u03c9 a finitary percolation on (G, \u03c1)}. (7.3.2)The invariant Cheeger constant is closely related to another quantity: mean degrees in finitarypercolations. Let deg\u03c9(\u03c1) denote the degree of \u03c1 in \u03c9 (seen as a subgraph; if \u03c1 6\u2208 \u03c9 we setdeg\u03c9(\u03c1) = 0) and let\u03b1((G, \u03c1)) = sup{E[deg\u03c9(\u03c1)]: \u03c9 a finitary percolation on (G, \u03c1)}.An easy application of the mass transport principle [7, Lemma 8.2] shows that, for any finitarypercolation \u03c9,E[deg\u03c9(\u03c1)] = E[\u2211v\u2208K\u03c9(\u03c1) deg\u03c9(v)|K\u03c9(\u03c1)|].It follows thatE[deg(\u03c1)] = iinv((G, \u03c1)) + \u03b1((G, \u03c1))so that if E[deg(\u03c1)] <\u221e then iinv((G, \u03c1)) is positive if and only if \u03b1((G, \u03c1)) is strictly smaller thanE[deg(\u03c1)].We say that an ergodic unimodular random rooted graph (G, \u03c1) is invariantly amenableif iinv((G, \u03c1)) = 0 and invariantly non-amenable otherwise. Note that this is a property ofthe law of (G, \u03c1) and not of an individual graph. We remark that what we are calling invariantamenability was called amenability when it was introduced by Aldous and Lyons [7, \u00a78]. We qualifyit as invariant to distinguish it from the more classical notion, which we also use below. Whileany invariantly amenable graph is trivially amenable, the converse is generally false. An exampleis a 3-regular tree where each edge is replaced by a path of independent length with unboundeddistribution; see [7] for a more detailed discussion.An important property of invariantly non-amenable graphs was first proved for Cayley graphsby Benjamini, Lyons and Schramm [45]. Aldous and Lyons [7] noted that the proof carriedthrough with minor modifications to the case of invariantly non-amenable unimodular randomrooted graphs, but did not provide a proof. As this property is crucial to our arguments, weprovide a proof for completeness, which the reader may wish to skip. When (G, \u03c1) is an ergodicunimodular random rooted graph, we say that a percolation \u03c9 on G is ergodic if (G, \u03c1, \u03c9) is er-godic as a unimodular random rooted marked graph. The following is stated slightly differentlyfrom both Theorem 3.2 in [45] and Theorem 8.13 in [7].1607.3. Background and definitionsTheorem 7.3.2. Let (G, \u03c1) be an invariantly non-amenable ergodic unimodular random rootedgraph with E[deg(\u03c1)] <\u221e. Then G admits an ergodic percolation \u03c9 so that iE(\u03c9) > 0 and verticesin \u03c9 have uniformly bounded degrees in G.Let us stress that the condition of uniformly bounded degrees is for the degrees in the full graphG, and not the degrees in the percolation.Remark 7.3.3. This theorem plays the same role for invariant non-amenability as Vira\u00b4g\u2019s oceansand islands construction [227] does for anchored expansion [16, 37, 227]. In particular, it gives usa percolation \u03c9 such that the induced network \u03c9\u00af is non-amenable (see the proof of Lemma 7.5.1).Proof. Let \u03c90 be the percolation induced by vertices of G of degree at most M and the edgesconnecting any two such vertices. By monotone convergence, and since \u03b1((G, \u03c1)) < Edeg(\u03c1), wecan take M to be large enough that Edeg\u03c90(\u03c1) > \u03b1((G, \u03c1)). This gives a percolation with boundeddegrees. We shall modify it further to get non-amenability as follows. Fix \u03b4 > 0 by3\u03b4 = E[deg\u03c90(\u03c1)]\u2212 \u03b1((G, \u03c1)).Construct inductively a decreasing sequence of site percolations \u03c9n as follows. Given \u03c9n, let\u03b7n be independent Bernoulli(1\/2) site percolations on \u03c9n, and for each set of vertices W let \u2202\u03c9nE Wdenote the set of edges of \u03c9n in the boundary of W . If K is a finite connected cluster of \u03b7n, withsmall boundary in \u03c9n, we remove it to construct \u03c9n+1. More precisely, let \u03c9n+1 = \u03c9n \\ \u03b3n, where\u03b3n is the subgraph of \u03c9n induced by the vertex set\u22c3{K : K a finite cluster in \u03b7n with |\u2202\u03c9nE (K)| < \u03b4|K|}.Let \u03c9 = \u2229\u03c9n be the limit percolation, which is clearly ergodic. We shall show below that \u03c9 6= \u2205.Any finite connected set in \u03c9 appears as a connected cluster in \u03b7n for infinitely many n. If such aset S has |\u2202\u03c9ES| < \u03b4|S| then it would have been removed at some step, and so \u03c9 has |\u2202\u03c9ES| \u2265 \u03b4|S|for all finite connected S. Since degrees are bounded by M , this implies iE(\u03c9) \u2265 \u03b4\/M > 0.It remains to show that \u03c9 6= \u2205. For some n, and any vertex u, let K(u) be its cluster in \u03b7n.Consider the mass transportfn(u, v) =\uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3deg\u03c9n(v)\/|K(u)| u \u2208 \u03b3n and v \u2208 K(u),E(v,K(u))\/|K(u)| u \u2208 \u03b3n and v \u2208 \u03c9n \\K(u),0 u \/\u2208 \u03b3n.Here E(v,K(u)) is the number of edges between v and K(u). We have that the total mass into vis the difference deg\u03c9n(v) \u2212 deg\u03c9n+1(v) (where the degree is 0 for vertices not in the percolation)while the mass sent from a vertex v \u2208 \u03b3n is twice the number of edges with either end in K(v),1617.3. Background and definitionsdivided by |K(v)|. Applying the mass transport principle we getE[deg\u03c9n(\u03c1)\u2212 deg\u03c9n+1(\u03c1)] = E[\u2211v\u2208K(\u03c1) deg\u03b3n(v) + 2|\u2202\u03c9nE (K(\u03c1))||K(\u03c1)| 1\u03c1\u2208\u03b3n]. (7.3.3)By a second transport, of deg\u03b3n(u)\/|K(u)| from every u \u2208 \u03b3n to each v \u2208 K(u), we see thatE[\u2211v\u2208K(\u03c1) deg\u03b3n(v)|K(\u03c1)| 1\u03c1\u2208\u03b3n]= E[deg\u03b3n(\u03c1)]. (7.3.4)Additionally, on the event {\u03c1 \u2208 \u03b3n}, we have by definition that|\u2202\u03c9nE (K(\u03c1))|\/|K(\u03c1)| \u2264 \u03b4.Plugging these two in (7.3.3) givesE[deg\u03c9n(\u03c1)\u2212 deg\u03c9n+1(\u03c1)] \u2264 E[deg\u03b3n(\u03c1)] + 2\u03b4P(\u03c1 \u2208 \u03b3n).Let \u03b3 = \u222an\u22651\u03b3n, which is a percolation since it is defined as a measurable, automorphism invari-ant function of (G, \u03c1) and the i.i.d. sequence of Bernoulli percolations (\u03b7n). Note the percolations\u03b3n are disjoint, so that the event \u03c1 \u2208 \u03b3n can occur for at most one n and that \u03b3 is a finitarypercolation. Thus E[deg\u03b3(\u03c1)] =\u2211n E[deg\u03b3n(\u03c1)] \u2264 \u03b1((G, \u03c1)). Also,\u2211n P(\u03c1 \u2208 \u03b3n) \u2264 1. Summingover n givesE[deg\u03c90(\u03c1)\u2212 deg\u03c9(\u03c1)] \u2264 \u03b1((G, \u03c1)) + 2\u03b4.The definition of \u03b4 leaves E[deg\u03c9(\u03c1)] \u2265 \u03b4. Thus \u03c9 is indeed non-empty as claimed, completing theproof.7.3.4 Circle packings and vertex extremal lengthRecall that a circle packing C is a collection of discs of disjoint interior in the plane C. Given acircle packing C, we define its tangency map as the map whose embedded vertex set V correspondsto the centres of circles in C and whose edges are given by straight lines between the centres oftangent circles. If C is a packing whose tangency map is isomorphic to G, we call C a packing of G.Theorem 7.3.4 (Koebe-Andreev-Thurston Circle Packing Theorem [156, 221]). Every finite simpleplanar map arises as the tangency map of a circle packing. If the map is a triangulation, the packingis unique up to Mo\u00a8bius transformations of the sphere.The carrier of a circle packing is the union of all the discs in the packing together with thecurved triangular regions enclosed between each triplet of circles corresponding to a face (theinterstices). Given some planar domain D, we say that a circle packing is in D if its carrier is D.1627.3. Background and definitionsTheorem 7.3.5 (Rigidity for Infinite Packings, Schramm [206]). Let G be a triangulation, circlepacked in either C or D. Then the packing is unique up to Mo\u00a8bius transformations preserving of Cor D respectively.It is often fruitful to think of packings in D as being circle packings in (the Poincare\u00b4 discmodel of) the hyperbolic plane. The uniqueness of the packing in D up to Mo\u00a8bius transformationsmay then be stated as uniqueness of the packing in the hyperbolic plane up to isometries of thehyperbolic plane.The vertex extremal length, defined in [121], from a vertex to infinity on an infinite graphG is defined to beVELG(v,\u221e) = supminf\u03b3:v\u2192\u221em(\u03b3)2\u2016m\u20162 , (7.3.5)where the supremum is over measuresm on V (G) such that \u2016m\u20162 = \u2211m(u)2 <\u221e, and the infimumis over paths from v to \u221e in G. A connected graph is said to be VEL parabolic if VEL(v \u2192\u221e) =\u221e for some vertex v (and hence for any vertex) and VEL hyperbolic otherwise. The VELtype is monotone in the sense that subgraphs of VEL parabolic graphs are also VEL parabolic. Asimple random walk on any VEL hyperbolic graph is transient. For graphs with bounded degreesthe converse also holds: Transient graphs with bounded degrees are VEL hyperbolic [121].Theorem 7.3.6 (He-Schramm [121, 122]). Let G be a one-ended, infinite, simple planar triangu-lation. Then G may be circle packed in either the plane C or the unit disc D, according to whetherit is VEL parabolic or hyperbolic respectively.The final classical fact about circle packing we will need is the following quantitative version,due to Hansen [116], of the Ring Lemma of Rodin and Sullivan [200], which will allow us to controlthe radii along a random walk.Theorem 7.3.7 (The Sharp Ring Lemma [116]). Let u and v be two adjacent vertices in a circlepacked triangulation, and r(u), r(v) the radii of the corresponding circles. There exists a universalpositive constant C such thatr(v)r(u)\u2264 eC deg(v).Measurability of Circle PackingAt several points throughout the paper, we will want to define mass transports in terms of circlepackings. In order for these to be measurable functions of the graph, we require measurability ofthe circle packing. Let (G, u, v) be a doubly rooted triangulation and let C(G, u, v) be the uniquecircle packing of G in D or C such that the circle corresponding to u is centred at 0, the circlecorresponding to v is centered on the positive real line and, in the parabolic case, the root circlehas radius one.Let Gk be an exhaustion of G by finite induced subgraphs with no cut-vertices and such thatthe complements G \\Gk are connected. Such an exhaustion exists by the assumption that G is a1637.4. Characterisation of the CP typeFigure 7.2: Circle packing induces an embedding of a triangulation with either hyperbolic orEuclidean geodesics, depending on CP type. By rigidity (Theorem 7.3.5), the angles between pairsof adjacent edges do not depend on the choice of packing.one-ended triangulation. Form a finite triangulation G\u2217k by adding an extra vertex \u2202k and an edgefrom \u2202k to each boundary vertex of Gk.Consider first the case when G is CP hyperbolic. By applying a Mo\u00a8bius transformation tosome circle packing of G\u2217k, we find a unique circle packing C\u2217k of G\u2217k in C\u221e such that the circlecorresponding to u is centred at the origin, the circle corresponding to v is centred on the positivereal line and \u2202k corresponds to the unit circle \u2202D. In the course of the proof of the He-SchrammTheorem, it is shown that this sequence of packings converges to the unique packing of G in D,normalised so that the circle corresponding to u is centred at 0 and the circle corresponding to vis centred on the positive real line.As a consequence, the centres and radii of the circles of C(G, u, v) are limits as r \u2192 \u221e of thecentre and radius of a graph determined by the ball of radius r around u. In particular, they arepointwise limits of continuous functions (with respect to the local topology on graphs) and henceare measurable.The hyperbolic radii are particularly nice to consider here. Since the circle packing in D isunique up to isometries of the hyperbolic plane (Mo\u00a8bius maps), the hyperbolic radii do not dependon the choice of packing, and we find that rh(v) is a function of (G, v).In the CP parabolic case, the same argument works except that the packing C\u2217k of G\u2217k must bechosen to map u to the unit circle and \u2202k to a larger circle also centred at 0.1647.4. Characterisation of the CP type7.4 Characterisation of the CP typeProof of Theorem 7.1.1. Since (G, \u03c1) is ergodic, and since the CP type does not depend on thechoice of root, the CP type of G is not random. We first relate the circle packing type to theaverage degree. Suppose (G, \u03c1) is CP hyperbolic and consider a circle packing of G in the unit disc.Embed G in D by drawing the hyperbolic geodesics between the hyperbolic centres of the circles inits packing, so that each triangle of G is represented by a hyperbolic triangle (see Figure 7.2). It iseasy to see that this is a proper embedding of G. By rigidity of the circle packing (Theorem 7.3.5),this drawing is determined by the isomorphism class of G, up to isometries of the hyperbolic plane.Define a mass transport as follows. For each face (u, v, w) of the triangulation with angle \u03b2 atu, transport \u03b2 from u to each of u, v, w. If u and v are adjacent, the transport from u to v hascontributions from both faces containing the edge, and the transport from u to itself has a termfor each face containing u. By rigidity (Theorem 7.3.5), these angles are independent of the choiceof circle packing, so that the mass sent from u to v is a measurable function of (G, u, v).For each face f of G, let \u03b8(f) denote the sum of the internal angles in f in the drawing. Thesum of the angles of a hyperbolic triangle is pi minus its area, so \u03b8(f) < pi for each face f . Eachvertex u sends each angle 3 times, for a total mass out of exactly 6pi. A vertex receives mass\u2211f :u\u2208f\u03b8(f) < pi deg(u).Applying the mass transport principle,6pi < piE[deg(\u03c1)].Thus if G is CP hyperbolic then E[deg(\u03c1)] > 6.In the CP parabolic case, we may embed G in C by drawing straight lines between the centresof the circles in its packing in the plane. By rigidity, this embedding is determined up to translationand scaling, and in particular all angles are determined by G. Since the sum of angles in a Euclideantriangle is pi, the same transport as above applied in the CP parabolic case shows that E[deg(\u03c1)] = 6.We now turn to amenability. Euler\u2019s formula implies that the average degree of any finite simpleplanar graph is at most 6. It follows that\u03b1((G, \u03c1)) = sup\uf8f1\uf8f2\uf8f3E[\u2211v\u2208K\u03c9(\u03c1) deg\u03c9(v)|K\u03c9(\u03c1)|]: \u03c9 a finitary percolation\uf8fc\uf8fd\uf8fe \u2264 6.If G is CP hyperbolic then E[deg(\u03c1)] > 6, so that \u03b1((G, \u03c1)) < E[deg(\u03c1)] and (G, \u03c1) is invariantlynon-amenable.Conversely, suppose G is invariantly non-amenable. By Theorem 7.3.2, G almost surely admitsa percolation \u03c9 which has positive Cheeger constant and bounded degrees. Such an \u03c9 is transientand since it has bounded degree it is also VEL hyperbolic. By monotonicty of the vertex extremal1657.4. Characterisation of the CP typelength, G is almost surely VEL hyperbolic as well. The He-Schramm Theorem then implies thatG is almost surely CP hyperbolic.Remark 7.4.1. In the hyperbolic case, let Area(u) be the total area of the triangles surrounding uin its drawing. Since the angle sum in a hyperbolic triangle is pi minus its area, the mass transportthat gives average degree greater than 6 in the hyperbolic case also givesE[deg(\u03c1)] = 6 +1piE[Area(\u03c1)],which relates the expected degree to the density of the circle packing.7.4.1 Completing the proof of the Benjamini-Schramm TheoremIn this section we complete our new proof of the following theorem of Benjamini and Schramm.Theorem 7.4.2 ([50]). Let (G, \u03c1) be a weak local limit of finite planar graphs Gn and suppose thatG has bounded degrees almost surely. Then (G, \u03c1) is almost surely recurrent.Recall that the number of ends of a graph G is the supremum over finite sets K of the numberof infinite connected components of G \\K. As explained in [50], it suffices to prove Theorem 7.4.2when the graphs Gn are simple triangulations. In this case Proposition 7.1.2 implies a specialcase of the Benjamini-Schramm Theorem: If (G, \u03c1) is a simple one-ended triangulation that is aBenjamini-Schramm limit of finite planar triangulations of uniformly bounded degree, then (G, \u03c1)is recurrent almost surely. At the time, this was the most difficult case.Thus, to complete the proof of Theorem 7.4.2 we need to consider the case in which the limit(G, \u03c1) has multiple ends. We describe below two different methods to handle this case.Method 1. This proof considers separately three cases, depending on the number of ends of G.First, by combining Proposition 6.10 and Theorem 8.13 of [7], we have the following.Proposition 7.4.3 ([7]). Let (G, \u03c1) be an ergodic unimodular random rooted graph. Then G hasone, two or infinitely many ends almost surely. If (G, \u03c1) has infinitely many ends almost surely, itis invariantly non-amenable.We rule out the case of infinitely many ends by showing that local limits of finite planar graphsare invariantly amenable. Recall the celebrated Lipton-Tarjan Planar Separator Theorem [167,Theorem 2] (which can also be proved using circle packing theory [182]).Theorem 7.4.4 ([167]). There exists a universal constant C such that for every m and every finiteplanar graph G, there exists a set S \u2282 V (G) of size at most Cm\u22121\/2|G| such that every connectedcomponent of G \\ S contains at most m vertices.Corollary 7.4.5. Let (G, \u03c1) be the local limit of a sequence of finite planar maps Gn and supposeE[deg(\u03c1)] <\u221e. Then (G, \u03c1) is invariantly amenable and hence has at most two ends.1667.4. Characterisation of the CP typeProof. Let \u03c9mn be a subset of V (Gn) such that Gn \\ \u03c9mn has size at most Cm\u22121\/2|Gn| and everyconnected component of \u03c9mn has size at most m. The sequence (Gn, \u03c1n, \u03c9mn ) is tight and thereforehas a subsequence converging to (G, \u03c1, \u03c9m) for some finitary percolation \u03c9m on (G, \u03c1). Since it isa limit of percolations on finite graphs with a uniform root, the limit is unimodular.We have thatP(\u03c1 \u2208 \u03c9m) \u2265 1\u2212 Cm\u22121\/2 \u2212\u2212\u2212\u2212\u2192m\u2192\u221e 1.Similarly, P(X1 \u2208 \u03c9m)\u2192 1. By integrability of deg(\u03c1), we have thatE[deg\u03c9m(\u03c1)] = E[1(\u03c1,X1 \u2208 \u03c9m) deg(\u03c1)]\u2192 E[deg(\u03c1)].Thus \u03b1((G, \u03c1)) = E[deg(\u03c1)] and hence (G, \u03c1) is invariantly amenable.Finally, we deal with the two-ended case.Proposition 7.4.6. Let (G, \u03c1) be a unimodular random rooted graph with exactly two ends almostsurely and suppose E[deg(\u03c1)] <\u221e. Then G is recurrent almost surely.Proof. We prove the equivalent statement for (G, \u03c1) reversible. We may also assume that (G, \u03c1)is ergodic. Say that a finite set S disconnects G if G \\ S has two infinite components. Since Gis two-ended almost surely, such a set S exists and each infinite component of G \\ S is necessarilyone-ended. We call these two components G1 and G2. Suppose for contradiction that G is transientalmost surely. In this case, a simple random walk Xn eventually stays in one of the Gi, and hencethe subgraph induced by this Gi must be transient.Now, since G is two-ended almost surely, there exist R and M such that, with positive proba-bility, the ball BR(Xn) disconnects G and |BR(Xn)| \u2264M . By the Ergodic Theorem this occurs forinfinitely many n almost surely. On the event that Xn eventually stays in Gi, since Gi is one-ended,this yields an infinite collection of disjoint cutsets of size at most M separating \u03c1 from infinity inGi. Thus, Gi is recurrent by the Nash-Williams criterion [173], a contradiction.Theorem 7.4.2 now follows by combining Theorem 7.1.1, Corollary 7.4.5, and Proposition 7.4.6.Method 2. This proof reduces Theorem 7.4.2 to Theorem 7.1.1 by taking universal covers. Givena (not necessarily planar) map M , a cover of M is a map M\u02dc together with a surjective graphhomomorphism pi : M\u02dc \u2192 M , such that for each vertex v, the homomorphism pi maps the edgesadjacent to v bijectively to the edges adjacent to pi(v) and preserves their cyclic ordering, and suchthat for each face f , pi maps the edges adjacent to f bijectively to the edges adjacent to pi(f). Theuniversal cover of M is a cover pi : M\u02dc \u2192M such that M\u02dc is simply connected. If M is drawn on asurface S, the universal cover M\u02dc of M may be constructed by taking every lift of every edge of Min S to the universal cover S\u02dc of S (see e.g. [119] for the topological notions of universal cover andpath lifting). Alternatively, the universal cover M\u02dc may be constructed directly as in [215]. Theuniversal cover is unique in the sense that if pi\u2032 : M\u02dc \u2032 \u2192M is also a universal cover of M then there1677.5. Boundary theoryexists an isomorphism of maps f : M\u02dc \u2032 \u2192 M\u02dc such that pi\u2032 = pi \u25e6 f . Note that if a cover pi : M\u02dc \u2192Mis a cover of a map M and M\u02dc is recurrent, the projection Xn = pi(X\u02dcn) of a simple random walkX\u02dcn on M\u02dc is a simple random walk on M , and it follows that M is also recurrent.Let (M,\u03c1) be a unimodular random rooted map with universal cover pi : M\u02dc \u2192 M . Let \u03c1\u02dc bechosen arbitrarily from the preimage pi\u22121(\u03c1); The isomorphism class of the rooted map (M\u02dc, \u03c1\u02dc) doesnot depend on this choice. We claim that the random rooted map (M\u02dc, \u03c1\u02dc) is unimodular. To seethis, recall that a random rooted graph is unimodular if and only if it is involution invariant [7,Proposition 2.2], meaning thatE\u2211v\u2208Vf(G, \u03c1, v) = E\u2211v\u2208Vf(G, v, \u03c1)whenever f is a mass-transport such that f(G, u, v) is zero unless u and v are adjacent in G.The equivalence of unimodularity and involution invariance extends immediately to random rootedmaps. Given such an f :M\u2022\u2022 \u2192 [0,\u221e], let g :M\u2022\u2022 \u2192 [0,\u221e] be defined to beg(M,u, v) =\u2211e: e\u2212=u, e+=vf(M\u02dc, e\u02dc\u2212, e\u02dc+),where pi : M\u02dc \u2192 M is the universal cover of M and e\u02dc is an arbitrary element of pi\u22121(e) for eachoriented edge e of M (by uniqueness of the universal cover, the value of g does not depend on thischoice). Then g is a mass transport and, letting V\u02dc denote the vertex set of M\u02dc , we have\u2211v\u2208Vg(M,\u03c1, v) =\u2211v\u02dc\u2208V\u02dcf(M\u02dc, \u03c1\u02dc, v\u02dc) and\u2211v\u2208Vg(M,v, \u03c1) =\u2211v\u02dc\u2208V\u02dcf(M\u02dc, v\u02dc, \u03c1\u02dc),so that we deduce involution invariance of (M\u02dc, \u03c1\u02dc) from involution invariance of (M,\u03c1). Furthermore,if (M,\u03c1) is ergodic then (M\u02dc, \u03c1\u02dc) is also ergodic. Indeed, for every invariance event A \u2286 M\u2022, theevent {(G, \u03c1) \u2208 M\u2022 : (G\u02dc, \u03c1\u02dc) \u2208 M\u2022} is also invariant to changing the root, and it follows that if(M,\u03c1) is ergodic then (M\u02dc, \u03c1\u02dc) is also ergodic.Alternative proof of Theorem 7.4.2. Let (G, \u03c1) be a simple, bounded degree, ergodic unimodularrandom rooted triangulation with E[deg(\u03c1)] = 6. The universal cover (G\u02dc, \u03c1\u02dc) of (G, \u03c1) has all theseproperties and is also one-ended, so that G\u02dc is CP parabolic almost surely by Theorem 7.1.1. Bythe He-Schramm Theorem [121], G\u02dc is recurrent almost surely, and so G is also recurrent almostsurely, completing the proof of the Benjamini-Schramm Theorem.7.5 Boundary theoryRecall that given a G and a vertex v we write PGv and EGv to denote the probability and expectationwith respect to random walk (Xn)n\u22650 on G started from v.1687.5. Boundary theory7.5.1 Convergence to the boundaryLet (G, \u03c1) be a one-ended, simple, CP hyperbolic reversible random triangulation. Recall that fora CP hyperbolic G with circle packing C in D, we write r(v) and z(v) for the Euclidean radius andcentre of the circle corresponding to the vertex v in C and zh(v), rh(v) for the hyperbolic centreand radius.Our first goal is to show that the Euclidean radii r(Xn) decay exponentially along a randomwalk (Xn). We initially prove only a bound, and will prove the existence of the limit rate of decaystated in Theorem 7.1.4 only after we have proven the exit measure is non-atomic.Lemma 7.5.1. Let (G, \u03c1) be a CP hyperbolic reversible random rooted triangulation with E[deg(\u03c1)] <\u221e and let C be a circle packing of G in the unit disc. Let (Xn)n\u22650 be a simple random walk on Gstarted from \u03c1. Then almost surelylim supn\u2192\u221elog r(Xn)n< 0.Proof. We may assume that (G, \u03c1) is ergodic, else we may take an ergodic decomposition. ByTheorem 7.1.1 (G, \u03c1) is invariantly non-amenable. By Theorem 7.3.2, there is an ergodic percolation\u03c9 on G such that deg(v) is bounded by some M for all v \u2208 \u03c9 and iE(\u03c9) > 0 almost surely.Recall the notion of an induced random walk on \u03c9: let Nm be the mth time X is in \u03c9 (thatis, N0 = inf{n \u2265 0 : Xn \u2208 \u03c9} and inductively Nm+1 = inf{n > Nm : Xn \u2208 \u03c9}). The inducednetwork \u03c9\u00af is defined to be the weighted graph on the vertices of \u03c9 with edge weights given byw\u00af(u, v) = deg(u)PGu (XN1 = v)so that XNm is the random walk on the weighted graph \u03c9\u00af. Note that \u03c9\u00af may have non-zero weightsbetween vertices which are not adjacent in G, so that \u03c9\u00af is no longer a percolation on G, and maynot even be planar.We first claim that \u03c9\u00af with the weights w\u00af of the induced random walk also has positive Cheegerconstant. Indeed, the weight of a vertex v \u2208 \u03c9\u00af is just its degree in G and so is between 1 and Mfor any vertex. The edge boundary in \u03c9\u00af of a set K is at least the number of edges connecting Kto V \\ K in \u03c9. Thus iE(\u03c9\u00af) \u2265 iE(\u03c9)\/M > 0. It follows that the induced random walk on \u03c9 hasspectral radius less than one [173].Now, as in (7.3.1), Cauchy-Schwarz gives that, for some c > 0,PG\u03c1 (XNm = v) \u2264M1\/2 exp(\u2212cm)for every vertex v almost surely. Since the total area of all circles in the packing is at most pi, with1697.5. Boundary theoryc as above, there exists at most ecm\/2 circles of radius greater than e\u2212cm\/4 for each m. HencePG\u03c1(r(XNm) \u2265 e\u2212cm\/4)=\u2211v:r(v)\u2265e\u2212cm\/4PG\u03c1 (XNm = v)\u2264\u2223\u2223\u2223\u2223{v : r(v) \u2265 e\u2212cm\/4}\u2223\u2223\u2223\u2223 \u00b7M1\/2e\u2212cm\u2264M1\/2e\u2212cm\/2.These probabilities are summable, and so Borel-Cantelli implies that almost surely for large enoughm,r(XNm) \u2264 e\u2212cm\/4.That is, we have exponential decay of the radii for the induced walk:lim supm\u2192\u221elog r(XNm)m\u2264 \u2212 c4. (7.5.1)It remains to prove that the exponential decay is maintained between visits to \u03c9. By stationarityand ergodicity of (G, \u03c1, \u03c9), the density of visits to \u03c9 is P(\u03c1 \u2208 \u03c9) 6= 0. That is,limm\u2192\u221eNmm= P(\u03c1 \u2208 \u03c9)\u22121almost surely. In particular,lim supm\u2192\u221elog r(XNm)Nm< 0. (7.5.2)Given n, let m be the number of visits to \u03c9 up to time n, so that Nm \u2264 n < Nm+1. Sincem\/Nm converges, n\/Nm \u2192 1. By the Sharp Ring Lemma (Theorem 7.3.7),r(Xn)r(XNm)\u2264 exp\uf8eb\uf8edC n\u2211i=Nmdeg(Xi)\uf8f6\uf8f8so thatlog r(Xn)n\u2264 log r(XNm)n+Cnn\u2211i=Nmdeg(Xi). (7.5.3)Again, the Ergodic Theorem gives us the almost sure limitlim1nn\u2211i=0deg(Xi) = E[deg(\u03c1)].1707.5. Boundary theoryThus almost surelylim1nn\u2211i=Nmdeg(Xi) = lim1nn\u2211i=0deg(Xi)\u2212 lim Nmnlim1NmNm\u2211i=0deg(Xi)= E[deg(\u03c1)]\u2212 E[deg(\u03c1)] = 0.Combined with (7.5.2) and (7.5.3) we getlim supn\u2192\u221elog r(Xn)n= lim supn\u2192\u221elog r(XNm)n= limn\u2192\u221eNmnlim supn\u2192\u221elog r(XNm)Nm< 0.Proof of Theorem 7.1.3, item 1. We prove the equivalent statement for (G, \u03c1) reversible with E[deg(\u03c1)] <\u221e, putting us in the setting of Lemma 7.5.1. The path formed by drawing straight lines betweenthe Euclidean centres of the circles along the random walk path has length r(\u03c1) + 2\u2211i\u22651 r(Xi),which is almost surely finite by Lemma 7.5.1. It follows that the sequence of Euclidean centres isCauchy almost surely and hence converges to some point, necessarily in the boundary. Because theradii of the circles r(Xn) converge to zero almost surely, the hyperbolic centres must also convergeto the same point.7.5.2 Full support and non-atomicity of the exit measureWe now prove item 2 of Theorem 7.1.3, which states that the exit measure on the unit circle hasfull support and no atoms almost surely. We start with a general observation regarding atoms inboundaries of stationary graphs.We say that two metrics d1 and d2 on the vertex set V of a graph G are compatible if theidentity map from V to itself extends to an isomorphism between the completions of the metricspaces (V, d1) and (V, d2) (or, equivalently, if the same sequences are Cauchy for d1 and d2). Forexample, the Euclidean distances between centres of circles corresponding to vertices in differentcircle packings of a one-ended planar triangulation in either the full plane or the unit disc arecompatible by Theorem 7.3.5. We define a compatible family of metrics to be a Borel functiond = dGu (v, w) from the space of triply rooted graphs (i.e., the set of isomoprhism classes of graphswith an ordered triple of distinguished vertices, equipped with an appropriate variant of the localtopology) to the positive reals such that for every locally finite, connected graph G = (V,E),1. dGu (\u00b7, \u00b7) is a metric on V for every vertex u of G, and2. the metrics dGu and dGv are compatible for each two vertices u and v of G.Given a compatible family of metrics d and a rooted graph (G, \u03c1), the completion V\u00af of V withrespect to dG\u03c1 has a topology that does not depend on the choice of root vertex \u03c1. Such a completionis called an invariant completion. Compatible families of metrics for maps are defined similarly.1717.5. Boundary theoryIn some cases of interest, a compatible family of metrics d might only be defined for graphs or mapsin some rerooting-invariant class (e.g. circle packings are defined for the class of one-ended simpleplanar triangulations). In this case, we may extend d arbitrarily to all graphs or maps by settingit to be the discrete metric where it is not defined.Lemma 7.5.2. Let d be a compatible family of metrics, let (G, \u03c1) be a stationary random rootedgraph or map, and let V\u00af be the completion of V with respect to dG\u03c1 . Suppose that the random walkon G converges almost surely to a point in the boundary \u2202V = V\u00af \\ V . Then the exit measure on\u2202V is either trivial (concentrated on a single point) or non-atomic almost surely.For each CP hyperbolic one-ended simple planar triangulation G, take a circle packing of G inD, normalized so that the circle corresponding to u is centred at the origin, and let d = dGu (v, w)be the Euclidean distance between the Euclidean centres of the circles corresponding to v and w.By circle packing rigidity (Theorem 7.3.5), this circle packing is unique up to rotations, so that themetric dGu is well defined. The metrics dGu and dGv are also compatible for every pair of vertices uand v in G. Thus, after an arbitrary extension to other maps, d is an compatible family of metrics.Another natural example, defined for all graphs, is the Martin compactification. As a conse-quence of this lemma, for any stationary random graph, the exit measure on the Martin boundary(which can be defined as the completion of V with respect to a compatible family of metrics, seee.g. [229]) is almost surely either non-atomic or trivial. In particular, this gives an alternative proofof a recent result of Benjamini, Paquette and Pfeffer [37], which states that for every stationaryrandom graph, the space of bounded harmonic functions on the graph is either one dimensional orinfinite dimensional almost surely. A straightforward extension of this lemma applies to randomfamilies of metrics.Proof. Condition on (G, \u03c1). For each atom \u03be of the exit measure, define the harmonic functionh\u03be(v) = PGv (limXn = \u03be). By Le\u00b4vy\u2019s 0-1 law,h\u03be(Xn)a.s.\u2212\u2212\u2192 1(limXn = \u03be)for each atom \u03be andPGXn(limXn is an atom) =\u2211\u03beh\u03be(Xn)a.s.\u2212\u2212\u2192 1(limXn is an atom).Define M(G, v) = max\u03be h\u03be(v) to be the maximal atom size. Since the topology of V\u00af does notdepend on the choice of root, the sequence M(G,Xn) is stationary. Combining the two abovelimits, we find the almost sure limitM(G,Xn)a.s.\u2212\u2212\u2192 1(limXn is an atom).Since M(G,Xn) is a stationary sequence with limit in {0, 1}, it follows that M(G, \u03c1) \u2208 {0, 1} almostsurely. That is, either there are no atoms in the exit measure or there is a single atom with weight1727.5. Boundary theory1 almost surely.Proof of Theorem 7.1.3, item 2. We may assume that (G, \u03c1) is ergodic. Applying Lemma 7.5.2 tothe deg(\u03c1)-biasing of (G, \u03c1), we deduce that the exit measure has at most a single atom. Next, werule out having a single atom. Suppose for contradiction that there is a single atom \u03be = \u03be(C) almostsurely for some (and hence every) circle packing C of G in D. Applying the Mo\u00a8bius transformation\u03a6(z) = \u2212iz + \u03bez \u2212 \u03be ,which maps D to the upper half-plane H = {=(z) > 0} and \u03be to \u221e, gives a circle packing of G inH such that the random walk tends to \u221e almost surely. Since circle packings in H are unique upto Mo\u00a8bius transformations and the boundary point \u221e is determined by the graph G, such a circlepacking in H is unique up to Mo\u00a8bius transformations of the upper half-plane that fix \u221e, namelyaz + b with real a \u2265 0 and b (translations and dilations).Inverting around the atom has therefore given us a way of canonically endowing G with Eu-clidean geometry: if we draw G in H using straight lines between the Euclidean centres of thecircles in the half-plane packing, the angles at the corners around each vertex u are independentof the original choice of packing C. Transporting each angle from u to each of the three verticesforming the corresponding face f as in the proof of Theorem 7.1.1 implies that E[deg(\u03c1)] = 6. Thiscontradicts Theorem 7.1.1 and the assumption that (G, \u03c1) is CP hyperbolic almost surely. Thiscompletes the proof that the exit measure is non-atomic.To finish, we show that the exit measure has support \u2202D. Suppose not. We will define a masstransport on G in which each vertex sends a mass of at most one but some vertices receive infinitemass, contradicting the mass transport principle.Consider the complement of the support of the exit measure, which is a union of disjoint openintervals\u22c3i\u2208I(\u03b8i, \u03c8i) in \u2202D. Since the exit measure is non-atomic, \u03b8i 6= \u03c8i mod 2pi for all i.For each such interval (\u03b8i, \u03c8i), let \u03b3i be the hyperbolic geodesic from ei\u03b8i to ei\u03c8i . That is, \u03b3i isthe intersection with D of the circle passing through both ei\u03b8i and ei\u03c8i that intersects \u2202D at rightangles. Let Ai be the set of vertices such that the circle corresponding to v is contained in the regionto the right of \u03b3i, i.e. bounded between \u03b3i and the boundary interval (\u03b8i, \u03c8i) (see Figure 7.3(a)).Each vertex is contained in at most one such Ai. For each vertex u in Ai, consider the hyperbolicgeodesic ray \u03b3u from the hyperbolic centre zh(u) to ei\u03b8i . Define a mass transport by sending massone from u \u2208 Ai to the vertex v corresponding to the first circle intersected by both \u03b3u and \u03b3i.There may be no such circle, in which case no mass is sent from u. Since the transport is definedin terms of the hyperbolic geometry and the support of the exit measure, it is a function of theisomorphism class of (G, u, v) by Theorem 7.3.5.Let \u03c6 \u2208 (\u03b8i, \u03c8i) and consider the set of vertices whose corresponding circles intersect both \u03b3iand the geodesic \u03b3\u03c6 from ei\u03c6 to ei\u03b8i . As \u03c6 increases from \u03b8i to \u03c8i, this set is increasing. It followsthat for each fixed v for which the circle corresponding to v intersects \u03b3i, the set Bv of \u03c6 \u2208 (\u03b8i, \u03c8i)1737.5. Boundary theory\u03c6i\u03b8i\u03b3i(a) A geodesic \u03b3i is drawn over each componentof the complement of the support. Circles con-tained in the shaded area are in Ai.\u03c6iBv\u03b8iv(b) The vertex v receives mass from circles withhyperbolic centres in the shaded area.Figure 7.3: An illustration of the mass transport used to show the exit measure has full support.for which the circle corresponding to v is the first circle intersected by \u03b3\u03c6 that also intersects \u03b3i isan interval (see Figure 7.3(b)).Since there are only countably many vertices, Bv must have positive length for some v. Thusthere is an open neighbourhood of the boundary in which all the circles send mass to this vertex.This vertex therefore receives infinite mass, contradicting the mass transport principle.7.5.3 The unit circle is the Poisson boundaryThe Poisson-Furstenberg boundary [91\u201393] (or simply the Poisson boundary) of a graph (or moregenerally, of a Markov chain) is a formal way to encode the asymptotic behaviour of random walkson G. We refer the reader to [142, 173, 195] for more detailed introductions.Recall that a function h : V (G)\u2192 R is said to be harmonic ifh(v) =1deg(v)\u2211u\u223cvh(u)for all v \u2208 V (G) \u2014 or, equivalently, if h(Xn) is a martingale. Let G\u00af = G\u222a\u2202G be a compactificationof G so that the random walk Xn converges almost surely. For each v \u2208 V (G) we let PGv denote thelaw of the limit of the random walk started at v. Every bounded Borel function g on \u2202G extendsto a harmonic functionh(v) := EGv[g(limXn)]on G. Such a compactification is called a realisation of the Poisson boundary of G if every1747.5. Boundary theorybounded harmonic funtion h on G may be represented as an extension of a boundary function inthis way.Harmonic functions can be used to encode asymptotic behaviour of the random walk as follows.Let GN be the space of sequences in G. The shift operator on GN is defined by \u03b8(x0, x1, . . . ) =(x1, x2, . . . ), and we write I for the \u03c3-algebra of shift-invariant events A = \u03b8A. Be careful to notethe distinction between invariant events for the random walk on G, just defined, and invariantevents for the sequence (G, (Xn+k)n\u2208Z)k\u2208Z as defined in Section 7.3.2.There is an isomorphism between the space of bounded harmonic functions on G and L\u221e(GN, I)given byh 7\u2192 g(x1, x2, . . . ) = limn\u2192\u221eh(xn), g 7\u2192 h(v) = EGv[g(v,X1, X2, . . . )].The limit here exists PGv -almost surely by the bounded Martingale Convergence Theorem, whilethe fact that these two mappings are inverses of one another is a consequence of Le\u00b4vy\u2019s 0-1 Law:If h(v) = EGv [g(X)] is the harmonic extension of some invariant function g, thenh(Xn)a.s.\u2212\u2212\u2192 g(\u03c1,X1, X2, . . . ). (7.5.4)As a consequence of this isomorphism, and since the span of simple functions is dense in L\u221e,the topological boundary \u2202G is a realisation of the Poisson boundary of G if and only if for everyinvariant event A there exists a Borel set B \u2282 \u2202G such that the symmetric difference A\u2206{limXn \u2208B} is PGv -null.For example, the Poisson boundary of a tree may be realised as its space of ends, and the one-point compactification of a transient graph G gives rise to a realisation of the Poisson boundaryif and only if G is Liouville (i.e. the only bounded harmonic functions on G are constant). ThePoisson boundary of any graph may be realised as the graph\u2019s Martin boundary [229], but this isnot always the most natural construction.Our main tools for controlling harmonic functions will be Le\u00b4vy\u2019s 0-1 Law and the followingconsequence of the Optional Stopping Theorem. For a set W \u2282 V of vertices, let TW be thefirst time the random walk visits W . If h is a positive, bounded harmonic function, the OptionalStopping Theorem impliesh(v) \u2265 EGv [h(XTW )1TW<\u221e] \u2265 PGv (Hit W ) inf{h(u) : u \u2208W}. (7.5.5)Lemma 7.5.3. Let (G, \u03c1) be a CP hyperbolic reversible random rooted triangulation with E[deg(\u03c1)] <\u221e, and let (Xn)n\u2208Z be the reversible bi-infinite random walk. Then almost surelyPGXn(hit {X\u22121, X\u22122, . . . })\u2192 0.Proof. Let C be a circle packing of G in D. Recall from Theorem 7.1.3, item 1 that for a random1757.5. Boundary theorywalk Xn, almost surely \u039e := lim z(Xn) exists, and its law is non-atomic and of full support on \u2202D.Since the exit measure is non-atomic, the limit points \u039e+ := lim z(Xn) and \u039e\u2212 := lim z(X\u2212n) arealmost surely distinct.Let {Ui}i\u2208I be a countable basis for the topology of \u2202D (say, intervals with rational endpoints)and for each i let hi be the harmonic functionhi(v) = PGv (\u039e \u2208 Ui) .By Le\u00b4vy\u2019s 0-1 law, hi(Xn)\u2192 1(\u039e+ \u2208 Ui) for every i almost surely. Thus there exists some i0 with\u039e\u2212 \u2208 Ui0 and \u039e+ \/\u2208 Ui0 . In particular there is almost surely some bounded harmonic functionh = hi0 \u2265 0 with h(Xn) \u2212\u2212\u2212\u2192n\u2192\u221e 0 anda := inf{h(X\u2212m) : m > 0} > 0.By (7.5.5)h(Xn) \u2265 a \u00b7 PGXn(hit {X\u22121, X\u22122, . . . }).Since h(Xn)\u2192 0, we almost surely havePGXn(Hit {X\u22121, X\u22122, . . . })\u2192 0.Proof of Theorem 7.1.3, item 3. We prove the equivalent statement for (G, \u03c1) reversible with E[deg(\u03c1)] <\u221e, and may assume that (G, \u03c1) is ergodic.We need to prove that for every invariant event A for the simple random walk on G withPG\u03c1 (A) > 0, there is a Borel set B \u2282 \u2202D such thatPG\u03c1(A\u2206 {\u039e+ \u2208 B})= 0,where \u039e+ = lim z(Xn). Let h be the harmonic function h(v) = PGv (A), and let B be the set of\u03be \u2208 \u2202D such that there exists a path (\u03c1, v1, v2, . . . ) in G such that for some c > 0,h(vi)\u2192 1, z(vi)\u2192 \u03be, and |\u03be \u2212 z(vi)| < 2e\u2212ci,where |\u00b7| denotes Euclidean length. The condition on exponential decay of |\u03be\u2212z(vi)| can be omittedby invoking the theory of universally measurable sets. We are spared from this by Lemma 7.5.1.With an explicit rate of convergence, it is straightforward to see that B is Borel: Let Bc,m,\u03b5,n bethe open set of \u03be \u2208 \u2202D such that there exists a path \u03c1, v1, . . . , vn in G such that h(vi) > 1 \u2212 \u03b5 forevery i \u2265 m, and with |\u03be\u2212z(vi)| < 2e\u2212ci. Then B =\u22c3c\u22c2\u03b5\u22c3m\u22c2nBc,m,\u03b5,n, where m,n are integersand c, \u03b5 are positive rationals, and it follows that B is Borel.If the random walk has (\u03c1,X1, . . . ) \u2208 A then, by Le\u00b4vy\u2019s 0-1 law and Lemma 7.5.1, the limitpoint \u039e+ is in B almost surely. In particular, if PG\u03c1 (A) > 0, then the exit measure of B is positive.It remains to show that (\u03c1,X1, . . . ) \u2208 A almost surely on the event that \u039e+ \u2208 B.1767.5. Boundary theoryLR\u039e+\u039e\u2212(vi)XnLR\u03c1Figure 7.4: For infinitely many n, a new random walk (red) started from Xn has probability atleast 1\/3 of hitting each of L and R, and probability at least 1\/4 of hitting the path (vi) (blue).Consider the two intervals L and R separating the almost surely distinct limit points \u039e+ and\u039e\u2212. Let pnL and pnR be the probabilities that a new, independent random walk started from Xnhits the boundary in the interval L or R respectively. Since the exit measure is non-atomic almostsurely, the eventEn = {min(pnL, pnR) > 1\/3}has positive probability (in fact, it is not hard to see that each of the random variables pnL isuniformly distributed on [0, 1] so that En has probability 1\/3). Moreover, the value of pnL doesnot depend on the choice of circle packing and is therefore a function of (G, (Xn+k)k\u2208Z). By thestationarity and ergodicity of (G, (Xn)n\u2208Z), the events En happen infinitely often almost surely(see Figure 7.4).Now condition on \u039e+ \u2208 B. Since the exit measure of B is positive, the events En still happeninfinitely often almost surely after conditioning. Let (vi)i\u22650 be a path from \u03c1 in G such thatz(vi)\u2192 \u039e+ \u2208 B and h(vi)\u2192 1. In particular,inf{h(vi) : i \u2265 1} > 0.The path (. . . , X\u22122, X\u22121, \u03c1, v1, v2, . . . ) disconnects Xn from at least one of the intervals L or R andsoPGXn(hit {. . . , X\u22121, \u03c1, v1, . . . }) \u2265 min(pnL, pnR) (7.5.6)which is greater than 1\/3 infinitely often almost surely. We stress that the expression refers to theprobability that an independent random walk started from Xn hits the path (. . . , X\u22121, \u03c1, v1, . . . ),and that this bound holds trivially if Xn is on the path (vm). By Lemma 7.5.3,PGXn(hit {. . . , X\u22121, \u03c1})a.s.\u2212\u2212\u2192 0, (7.5.7)and hence PGXn(hit {v1, v2, . . . }) > 1\/4 infinitely often almost surely (see Figure 7.4). Note that,1777.6. Hyperbolic speed and decay of radiisince the choice of vi could depend on the whole trajectory of X, we have not shown that X hitsthe path (vi) infinitely often. Nevertheless, by (7.5.5), almost surely infinitely oftenh(Xn) >14inf{h(vi) : i \u2265 1} > 0. (7.5.8)By Le\u00b4vy\u2019s 0-1 law, limn\u2192\u221e h(Xn) = 1 almost surely as desired.7.6 Hyperbolic speed and decay of radiiWe now use the fact that the exit measure is almost surely non-atomic to strengthen Lemma 7.5.1and deduce that the limit rate of decay of the Euclidean radii along the random walk exists. The keyidea is to use a circle packing in the upper half-plane normalised by the limits of two independentrandom walks.Fix some circle packing C in D, so that, by Theorem 7.1.3, the limit points \u039e\u00b1 = limn\u2192\u00b1\u221e z(Xn)exist and are distinct almost surely. Let \u03a6X be a Mo\u00a8bius transformation that maps D to the upperhalf-plane H and sends \u039e+ to 0 and \u039e\u2212 to\u221e. We consider the upper half-plane packing C\u0302 = \u03a6X(C).Similarly to the proof of non-atomicity in Section 7.5.2, we now have two boundary points 0and \u221e fixed by the graph G and the path (Xn), so that the resulting circle packing is unique upto scaling. Now, however, the packing depends on both G and the random walk, so that this newsituation is not paradoxical (as it was in Section 7.5.2 where we ruled out the possibility that theexit measure has a single atom).Proof of Theorem 7.1.4. We prove the equivalent statement for (G, \u03c1) reversible with E[deg(\u03c1)] <\u221e, and may assume that (G, \u03c1) is ergodic. We fix a circle packing C\u0302 = \u03a6X(C) in H as above,with the doubly infinite random walk from \u221e to 0. Let r\u02c6(v) be the Euclidean radius of the circlecorresponding to v in C\u0302. The ratio of radii r\u02c6(Xn)\/r\u02c6(Xn\u22121) does not depend on the choice of C\u0302, sothese ratios form a stationary ergodic sequence. By the Sharp Ring Lemma, E[| log(r\u02c6(X1)\/r\u02c6(\u03c1))|] \u2264CE[deg(\u03c1)]<\u221e, so that the Ergodic Theorem implies that\u2212 1nlogr\u02c6(Xn)r\u02c6(\u03c1)= \u2212 1nn\u22111logr\u02c6(Xi)r\u02c6(Xi\u22121)a.s.\u2212\u2212\u2212\u2192n\u2192\u221e \u2212E[logr\u02c6(X1)r\u02c6(\u03c1)]. (7.6.1)Now, since C\u0302 is the image of C through the Mo\u00a8bius map \u03a6X , and since \u03a6X is conformal at \u039e+,r\u02c6(Xn)r(Xn)\u2192 \u2223\u2223\u03a6\u2032X (\u039e+) \u2223\u2223 > 0. (7.6.2)Thereforelim\u2212 log r(Xn)n= E[\u2212 log r\u02c6(X1)r\u02c6(\u03c1)](7.6.3)and by Lemma 7.5.1 this limit must be positive. This establishes the rate of decay of the radii.Next, we relate this to the distance of z(Xn) from \u2202D. By the triangle inequality, 1 \u2212 |z(Xn)|1787.6. Hyperbolic speed and decay of radiiis at most the length of the path formed by drawing straight lines between the Euclidean centresof the circles along the random walk path starting at Xn:1\u2212 |zh(Xn)| \u2264 1\u2212 |z(Xn)| \u2264\u2211i\u2265n2r(Xi).Since the radii decay exponentially, taking the limits of the logarithms,lim inf\u2212 log (1\u2212 |zh(Xn)|)n\u2265 lim \u2212 log r(Xn)n. (7.6.4)To get a corresponding upper bound, note that, since every circle neighbouring Xn is containedin the open unit disc, 1\u2212 |zh(Xn)| is at least the radius of the smallest neighbour of Xn. Applyingthe Sharp Ring Lemma, we have1\u2212 |zh(Xn)| \u2265 r(Xn) exp(\u2212C deg(Xn)).Taking logarithms and passing to the limit,lim sup\u2212 log (1\u2212 |zh(Xn)|)n\u2264 lim \u2212 log r(Xn)n+ limC deg(Xn)n= lim\u2212 log r(Xn)n, (7.6.5)where the almost sure limit deg(Xn)\/n \u2192 0 follows from E[deg(\u03c1)] < \u221e and Borel-Cantelli.Combining (7.6.4) and (7.6.5) gives the almost sure limitlim\u2212 log (1\u2212 |zh(Xn)|)n= lim\u2212 log r(Xn)n.Finally, to relate this to the speed in the hyperbolic metric, recall that distances from the origin inthe hyperbolic metric are given bydhyp(0, z) = 2 tanh\u22121 |z|and hencelim1ndhyp(zh(\u03c1), zh(Xn)) = lim1ndhyp(0, zh(Xn))= lim2ntanh\u22121 |zh(Xn)|= lim\u2212 1nlog(1\u2212 |zh(Xn)|).1797.7. Extensions7.7 ExtensionsWe now discuss two basic extensions of our main results beyond simple triangulations. These areto weighted and to non-simple triangulations. The latter are of particular interest since the PSHTis not simple. Some of our results hold for much more general planar maps, which are treated in[20].Weighted networks. Suppose (G, \u03c1,w) is a unimodular random rooted weighted triangulation.As in the unweighted case, if E[w(\u03c1)] is finite then biasing by w(\u03c1) gives an equivalent randomrooted weighted triangulation which is reversible for the weighted simple random walk [7, Theorem4.1]. Our arguments generalise with no change to recover all our main results in the weightedsetting provided the following conditions are satisfied.1. E[w(\u03c1)] <\u221e. This allows us to bias to get a reversible random rooted weighted triangulation.2. E[w(\u03c1) deg(\u03c1)] <\u221e. After biasing by w(\u03c1), the expected degree is finite, allowing us to applythe Ring Lemma together with the Ergodic Theorem as in the proofs of Lemma 7.5.1 andTheorem 7.1.4.3. A version of Theorem 7.3.2 holds. That is, there exists a percolation \u03c9 such that the inducednetwork \u03c9\u00af has positive Cheeger constant almost surely. Two natural situations in which thisoccurs are(a) when all the weights are non-zero almost surely. In this situation, we may adapt theproof of Theorem 7.3.2 by first deleting all edges of weight less than 1\/M and all verticesof total weight greater than M before continuing the construction as before.(b) when the subgraph formed by the edges of non-zero weight is connected and is itselfinvariantly non-amenable. This occurs when we circle pack planar maps that are nottriangulations by adding edges of weight 0 in non-triangular faces to triangulate them.Non-simple triangulations. Suppose G is a one-ended planar map. The endpoints of anydouble edge or loop in G disconnect G into connected components exactly one of which is infinite.The simple core of G, denoted core(G), is defined by deleting the finite component containedwithin each double edge or loop of G before gluing the double edges together or deleting the loopas appropriate. See Figure 7.5 for an example, and [24] for a more detailed description. WhenG is a triangulation, so is its core. The core can be seen as a subgraph of G, with some verticesremoved, and multiple edges replaced by a single edge. The induced random walk on the core, istherefore a random walk on a weighted simple triangulation.In general, it is possible that all of G is deleted by this procedure, but in this case there areinfinitely many disjoint vertex cut-sets of size 2 separating each vertex from infinity, implying thatG is VEL parabolic and hence invariantly amenable. When G is invariantly non-amenable, theconclusions of Theorem 7.1.3 hold with the necessary modifications.1807.7. ExtensionsFigure 7.5: Extracting the core of a non-simple map. Left: part of a map. Right: correspondingpart of its core.Theorem 7.7.1. Let (G, \u03c1) be an invariantly non-amenable, one-ended, unimodular random rootedplanar triangulation with E[deg2(\u03c1)] <\u221e. Then core(G) is CP hyperbolic. Let C be a circle packingof core(G) in D, and let (Yn)n\u2208N be the induced random walk on core(G). The following holdconditional on (G, \u03c1) almost surely:1. z(Yn) and zh(Yn) both converge to a (random) point denoted \u039e \u2208 \u2202D,2. The law of \u039e has full support and no atoms.3. \u2202D is a realisation of the Poisson boundary of G. That is, for every bounded harmonicfunction h on G there exists a bounded measurable function g : \u2202D\u2192 R such thath(v) = EGv[g(\u039e)].Since the additional components needed to prove this are straightforward, we omit some of thedetails.Sketch of proof. First, (core(G), \u03c1) is unimodular when sampled conditional on \u03c1 \u2208 core(G): essen-tially, a mass transport on core(G) gives a mass transport on G which is 0 for all deleted vertices.The mass transport principle for G implies the principle for core(G).Second, core(G) is CP hyperbolic. Since (G, \u03c1) is invariantly non-amenable, it is VEL hyperbolic(see the proof of Theorem 7.1.1). Because the infimum over paths in the definition of the vertexextremal length is the same as the infimum over paths in the core, the vertex extremal length fromv \u2208 core(G) to \u221e is the same in G and core(G). (Alternatively, one could deduce non-amenabilityof core(G) from non-amenability of G, and apply Theorem 7.1.1.)Now, since core(G) is a weighted CP hyperbolic unimodular simple triangulation (and thesecond moment of the degree of the root is finite), by Theorem 7.1.3 the random walk on core(G)converges to a point in the boundary, the exit measure has full support and no atoms, and \u2202D is arealisation of the Poisson boundary of core(G).Finally, by the Optional Stopping Theorem, the bounded harmonic functions on core(G) are in1817.8. Open problemsone-to-one correspondence with the bounded harmonic functions on G by restriction and extension:hG 7\u2192 hcore(G) = hG|core(G), hcore(G) 7\u2192 hG(v) = EGv [hcore(G)(XN0)].Thus, the realisation of \u2202D as the Poisson boundary of core(G) extends to G.7.8 Open problemsProblem 7.8.1. Can the identification of the Poisson and geometric boundaries be strengthenedto an identification of the Martin boundary? This was done in [17] for CP hyperbolic triangulationswith bounded degrees. Specifically, we believe the following.Conjecture 7.8.2. Let (G, \u03c1) be an infinite simple, one-ended, CP hyperbolic unimodular randomrooted planar triangulation with E[deg(\u03c1)] <\u221e, and let C be a circle packing of G in the unit disc.Then almost surely for every point \u03be \u2208 \u2202D there exists a unique positive harmonic function h\u03be onG such that h\u03be(\u03c1) = 1 and h\u03be is bounded on {v : |z(v) \u2212 \u03be| \u2265 \u03b5} for every \u03b5 > 0. Moreover,the function \u03be 7\u2192 h\u03be almost surely extends to a homeomorphism from z(V ) \u222a \u2202D to the Martincompactification of G.Problem 7.8.3 (Ho\u00a8lder continuity of the exit measures). In the setting of Theorem 7.1.3, do thereexist positive constants c and C such thatPG\u03c1 (\u039e \u2208 I) \u2264 C|I|cfor every interval I \u2282 \u2202D?Problem 7.8.4 (Dirichlet energy of z). In the bounded degree case, by applying the main theoremof [12], convergence to the boundary may be shown by observing that the Dirichlet energy of thecentres function z is finite:E(z) =\u2211u\u223cv(z(u)\u2212 z(v))2 \u2264\u22112 deg(v)r(v)2 \u2264 2 max{deg(v)}.Is the Dirichlet energy of z almost surely finite for a unimodular random rooted CP hyperbolictriangulation? This may provide a route to weakening the moment assumption in our results.Problem 7.8.5 (Other embeddings). How does the canonical embedding of the Poisson-Delaunaytriangulation differ from the embedding given by the circle packing? Is there a circle packing sothat dhyp(v, zh(v)) is stationary?The conformal embedding of a triangulation is defined by forming a Riemann surface bygluing equilateral triangles according to the combinatorics of the triangulation before mapping theresulting surface conformally to D or C. Is it possible to control the large scale distortion betweenthe conformal embedding and the circle packing? In general the answer is no, but in the unimodularcase there is hope.1827.8. Open problemsRegardless of the answer to this question, our methods should extend without too much difficultyto establish analogues of Theorem 7.1.3 for these other embeddings, the main obstacle being toshow almost sure convergence of the random walk to a point in the boundary \u2202D.Problem 7.8.6. Reduce the moment assumption on deg(\u03c1) in Theorems 7.1.3 and 7.1.4. Finiteexpectation is needed to switch to a reversible distribution on rooted maps, but perhaps the secondmoment is not needed.183Chapter 8Boundaries of planar graphs: aunified approachSummary. We give a new proof that the Poisson boundary of a planar graph coincides with theboundary of its square tiling and with the boundary of its circle packing, originally proven byGeorgakopoulos [100] and Angel, Barlow, Gurel-Gurevich and Nachmias [17] respectively. Ourproof is robust, and also allows us to identify the Poisson boundaries of graphs that are rough-isometric to planar graphs.We also prove that the boundary of the square tiling of a bounded degree plane triangulationcoincides with its Martin boundary. This is done by comparing the square tiling of the triangulationwith its circle packing.8.1 IntroductionSquare tilings of planar graphs were introduced by Brooks, Smith, Stone and Tutte [60], andare closely connected to random walk and potential theory on planar graphs. Benjamini andSchramm [48] extended the square tiling theorem to infinite, uniquely absorbing plane graphs (seeSection 8.2.2). These square tilings take place on the cylinder R\/\u03b7Z\u00d7 [0, 1], where \u03b7 is the effectiveconductance to infinity from some fixed root vertex \u03c1 of G. They also proved that the randomwalk on a transient, bounded degree, uniquely absorbing plane graph converges to a point in theboundary of the cylinder R\/\u03b7Z \u00d7 {1}, and that the limit point of a random walk started at \u03c1 isdistributed according to the Lebesgue measure on the boundary of the cylinder.Benjamini and Schramm [48] applied their convergence result to deduce that every transient,bounded degree planar graph admits non-constant bounded harmonic functions. Recall that afunction h : V \u2192 R on the state space of a Markov chain (V, P ) is harmonic ifh(u) =\u2211v\u223cuP (u, v)h(v)for every vertex u \u2208 V , or equivalently if \u3008h(Xn)\u3009n\u22650 is a martingale when \u3008Xn\u3009n\u22650 is a trajectoryof the Markov chain. If G is a transient, uniquely absorbing, bounded degree plane graph, then for1848.1. Introductioneach bounded Borel function f : R\/\u03b7Z\u2192 R, we define a harmonic function h on G by settingh(v) = Ev[f(limn\u2192\u221e \u03b8(Xn))]for each v \u2208 V , where Ev denotes the expectation with respect to a random walk \u3008Xn\u3009n\u22650 startedat v and \u03b8(v) is the horizontal coordinate associated to the vertex v by the square tiling of G (seeSection 8.2.3). Georgakopoulos [100] proved that moreover every bounded harmonic function onG may be represented this way, answering a question of Benjamini and Schramm [48]. In otherwords, Georgakopoulos\u2019s theorem identifies the geometric boundary of the square tiling of G withthe Poisson boundary of G (see Section 8.3). Probabilistically, this means that the tail \u03c3-algebraof the random walk \u3008Xn\u3009n\u22650 is trivial conditional on the limit of \u03b8(Xn).In this paper, we give a new proof of Georgakopoulos\u2019s theorem. We state our result in thenatural generality of plane networks. Recall that a network (G, c) is a connected, locally finitegraph G = (V,E), possibly containing self-loops and multiple edges, together with a functionc : E \u2192 (0,\u221e) assigning a positive conductance to each edge of G. The conductance c(v) of avertex v is defined to be the sum of the conductances of the edges emanating from v, and for eachpair of vertices u, v the conductance c(u, v) is defined to be the sum of the conductances of theedges connecting u to v. The random walk on the network is the Markov chain with transitionprobabilities p(u, v) = c(u, v)\/c(u). Graphs without specified conductances are considered networksby setting c(e) \u2261 1. We will usually suppress the notation of conductances, and write simply Gfor a network. Instead of square tilings, general plane networks are associated to rectangle tilings,see Section 8.2.3. See Section 8.2.2 for detailed definitions of plane graphs and networks. For eachvertex v of G, I(v) \u2286 R\/\u03b7Z is an interval associated to v by the rectangle tiling of G.Theorem 8.1.1 (Identification of the Poisson boundary). Let G be a plane network and let S\u03c1 bethe rectangle tiling of G in the cylinder R\/\u03b7Z\u00d7 [0, 1]. Suppose that \u03b8(Xn) converges to a point inR\/\u03b7Z and that length(I(Xn)) converges to zero almost surely as n tends to infinity. Then for everybounded harmonic function h on G, there exists a bounded Borel function f : R\/\u03b7Z\u2192 R such thath(v) = Ev[f(limn\u2192\u221e \u03b8(Xn))].for every v \u2208 V . That is, the geometric boundary of the rectangle tiling of G coincides with thePoisson boundary of G.The convergence theorem of Benjamini and Schramm [48] implies that the hypotheses of The-orem 8.1.1 are satisfied when G has bounded degrees.8.1.1 Circle packingAn alternative framework in which to study harmonic functions on planar graphs is given bythe circle packing theorem. A circle packing is a collection C of non-overlapping (but possibly1858.1. IntroductionFigure 8.1: The square tiling and the circle packing of the 7-regular hyperbolic triangulation.tangent) discs in the plane. Given a circle packing C, the tangency graph of C is defined to bethe graph with vertices corresponding to the discs of C and with two vertices adjacent if and onlyif their corresponding discs are tangent. The tangency graph of a circle packing is clearly planar,and can be drawn with straight lines between the centres of tangent discs in the packing. TheKoebe-Andreev-Thurston Circle Packing Theorem [156, 221] states conversely that every finite,simple (i.e., containing no self-loops or multiple edges), planar graph may be represented as thetangency graph of a circle packing. If the graph is a triangulation (i.e., every face has three sides),its circle packing is unique up to Mo\u00a8bius transformations and reflections. We refer the reader to[215] and [202] for background on circle packing.The carrier of a circle packing is defined to be the union of all the discs in the packing togetherwith the bounded regions that are disjoint from the discs in the packing and are enclosed by thesome set of discs in the packing corresponding to a face of the tangency graph. Given some planardomain D, we say that a circle packing is in D if its carrier is D.The circle packing theorem was extended to infinite planar graphs by He and Schramm [121],who proved that every proper plane triangulation admits a locally finite circle packing in the planeor the disc, but not both. We call a triangulation of the plane CP parabolic if it can be circlepacked in the plane and CP hyperbolic otherwise. He and Schramm also proved that a boundeddegree simple triangulation of the plane is CP parabolic if and only if it is recurrent for the simplerandom walk.Benjamini and Schramm [47] proved that, when a bounded degree, CP hyperbolic triangulationis circle packed in the disc, the simple random walk converges to a point in the boundary of the discand the law of the limit point is non-atomic and has full support. Angel, Barlow, Gurel-Gurevichand Nachmias [17] later proved that, under the same assumptions, the boundary of the disc is arealisation of the Poisson boundary of the triangulation. These results were extended to unimodular1868.1. Introductionrandom rooted triangulations of unbounded degree by Angel, Hutchcroft, Nachmias and Ray [21].Our proof of Theorem 8.1.1 is adapted from the proof of [21], and also yields a new proof of thePoisson boundary result of [17], which follows as a special case of both Theorems 8.1.2 and 8.1.3below.8.1.2 Robustness under rough isometriesThe proof of Theorem 8.1.1 is quite robust, and also allows us to characterise the Poisson boundariesof certain non-planar networks.Benjamini and Schramm [47] proved that every transient network of bounded local geometrythat is rough isometric to a planar graph admits non-constant bounded harmonic functions. Ingeneral, however, rough isometries do not preserve the property of admitting non-constant boundedharmonic functions [47, Theorem 3.5], and consequently do not preserve Poisson boundaries.Our next theorem establishes that, for a bounded degree graphG roughly isometric to a boundeddegree proper plane graph G\u2032, the Poisson boundary of G coincides with the geometric boundaryof a suitably chosen embedding of G\u2032, so that the same embedding gives rise to a realisation of thePoisson boundaries of both G and G\u2032. See Section 8.2.2 for the definition of an embedding of aplanar graph.Theorem 8.1.2 (Poisson boundaries of roughly planar networks). Let G be a transient networkwith bounded local geometry such that there exists a proper plane graph G\u2032 with bounded degrees anda rough isometry \u03c6 : G\u2192 G\u2032. Let \u3008Xn\u3009n\u22650 be a random walk on G. Then there exists an embeddingz of G\u2032 into D such that z \u25e6 \u03c6(Xn) converges to a point in \u2202D and the law of the limit point isnon-atomic. Moreover, for every such embedding z and for every bounded harmonic function h onG, there exists a bounded Borel function f : \u2202D\u2192 R such thath(v) = Ev[f(limn\u2192\u221e z \u25e6 \u03c6(Xn))].for every v \u2208 V . That is, the geometric boundary of the disc coincides with the Poisson boundaryof G.The part of Theorem 8.1.2 concerning the existence of an embedding is implicit in [47].A further generalisation of Theorem 8.1.1 concerns embeddings of possibly irreversible planarMarkov chains: The only changes required to the proof of Theorem 8.1.1 in order to prove thefollowing are notational.Theorem 8.1.3. Let (V, P ) be a Markov chain such that the graphG =(V, {(u, v) \u2208 V 2 : P (u, v) > 0 or P (v, u) > 0})is planar. Suppose further that there exists a vertex \u03c1 \u2208 V such that for every v \u2208 V there existsn such that Pn(\u03c1, v) > 0, and let \u3008Xn\u3009n\u22650 be a trajectory of the Markov chain. Let z be a (not1878.1. Introductionnecessarily proper) embedding of G into the unit disc D such that \u3008z(Xn)\u3009n\u22650 converges to a pointin \u2202D almost surely and the law of the limit point is non-atomic. Then for every bounded harmonicfunction h on (V, P ), there exists a bounded Borel function f : \u2202D\u2192 R such thath(v) = Ev[f(limn\u2192\u221e z(Xn))].for every v \u2208 V .8.1.3 The Martin boundaryIn [17] it was also proven that the Martin boundary of a bounded degree CP hyperbolic triangulationcan be identified with the geometric boundary of its circle packing. Recall that a function g : V \u2192 Ron a network G is superharmonic ifg(u) \u2265 1c(u)\u2211v\u223cuc(u, v)g(v)for every vertex u \u2208 V . Let \u03c1 be a fixed vertex of G and consider the space S+ of positivesuperharmonic functions g on G such that g(\u03c1) = 1, which is a convex, compact subset of thespace of functions V \u2192 R equipped with the product topology (i.e. the topology of pointwiseconvergence). For each v \u2208 V , let Pv be the law of the random walk on G started at v. We canembed V into S+ by sending each vertex u of G to its Martin kernelMu(v) :=Ev[#(visits to u)]E\u03c1[#(visits to u)] = Pv(hit u)P\u03c1(hit u).The Martin compactificationM(G) of the network G is defined as the closure of {Mu : u \u2208 V }in S+, and the Martin boundary \u2202M(G) of the network G is defined to be the complement ofthe image of V ,\u2202M(G) :=M(G) \\ {Mu : u \u2208 V }.See [84, 201, 229] for background on the Martin boundary.Our next result is that, for a triangulation of the plane with bounded local geometry, thegeometric boundary of the square tiling coincides with the Martin boundary.Theorem 8.1.4 (Identification of the Martin boundary). Let T be a transient, simple, proper planetriangulation with bounded local geometry. Let S\u03c1 be a square tiling of T in a cylinder R\/\u03b7Z\u00d7 [0, 1].Then1. A sequence of vertices \u3008vn\u3009n\u22650 in T converges to a point in the Martin boundary of T if andonly if y(vn)\u2192 1 and \u03b8(vn) converges to a point in R\/\u03b7Z.2. The mapM : \u03b8 7\u2212\u2192M\u03b8 := limn\u2192\u221eMvn where(\u03b8(vn), y(vn))\u2192 (\u03b8, 1),1888.1. Introductionwhich is well-defined by (1), is a homeomorphism from R\/\u03b7Z to the Martin boundary \u2202M(T )of T .That is, the geometric boundary of the rectangle tiling of G coincides with the Martin boundary ofG.This will be deduced from the analogous statement for circle packings [17, Theorem 1.2] togetherwith the following theorem, which states that for a bounded degree triangulation T , the squaretiling and circle packing of T define equivalent compactifications of T .Theorem 8.1.5 (Comparison of square tiling and circle packing). Let T be a transient, simple,proper plane triangulation with bounded local geometry. Let S\u03c1 be a rectangle tiling of T in thecylinder R\/\u03b7Z\u00d7 [0, 1] and let C be a circle packing of T in D with associated embedding z. Then1. A sequence of vertices \u3008vn\u3009n\u22650 in T converges to a point in \u2202D if and only if y(vn)\u2192 1 and\u03b8(vn) converges to a point in R\/\u03b7Z.2. The map\u03be 7\u2192 limi\u2192\u221e\u03b8(vi) where z(vi)\u2192 \u03be,which is well-defined by (1), is a homeomorphism from \u2202D to R\/\u03b7Z.Theorem 8.1.5 also allows us to deduce the Poisson boundary results of [100] and [17] from eachother in the case of bounded degree triangulations.Theorem 8.1.4 has the following immediate corollaries by standard properties of the Martincompactification.Corollary 8.1.6 (Continuity of harmonic densities). Let T be a transient proper simple planetriangulation with bounded local geometry, and let S\u03c1 be a rectangle tiling of T in the cylinderR\/\u03b7Z\u00d7 [0, 1]. For each vertex v of T , let \u03c9v denote the harmonic measure from v, defined by\u03c9v(A ) := Pv(limn\u2192\u221e \u03b8(Xn) \u2208 A)for each Borel set A \u2286 R\/\u03b7Z, and let \u03bb = \u03c9\u03c1 denote the Lebesgue measure on R\/\u03b7Z. Then forevery v of T , the density of the harmonic measure from v with respect to Lebesgue is given byd\u03c9vd\u03bb(\u03b8) = M\u03b8(v)which is continuous with respect to \u03b8.Corollary 8.1.7 (Representation of positive harmonic functions). Let T be a transient propersimple plane triangulation with bounded local geometry, and let S\u03c1 be a rectangle tiling of T in thecylinder R\/\u03b7Z \u00d7 [0, 1]. Then for every positive harmonic function h on T , there exists a uniquemeasure \u00b5 on R\/\u03b7Z such thath(v) =\u222bR\/\u03b7ZM\u03b8(v) d\u00b5(\u03b8).1898.2. Backgroundfor every v \u2208 V .8.2 Background8.2.1 NotationWe use e to denote both oriented and unoriented edges of a graph or network. An oriented edge eis oriented from its tail e\u2212 to its head e+. Given a network G and a vertex v of G, we write Pv forthe law of the random walk on G started at v and Ev for the associated expectation operator.8.2.2 Embeddings of planar graphsLet G = (V,E) be a graph. For each edge e, choose an orientation of e arbitrarily and let I(e) be anisometric copy of the interval [0, 1]. The metric space G = G(G) is defined to be the quotient of theunion\u22c3e I(e)\u222aV , where we identify the endpoints of I(e) with the vertices e\u2212 and e+ respectively,and is equipped with the path metric. An embedding of G into a surface S is a continuous,injective map z : G\u2192 S. The embedding is proper if every compact subset of S intersects at mostfinitely many edges and vertices of z(G). A graph is planar if and only if it admits an embeddinginto R2. A plane graph is a planar graph together with an embedding G(G)\u2192 R2 or some othersurface homeomorphic to R2 such as the open disc; it is a proper plane graph if the embeddingz is proper. A (proper) plane network is a (proper) plane graph together with an assignment ofconductances c : E \u2192 (0,\u221e). A proper plane triangulation is a proper plane network in whichevery face (i.e connected component of R2 \\ z(G)) has three sides.A set of vertices W \u2286 V is said to be absorbing if with positive probability the random walk\u3008Xn\u3009n\u22650 on G is contained in W for all n greater than some random N . A plane graph G is saidto be uniquely absorbing if for every finite subgraph G0 of G, there is exactly one connectedcomponent D of R2 \\ \u22c3{z(I(e)) : e \u2208 G0} such that the set of vertices {v \u2208 V : z(v) \u2208 D} isabsorbing.8.2.3 Square tilingLet G be a transient, uniquely absorbing plane network and let \u03c1 be a vertex of G. For each v \u2208 Vlet y(v) denote the probability that the random walk on G started at v never visits \u03c1, and let\u03b7 :=\u2211u\u223c\u03c1c(\u03c1, u)y(u).Let R\/\u03b7Z be the circle of length \u03b7. Then there exists a setS\u03c1 = {S(e) : e \u2208 E}such that:1908.3. The Poisson boundary1. For each oriented edge e of G such that y(e+) \u2265 y(e\u2212), S(e) \u2286 R\/\u03b7Z\u00d7 [0, 1) is a rectangle ofthe formS(e) = I(e)\u00d7[y(e\u2212), y(e+)]where I(e) \u2286 R\/\u03b7Z is an interval of lengthlength(I(e)):= c(e)(y(e+)\u2212 y(e\u2212)).If e is such that y(e+) < y(e\u2212), we define I(e) = I(\u2212e) and S(e) = S(\u2212e). In particular, theaspect ratio of S(e) is equal to the conductance c(e) for every edge e \u2208 E.2. The interiors of the rectangles S(e) are disjoint, and the union\u22c3e S(e) = R\/\u03b7Z\u00d7 [0, 1).3. For every vertex v \u2208 V , the set I(v) = \u22c3e\u2212=v I(e) is an interval and is equal to \u22c3e+=v I(e).4. For almost every \u03b8 \u2208 C and for every t \u2208 [0, 1), the line segment {\u03b8} \u00d7 [0, t] intersects onlyfinitely many rectangles of S\u03c1.Note that the rectangle corresponding to an edge through which no current flows is degenerate,consisting of a single point. The existence of the above tiling was proven by Benjamini and Schramm[48]. Their proof was stated for the case c \u2261 1 but extends immediately to our setting, see [100].Let us also note the following property of the rectangle tiling, which follows from the construc-tion given in [48].(5) For each two edges e1 and e2 of G, the interiors of the vertical sides of the rectangles S(e1)and S(e2) have a non-trivial intersection only if e1 and e2 both lie in the boundary of somecommon face f of G.For each v \u2208 V , we let \u03b8(v) be a point chosen arbitrarily from I(v).Let G be a uniquely absorbing proper plane network. Benjamini and Schramm [48] provedthat if G has bounded local geometry and \u3008Xn\u3009n\u22650 is a random walk on G started at \u03c1, then\u03b8(Xn) converges to a point in R\/\u03b7Z and the law of the limit point is Lebesgue (their proof isgiven for bounded degree plane graphs but extends immediately to this setting). An observationof Georgakopoulos [100, Lemma 6.2] implies that, more generally, whenever G is such that \u03b8(Xn)converges to a point in R\/\u03b7Z and length(I(Xn)) converges to zero almost surely, the law of thelimit point is Lebesgue.8.3 The Poisson boundaryLet (V, P ) be a Markov chain. Harmonic functions on (V, P ) encode asymptotic behaviours of atrajectory \u3008Xn\u3009n\u22650 as follows. Let \u2126 denote the path space\u2126 ={\u3008xi\u3009i\u22650 \u2208 V N : p(xi, xi+1) > 0 \u2200i \u2265 0}1918.3. The Poisson boundaryand let B denote the Borel \u03c3-algebra for the product topology on \u2126. Let I denote the invariant\u03c3-algebraI = {A \u2208 B : \u3008xi\u3009i\u22650 \u2208 A \u21d0\u21d2 \u3008xi+1\u3009i\u22650 \u2208 A \u2200\u3008xi\u3009i\u22650 \u2208 \u2126} .Assume that there exists a vertex \u03c1 \u2208 V from which all other vertices are reachable:\u2200v \u2208 V \u2203k \u2265 0 such that pk(\u03c1, v) > 0which is always satisfied when P is the transition operator of the random walk on a network G.Then there exists an invertible linear transformation H between L\u221e(\u2126, I, P\u03c1) and the space ofbounded harmonic functions on G defined as follows [55, 173, Proposition 14.12]:H : f 7\u2212\u2192 Hf(v) = Ev[f(\u3008Xn\u3009n\u22650)]H\u22121 : h 7\u2212\u2192 h\u02dc (\u3008xi\u3009i\u22650) = limi\u2192\u221eh(xi) (8.3.1)where the above limit exists for P\u03c1-a.e. sequence \u3008xi\u3009i\u22650 by the martingale convergence theorem.Proposition 8.3.1 (Path-hitting criterion for the Poisson boundary). Let (V, P ) be a Markov chainand let \u3008Xn\u3009n\u22650 be a trajectory of the Markov chain. Suppose that \u03c8 : V \u2192 M is a function fromV to a metric space M such that \u03c8(Xn) converges to a point in M almost surely. For each k \u2265 0,let \u3008Zkj \u3009j\u22650 be a trajectory of the Markov chain started at Xk that is conditionally independent of\u3008Xn\u3009n\u22650 given Xk, and let P denote the joint distribution of \u3008Xn\u3009n\u22650 and each of the \u3008Zkm\u3009m\u22650.Suppose that almost surely, for every path \u3008vi\u3009i\u22650 \u2208 \u2126 started at \u03c1 such that limi\u2192\u221e \u03c8(vi) =limn\u2192\u221e \u03c8(Xn), we have thatlim supk\u2192\u221eP(\u3008Zkm\u3009m\u22650 hits {vi : i \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650) > 0. (8.3.2)Then for every bounded harmonic function h on G, there exists a bounded Borel function f : M\u2192 Rsuch thath(v) = Ev[f(limn\u2192\u221e\u03c8(Xn))]for every v \u2208 V .Note that it suffices to define f on the support of the law of limn\u2192\u221e \u03c8(Xn), which is containedin the set of accumulation points of {\u03c8(v) : v \u2208 V }.A consequence of the correspondence (8.3.1) is that, to prove Proposition 8.3.1, it suffices toprove that for every invariant event A \u2208 I, there exists a Borel set B \u2286M such thatPv(A4{ limn\u2192\u221e\u03c8(Xn) \u2208 B})= 0 for every vertex v \u2208 V .1928.3. The Poisson boundaryProof of Proposition 8.3.1. Let A \u2208 I be an invariant event and let h be the harmonic functionh(v) := Pv(\u3008Xn\u3009n\u22650 \u2208 A ).Le\u00b4vy\u2019s 0-1 law implies thath(Xn)a.s.\u2212\u2212\u2212\u2192n\u2192\u221e 1(\u3008Xn\u3009n\u22650 \u2208 A )and so it suffices to exhibit a Borel set B \u2286M such thatP\u03c1(A4{ limn\u2192\u221e\u03c8(Xn) \u2208 B})= P\u03c1({lim supn\u2192\u221eh(Xn) > 0}4{ limn\u2192\u221e\u03c8(Xn) \u2208 B})= 0.We may assume that P\u03c1(A ) > 0, otherwise the claim is trivial.Let dM denote the metric of M. For each natural number m > 0, let N(m) be the smallestnatural number such thatP\u03c1(\u2203n \u2265 N(m) such that dM(\u03c8(Xn), limk\u2192\u221e\u03c8(Xk))\u2265 1m)\u2264 2\u2212m.For each n, let m(n) be the largest m such that n \u2265 N(m), so that m(n) \u2192 \u221e as n \u2192 \u221e.Borel-Cantelli implies that, for all but finitely many m, there does not exist an n \u2265 N(m) suchthatdM(\u03c8(Xn), limk\u2192\u221e\u03c8(Xk))>1m,and it follows thatdM(\u03c8(Xn), limk\u2192\u221e\u03c8(Xk))\u2264 1m(n)for all but finitely many n almost surely.Define the set B \u2286M byB :=\uf8f1\uf8f2\uf8f3x \u2208M : \u2203 a path \u3008vi\u3009i\u22650 in G with v0 = \u03c1 such that dM(\u03c8(vi), x) \u2264 1\/m(i)for all but finitely many i and infi\u22650 h(vi) > 0\uf8fc\uf8fd\uf8fe .To see that B is Borel, observe that it may be written in terms of closed subsets of M as followsB =\u22c3k\u22650\u22c3j\u22650\u22c2I\u2265j\uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3x \u2208M :\u2203 a path \u3008vi\u3009Ii=0 in G with v0 = \u03c1 such thatdM(\u03c8(vi), x) \u2264 1\/m(i) for all j \u2264 i \u2264 I andh(vi) \u2265 1\/k for all 0 \u2264 i \u2264 I\uf8fc\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8fe .It is immediate that limn\u2192\u221e \u03c8(Xn) \u2208 B almost surely on the event that h(Xn) converges to 1:simply take \u3008vi\u3009i\u22650 = \u3008Xi\u3009i\u22650 as the required path. In particular, the event {limn\u2192\u221e \u03c8(Xn) \u2208 B}1938.3. The Poisson boundaryhas positive probability.We now prove conversely that lim infn\u2192\u221e h(Xn) > 0 almost surely on the event that limn\u2192\u221e \u03c8(Xn) \u2208B. Condition on this event, so that there exists a path \u3008vi\u3009i\u22650 in G starting at \u03c1 such thatlimi\u2192\u221e \u03c8(vi) = limn\u2192\u221e \u03c8(Xn) and infi\u22650 h(vi) > 0. Fix one such path. Applying the optionalstopping theorem to \u3008h(Zkm)\u3009m\u22650, we haveh(Xk) \u2265 P(\u3008Zkm\u3009m\u22650 hits {vi : i \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650) \u00b7 inf{h(vi) : i \u2265 0}and so, by our assumption (8.3.2), we have thatlim supk\u2192\u221eh(Xk) \u2265 lim supk\u2192\u221eP(\u3008Zkm\u3009m\u22650 hits {vi : i \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650) \u00b7 inf{h(vi) : i \u2265 0}is positive almost surely.We remark that controlling the rate of convergence of the path in the definition of B can beavoided by invoking the theory of universally measurable sets.We now apply the criterion given by Proposition 8.3.1 to prove Theorem 8.1.1 and Theo-rem 8.1.2.Proof of Theorem 8.1.1. Let \u3008Xn\u3009n\u22650 and \u3008Yn\u3009n\u22650 be independent random walks on G started at\u03c1, and for each k \u2265 0 let \u3008Zkj \u3009j\u22650 be a random walk on G started at Xk that is conditionallyindependent of \u3008Xn\u3009n\u22650 and \u3008Y \u3009n\u22650 given Xk. Let P denote the joint distribution of \u3008Xn\u3009n\u22650,\u3008Yn\u3009n\u22650 and all of the random walks \u3008Zkm\u3009m\u22650. Given two points \u03b81, \u03b82 \u2208 R\/\u03b7Z, we denote by(\u03b81, \u03b82) \u2282 R\/\u03b7Z the open arc between \u03b81 and \u03b82 in the counter-clockwise direction. For each suchinterval (\u03b81, \u03b82) \u2208 R\/\u03b7Z, letq(\u03b81,\u03b82)(v) := Pv(limn\u2192\u221e \u03b8(Xn) \u2208 (\u03b81, \u03b82)).be the probability that a random walk started at v converges to a point in the interval (\u03b81, \u03b82).Since the law of limn\u2192\u221e \u03b8(Xn) is Lebesgue and hence non-atomic, the two random variables\u03b8+ := limn\u2192\u221e \u03b8(Xn) and \u03b8\u2212 := limn\u2192\u221e \u03b8(Yn) are almost surely distinct. We can therefore writeR\/\u03b7Z \\ {\u03b8+, \u03b8\u2212} as the union of the two disjoint non-empty intervals R\/\u03b7Z \u00d7 {1} \\ {\u03b8+, \u03b8\u2212} =(\u03b8+, \u03b8\u2212) \u222a (\u03b8\u2212, \u03b8+). LetQk := q(\u03b8\u2212,\u03b8+)(Xk) = P(limm\u2192\u221e \u03b8(Zkm) \u2208 (\u03b8\u2212, \u03b8+)\u2223\u2223\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650)be the probability that a random walk started at Xk, that is conditionally independent of \u3008Xn\u3009n\u22650and \u3008Yn\u3009n\u22650 given Xk, converges to a point in the interval (\u03b8\u2212, \u03b8+).We claim that the random variable Qk is uniformly distributed on [0, 1] conditional on \u3008Xn\u3009kn=0and \u3008Yn\u3009n\u22650. Indeed, since the law of \u03b8+ given Xk is non-atomic, for each s \u2208 [0, 1] there exists99In the present setting, \u03b8s is unique since the ;aw of \u03b8+ given Xk has full support.1948.3. The Poisson boundaryR\/\u03b7Z\u00d7 [0, 1]\u3008vi\u3009\u3008Xn\u3009\u3008Yn\u3009Xk\u03b8\u2212\u03b8+ (\u03b8+, \u03b8\u2212)(\u03b8\u2212, \u03b8+)\u3008Zkm\u3009Figure 8.2: Illustration of the proof. Conditioned on the random walk \u3008Xn\u3009n\u22650, there exists arandom \u03b5 > 0 such that almost surely, for infinitely many k, a new random walk \u3008Zkm\u3009m\u22650 (red)started from Xk has probability at least \u03b5 of hitting the path \u3008vi\u3009i\u22650 (blue).\u03b8s = \u03b8s(Xk, \u03b8\u2212) \u2208 R\/\u03b7Z such thatP(\u03b8+ \u2208 (\u03b8\u2212, \u03b8s)\u2223\u2223\u2223Xk, \u03b8\u2212) = s.The claim follows by observing thatP(Qk \u2208 [0, s]\u2223\u2223\u2223 \u3008Xn\u3009kn=0, \u3008Yn\u3009n\u22650) = P(\u03b8+ \u2208 (\u03b8\u2212, \u03b8s) | Xk, \u03b8\u2212) = s.As a consequence, Fatou\u2019s lemma implies that for every \u03b5 > 0,P(Qk \u2208 [\u03b5, 1\u2212 \u03b5] infinitely often) = E[lim supk\u2192\u221e1(Qk \u2208 [\u03b5, 1\u2212 \u03b5])]\u2265 lim supk\u2192\u221eP(Qk \u2208 [\u03b5, 1\u2212 \u03b5])= 1\u2212 2\u03b5.and solim supk\u2192\u221emin{Qk, 1\u2212Qk} > 0 almost surely. (8.3.3)Let \u3008vi\u3009i\u22650 be a path in G started at \u03c1 such that limi\u2192\u221e \u03b8(vi) = \u03b8+ and the height y(vi) tendsto 1. Observe (Figure 1) that for every k \u2265 0, the union of the traces {vi : i \u2265 0} \u222a {Yn : n \u2265 0}either contains Xk or disconnects Xk from at least one of the two intervals (\u03b8\u2212, \u03b8+) or (\u03b8+, \u03b8\u2212).That is, for at least one of the intervals (\u03b8\u2212, \u03b8+) or (\u03b8+, \u03b8\u2212), any path in G started at Xk that1958.3. The Poisson boundaryconverges to this interval must intersect {vi : i \u2265 0} \u222a {Yn : n \u2265 0}. It follows thatP(\u3008Zkm\u3009m\u22650 hits {vi : i \u2265 0} \u222a {Yn : n \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650) \u2265 min{Qk, 1\u2212Qk}.and so, applying (8.3.3),lim supk\u2192\u221eP(\u3008Zkm\u3009m\u22650 hits {vi : i \u2265 0} \u222a {Yn : n \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650) > 0 (8.3.4)almost surely. We next claim thatP(\u3008Zkm\u3009m\u22650 hits {Yn : n \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650) a.s.\u2212\u2212\u2212\u2192k\u2192\u221e0. (8.3.5)Indeed, we have thatP(\u3008Zkm\u3009m\u22650 hits {Yn : n \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650)= P(\u3008Zkm\u3009m\u22650 hits {Yn : n \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009kn=0, \u3008Yn\u3009n\u22650)= P(\u3008Xm\u3009m\u2265k hits {Yn : n \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009kn=0, \u3008Yn\u3009n\u22650) .The rightmost expression converges to zero almost surely by an easy application of Le\u00b4vy\u2019s 0-1 law:For each k0 \u2264 k, we have thatP(\u3008Zkm\u3009m\u22650 hits {Yn : n \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650)\u2264 P(\u3008Xm\u3009m\u2265k0 hits {Yn : n \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009kn=0, \u3008Yn\u3009n\u22650)a.s.\u2212\u2212\u2212\u2192k\u2192\u221e1(\u3008Xm\u3009m\u2265k0 hits {Yn : n \u2265 0}),and the claim follows since 1(\u3008Xm\u3009m\u2265k0 hits {Yn : n \u2265 0})\u2192 0 a.s. as k0 \u2192\u221e.Combining (8.3.4) and (8.3.5), we deduce thatlim supk\u2192\u221eP(\u3008Zkm\u3009m\u22650 hits {vi : i \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650)= lim supk\u2192\u221eP(\u3008Zkm\u3009m\u22650 hits {vi : i \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650) > 0almost surely. Applying Proposition 8.3.1 with \u03c8 = (\u03b8, y) : V \u2192 R\/\u03b7Z \u00d7 [0, 1] completes theproof.8.3.1 Proof of Theorem 8.1.2Proof. Let G = (V,E) be a transient network with bounded local geometry such that there existsa proper plane graph G\u2032 = (V \u2032, G\u2032) with bounded degrees and a rough isometry \u03c6 : V \u2192 V \u2032.Then G\u2032 is also transient [173, \u00a72.6]. We may assume that G\u2032 is simple, since \u03c6 remains a rough1968.3. The Poisson boundaryisometry if we modify G\u2032 by deleting all self-loops and identifying all multiple edges between eachpair of vertices. In this case, there exists a simple, bounded degree, proper plane triangulationT \u2032 containing G\u2032 as a subgraph [47, Lemma 4.3]. The triangulation T \u2032 is transient by Rayleighmonotonicty, and since it has bounded degrees it is CP hyperbolic by the He-Schramm Theorem.Let C be a circle packing of T \u2032 in the disc D and let z be the embedding of G\u2032 defined by this circlepacking, so that for each v \u2208 V \u2032, z(v\u2032) is the center of the circle of C corresponding to v\u2032, and eachedge of G\u2032 is embedded as a straight line between the centers of the circles corresponding to itsendpoints.Recall that the Dirichlet energy of a function f : V \u2192 R is defined byEG(f) = 12\u2211ec(e)\u2223\u2223\u2223f(e+)\u2212 f(e\u2212)\u2223\u2223\u22232 .The following lemma is implicit in [212].Lemma 8.3.2. Let \u03c6 : V \u2192 V \u2032 be a rough isometry from a network G with bounded local geometryto a network G\u2032 with bounded local geometry. Then there exists a constant C such thatEG(f \u25e6 \u03c6) \u2264 CEG\u2032(f). (8.3.6)for all functions f : V \u2032 \u2192 R.For each pair of adjacent vertices u and v in G, we have that d\u2032(\u03c6(u), \u03c6(v)) \u2264 \u03b1 + \u03b2, and sothere exists a path in G\u2032 from \u03c6(u) to \u03c6(v) of length at most \u03b1+ \u03b2. Fix one such path \u03a6(u, v) foreach u and v.Proof. Since there at most \u03b1 + \u03b2 edges of G in each path \u03a6(u, v), and the conductances of G arebounded, there exists a constant C1 such thatEG(f \u25e6 \u03c6) =\u2211ec(e)(f \u25e6 \u03c6(e+)\u2212 f \u25e6 \u03c6(e\u2212))2=\u2211ec(e)( \u2211e\u2032\u2208\u03a6(e)f(e\u2032+)\u2212 f(e\u2032\u2212))2\u2264 C1\u2211e\u2211e\u2032\u2208\u03a6(e)(f(e\u2032+)\u2212 f(e\u2032\u2212))2. (8.3.7)Let e\u2032 be an edge of G\u2032. If e1 and e2 are two edges of G such that \u03a6(e\u22121 , e+1 ) and \u03a6(e\u22121 , e+1 ) bothcontain e\u2032, then d\u2032(\u03c6(e\u22121 ), \u03c6(e\u22122 )) \u2264 2(\u03b1+ \u03b2) and sod(e\u22121 , e\u22122 ) \u2264 \u03b1(d\u2032(\u03c6(e\u22121 ), \u03c6(e\u22122 ))+ \u03b2)\u2264 \u03b1(2\u03b1+ 3\u03b2).Thus, the set of edges e of G such that \u03a6(e\u2212, e+) contains e\u2032 is contained a ball of radius \u03b1(2\u03b1+3\u03b2)in G. Since G has bounded degrees, the number of edges contained in such a ball is bounded bysome constant C2. Combining this with the assumption that the conductances of G\u2032 are bounded1978.3. The Poisson boundarybelow by some constant C3, we have thatEG(f \u25e6 \u03c6) \u2264 C1\u2211e\u2211e\u2032\u2208\u03a6(e)(f(e\u2032+)\u2212 f(e\u2032\u2212))2\u2264 C1C2\u2211e\u2032(f(e\u2032+)\u2212 f(e\u2032\u2212))2 \u2264 C1C2C3EG\u2032(f).The proofs of Lemmas 8.3.3 and 8.3.4 below are adapted from [17].Lemma 8.3.3. Let \u3008Xn\u3009n\u22650 denote the random walk on G. Then z \u25e6 \u03c6(Xn) converges to a pointin \u2202D almost surely.Proof. Ancona, Lyons and Peres [12] proved for every function f of finite Dirichlet energy on atransient network G, the sequence f(Xn) converges almost surely as n \u2192 \u221e. Thus, it suffices toprove that each coordinate of z \u25e6 \u03c6 has finite Dirichlet energy. Applying the inequality (8.3.6),it suffices to prove that each coordinate of z has finite energy. For each vertex v\u2032 of G\u2032, let r(v\u2032)denote the radius of the circle corresponding to v\u2032 in C. Then, letting z = (z1, z2),EG\u2032(z1) + EG\u2032(z2) =\u2211e\u2032\u2223\u2223\u2223z(e+)\u2212 z(e\u2212)\u2223\u2223\u22232 = C\u2211e\u2032(r(e+) + r(e\u2212))2\u2264 2 max (deg(v\u2032))C\u2211v\u2032r(v)2 \u2264 2C max (deg(v\u2032)) <\u221esince\u2211pir(v)2 \u2264 pi is the total area of all the circles in the packing.Lemma 8.3.4. The law of limn\u2192\u221e z \u25e6 \u03c6(Xn) does not have any atoms.Proof. Let Bk(\u03c1) be the set of vertices of G at graph distance at most k from \u03c1, and let Gk bethe subnetwork of G induced by Bk(\u03c1) (i.e. the induced subgraph together with the conductancesinherited from G). Recall that the free effective conductance between a set A and a set B inan infinite graph G is given byC Feff(A\u2194 B ; G) = min{E(F ) : F (a) = 1 \u2200a \u2208 A, F (b) = 0 \u2200b \u2208 B} .The same variational formula also defines the effective conductance between two sets A and B ina finite network G, denoted Ceff(A \u2194 B;G). A related quantity is the effective conductance froma vertex v to infinity in GCeff(v \u2192\u221e;G) := limk\u2192\u221eC Feff(v \u2194 V \\Bk(\u03c1);G) = limk\u2192\u221eC Feff(v \u2194 V \\Bk(\u03c1);Gk+1),which is positive if and only if G is transient. See [173, \u00a72 and \u00a79] for background on electricalnetworks. The inequality (8.3.6) above implies that there exists a constant C such thatC Feff(A\u2194 B ; G) \u2264 CC Feff(\u03c6(A)\u2194 \u03c6(B) ; G\u2032) (8.3.8)1988.3. The Poisson boundaryfor each two sets of vertices A and B in G.Let \u03c1 \u2208 V and \u03be \u2208 \u2202D be fixed, and letA\u03b5(\u03be) := {v \u2208 V : |z \u25e6 \u03c6(v)\u2212 \u03be| \u2264 \u03b5}.In [17, Corollary 5.2], it is proven thatlim\u03b5\u21920C Feff(\u03c6(\u03c1)\u2194 \u03c6(A\u03b5(\u03be)) ; T \u2032)= 0.Applying (8.3.8) and Rayleigh monotonicity, we have thatC Feff(\u03c1\u2194 A\u03b5(\u03be) ; G) \u2264 CC Feff (\u03c6(\u03c1)\u2194 \u03c6(A\u03b5(\u03be)) ; G\u2032)\u2264 CC Feff(\u03c6(\u03c1)\u2194 \u03c6(A\u03b5(\u03be)) ; T \u2032) \u2212\u2212\u2212\u2192\u03b5\u219200. (8.3.9)The well-known inequality of [173, Exercise 2.34] then implies thatP\u03c1(hit A\u03b5(\u03be) before V \\Bk(\u03c1)) \u2264Ceff(\u03c1\u2194 A\u03b5(\u03be) \u2229Bk(\u03c1);Gk+1)Ceff(\u03c1\u2194 A\u03b5(\u03be) \u222a V \\Bk(\u03c1);Gk+1)\u2264 Ceff(\u03c1\u2194 A\u03b5(\u03be) \u2229Bk(\u03c1);Gk+1)Ceff(\u03c1\u2194 V \\Bk(\u03c1);Gk+1) .Applying the exhaustion characterisation of the free effective conductance [173, \u00a79.1], which statesthatC Feff(A\u2194 B;G) = limk\u2192\u221eCeff(A\u2194 B;Gk),we have thatP\u03c1(hit A\u03b5(\u03be))= limk\u2192\u221eP\u03c1(hit A\u03b5(\u03be) before V \\Bk(\u03c1))\u2264 limk\u2192\u221eCeff(\u03c1\u2194 A\u03b5(\u03be) \u2229Bk(\u03c1);Gk+1)Ceff(\u03c1\u2194 V \\Bk(\u03c1);Gk+1) = C Feff (\u03c1\u2194 A\u03b5(\u03be);G)Ceff (\u03c1\u2192\u221e;G) . (8.3.10)Combining (8.3.9) and (8.3.10) we deduce thatP\u03c1(limn\u2192\u221e z \u25e6 \u03c6(Xn) = \u03be)\u2264 lim\u03b5\u21920P\u03c1(hit A\u03b5(\u03be))= 0.This concludes the part of Theorem 8.1.2 concerning the existence of an embedding.Remark 8.3.5. The statement that P(hit A\u03b5(\u03be)) converges to zero as \u03b5 tends to zero for every\u03be \u2208 \u2202D can also be used to deduce convergence of z \u25e6 \u03c6(Xn) without appealing to the results of[12]: Suppose for contradiction that with positive probability z \u25e6 \u03c6(Xn) does not converge, so thatthere exist two boundary points \u03be1, \u03be2 \u2208 \u2202D in the closure of {z \u25e6 \u03c6(Xn) : n \u2265 0}. It is not hard tosee that in this case the closure of {z \u25e6 \u03c6(Xn) : n \u2265 0} must contain one of the boundary intervals[\u03be1, \u03be2] or [\u03be2, \u03be1]. However, if the closure of {z \u25e6 \u03c6(Xn) : n \u2265 0} contains an interval of positive1998.3. The Poisson boundarylength with positive probability, then there exists a point \u03be \u2208 \u2202D that is contained in the closureof {z \u25e6 \u03c6(Xn) : n \u2265 0} with positive probability. This contradicts the convergence of P(hit A\u03b5(\u03be))to zero.Proof of Theorem 8.1.2, identification of the Poisson boundary. Let \u3008Xn\u3009n\u22650 and \u3008Yn\u3009n\u22650 beindependent random walks on G started at \u03c1, and for each k \u2265 0 let \u3008Zkj \u3009j\u22650 be a random walk onG started at Xk that is conditionally independent of \u3008Xn\u3009n\u22650 and \u3008Y \u3009n\u22650 given Xk. Let P denotethe joint distribution of \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650 and all of the random walks \u3008Zkm\u3009m\u22650.Suppose that z is an embedding of G\u2032 in D such that z\u25e6\u03c6(Xn) converges to a point in \u2202D almostsurely and the law of the limit is non-atomic. Since the law of limn\u2192\u221e z \u25e6\u03c6(Xn) is non-atomic, therandom variables \u03be+ = limn\u2192\u221e z \u25e6 \u03c6(Xn) and \u03be\u2212 = limn\u2192\u221e z \u25e6 \u03c6(Yn) are almost surely distinct.LetQk := P(limm\u2192\u221e z \u25e6 \u03c6(Zkm) \u2208 (\u03be\u2212, \u03be+)\u2223\u2223\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650) .Using the non-atomicity of the law of limn\u2192\u221e z \u25e6 \u03c6(Xn), the same argument as in the proof ofTheorem 8.1.1 also shows thatlim supk\u2192\u221emin{Qk, 1\u2212Qk} > 0 almost surely. (8.3.11)We now come to a part of the proof that requires more substantial modification. Let \u3008vi\u3009i\u22650 bea path in G starting at \u03c1 such that z \u25e6 \u03c6(vi)\u2192 \u03be+.Given a path \u3008ui\u3009i\u22650 in G, let \u03a6(\u3008ui\u3009i\u22650) denote the path in G\u2032 formed by concatenating thepaths \u03a6(ui, ui+1) for i \u2265 0. For each k \u2265 0, let \u03c4k be the first time t such that the path \u03a6(Zkt\u22121, Zkt )intersects the union of the traces of the paths \u03a6(\u3008vi\u3009i\u22650) and \u03a6(\u3008Yn\u3009n\u22650). Since the union of thetraces of the paths \u03a6(\u3008vi\u3009i\u22650) and \u03a6(\u3008Yn\u3009n\u22650) either contains \u03c6(Xk) or disconnects \u03c6(Xk) from atleast one of the two intervals (\u03be\u2212, \u03be+) or (\u03be+, \u03be\u2212), we have thatP(\u03c4k <\u221e\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650) \u2265 min{Qk, 1\u2212Qk}and solim supk\u2192\u221eP(\u03c4k <\u221e\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650) > 0 (8.3.12)almost surely by (8.3.11).On the event that \u03c4k is finite, by definition of \u03a6, there exists a vertex u \u2208 {vi : i \u2265 0} \u222a {Yn :n \u2265 0} such that d\u2032(\u03c6(Z\u03c4 ), \u03c6(u)) \u2264 2\u03b1 + 2\u03b2, and consequently d(Z\u03c4 , u) \u2264 \u03b1(2\u03b1 + 3\u03b2) since \u03c6 is arough isometry. Since G has bounded degrees and edge conductances bounded above and below,c(e)\/c(u) \u2265 \u03b4 for some \u03b4 > 0. Thus, by the strong Markov property,P(\u3008Zkm\u3009m\u22650 hits {vi : i \u2265 0} \u222a {Yn : n \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650, {\u03c4k <\u221e})\u2265 \u03b4\u03b1(2\u03b1+3\u03b2) > 0 (8.3.13)2008.4. Identification of the Martin boundaryfor all k \u2265 0. Combining (8.3.12) and (8.3.13) yieldslim supk\u2192\u221eP(\u3008Zkm\u3009m\u22650 hits {vi : i \u2265 0} \u222a {Yn : n \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650) > 0 (8.3.14)almost surely. The same argument as in the proof of Theorem 8.1.1 also implies thatP(\u3008Zkm\u3009m\u22650 hits {Yn : n \u2265 0}\u2223\u2223\u2223 \u3008Xn\u3009n\u22650, \u3008Yn\u3009n\u22650) a.s.\u2212\u2212\u2212\u2192k\u2192\u221e0. (8.3.15)We conclude by combining (8.3.14) with (8.3.15) and applying Proposition 8.3.1 to \u03c8 = z \u25e6\u03c6 : V \u2192D \u222a \u2202D.8.4 Identification of the Martin boundaryIn this section, we prove Theorem 8.1.5 and deduce Theorem 8.1.4. We begin by proving thatthe rectangle tiling of a bounded degree triangulation with edge conductances bounded above andbelow does not have any accumulations of rectangles other than at the boundary circle R\/\u03b7Z\u00d7{1}.Proposition 8.4.1. Let T be a transient, simple, proper plane triangulation with bounded localgeometry. Then for every vertex v of T and every \u03b5 > 0, there exist at most finitely many verticesu of T such that the probability that a random walk started at u visits v is greater than \u03b5.Proof. Let C be a circle packing of T in the unit disc D, with associated embedding z of T . Benjaminiand Schramm [47, Lemma 5.3] proved that for every \u03b5 > 0 and \u03ba > 0, there exists \u03b4 > 0 such thatfor any v \u2208 V\u02dc with |z(v)| \u2265 1\u2212 \u03b4, the probability that a random walk from v ever visits a vertex usuch that |z(u)| \u2264 1 \u2212 \u03ba is at most \u03b5. (Their proof is given for c \u2261 1 but extends immediately tothis setting.) By setting \u03ba = 1\u2212 |z(v)|, it follows that for every \u03b5 > 0, there exists \u03b4 > 0 such thatfor every vertex u with |z(u)| \u2265 1\u2212 \u03b4, the probability that a random walk started at u hits v is atmost \u03b5. The claim follows since |z(u)| \u2265 1\u2212 \u03b4 for all but finitely many vertices u of T .Corollary 8.4.2. Let T be a transient proper plane triangulation with bounded local geometry, andlet S\u03c1 be a rectangle tiling of T . Then for every t \u2208 [0, 1), the cylinder R\/\u03b7Z\u00d7 [0, t] intersects onlyfinitely many rectangles S(e) \u2208 S\u03c1.We also require the following simple geometric lemma.Lemma 8.4.3. Let T be a transient proper plane triangulation with bounded local geometry, andlet S\u03c1 be the square tiling of T in the cylinder R\/\u03b7Z \u00d7 [0, 1]. Then for every sequence of vertices\u3008vn\u3009n\u22650 such that y(vn) \u2192 1 and \u03b8(vn) \u2192 \u03b80 for some (\u03b80, y0) as n \u2192 \u221e, there exists a path\u3008\u03b3n\u3009n\u22650 containing {vn : n \u2265 0} such that y(\u03b3n)\u2192 1 and \u03b8(\u03b3n)\u2192 \u03b80 as n\u2192\u221e.Proof. Define a sequence of vertices \u3008v\u2032n\u3009n\u22650 as follows. If the interval I(vn) is non-degenerate (i.e.has positive length), let v\u2032n = vn. Otherwise, let \u3008\u03b7j\u3009j\u22650 be a path from vn to \u03c1 in T , let j(n) be thesmallest j > 0 such that the interval I(\u03b7j(n)) is non-degenerate, and set v\u2032n = \u03b7j(n). Observe that2018.4. Identification of the Martin boundaryy(v\u2032n) = y(vn), that the interval I(v\u2032n) contains the singleton I(vn), and that the path \u03b7\u02dcn = \u3008\u03b7nj \u3009j(n)j=0from vn to v\u2032n in T satisfies (I(\u03b7\u02dcj), y(\u03b7\u02dcj)) = (I(vn), y(vn)) for all j < j(n).Let \u03b8\u2032(v\u2032n) \u2208 I(vn) be chosen so that the line segment `n between the points (\u03b8\u2032(v\u2032n), y(v\u2032n)) and(\u03b8\u2032(v\u2032n+1), y(v\u2032n+1)) in the cylinder R\/\u03b7Z\u00d7[0, 1] does not intersect any degenerate rectangles of S\u03c1 ora corner of any rectangles of S\u03c1 (this is a.s. the case if \u03b8\u2032(v\u2032n) is chosen uniformly from I(vn) for eachn). The line segment `n intersects some finite sequence of squares corresponding to non-degeneraterectangles of S\u03c1, which we denote en1 , . . . , enl(n). Let eni be oriented so that y(en+i ) > y(en\u2212i ) forevery i and n. For each 1 \u2264 i \u2264 l(n)\u2212 1, either1. the vertical sides of the rectangles S(eni ) and S(eni+1) have non-disjoint interiors, in whichcase eni and eni+1 lie in the boundary of a common face of T , or2. the horizontal sides of the rectangles S(eni ) and S(eni+1) have non-disjoint interiors, in whichcase eni and eni+1 share a common endpoint.In either case, since T is a triangulation, there exists an edge in T connecting en+i to en\u2212i+1 for each iand n. We define a path \u03b3n by alternatingly concatenating the edges eni and the edges connectingen+i to en\u2212i+1 as i increases from 1 to l(n). Define the path \u03b3 by concatenating all of the paths\u03b7\u02dcn \u25e6 \u03b3n \u25e6 (\u2212\u03b7\u02dcn+1)where \u2212\u03b7\u02dcn+1 denotes the reversal of the path \u03b7\u02dcn+1.Let M be an upper bound for the degrees of T and for the conductances and resistances of theedges of T . For each vertex v of T , there are at most M rectangles of S\u03c1 adjacent to I(v) fromabove, so that at least one of these rectangles has width at least length(I(v))\/M . This rectanglemust have height at least length(I(v))\/M2. It follows thatlength(I(v)) \u2264M\u22122 (1\u2212 y(v)) (8.4.1)for all v \u2208 T , and so for every edge e of T ,|y(e\u2212)\u2212 y(e+)| \u2264 1c(e)length(I(e\u2212))\u2264M\u22123(1\u2212 y(e\u2212))(8.4.2)By construction, every vertex w visited by the path \u03b7\u02dcn\u25e6\u03b3n\u25e6(\u2212\u03b7\u02dcn+1) has an edge emanating fromit such that the associated rectangle intersects the line segment `n, and consequently a neighbouringvertex w\u2032 such that y(w\u2032) \u2265 min{y(vn), y(vn+1)}. Applying (8.4.2), we deduce thaty(w) \u2265 (1 +M\u22123) min{y(vn), y(vn+1)} \u2212M\u22123for all vertices w visited by the path \u03b7\u02dcn \u25e6 \u03b3n \u25e6 (\u2212\u03b7\u02dcn+1), and consequently y(\u03b3k) \u2192 1 as k \u2192 \u221e.The estimate (8.4.2) then implies that length(I(\u03b3k)) \u2192 0 as k \u2192 \u221e. Since I(w) intersects theprojection to the boundary circle of the line segment `n for each vertex w visited by the path\u03b7\u02dcn \u25e6 \u03b3n \u25e6 (\u2212\u03b7\u02dcn+1), we deduce that \u03b8(\u03b3k)\u2192 \u03b80 as k \u2192\u221e.2028.4. Identification of the Martin boundaryWe also have the following similar lemma for circle packings.Lemma 8.4.4. Let T be a CP hyperbolic plane triangulation, and let C be a circle packing of Tin D with associated embedding z. Then for every sequence \u3008vn\u3009n\u22650 such that z(vn) converges asn \u2192 \u221e, there exists a path \u3008\u03b3n\u3009n\u22650 in T containing {vn : n \u2265 0} such that z(\u03b3n) converges asn\u2192\u221e.Proof. The proof is similar to that of Lemma 8.4.3, and we provide only a sketch. Draw a straightline segment in D between the centres of the circles corresponding to each consecutive pair ofvertices vn and vn+1. The set of circles intersected by the line segment contains a path \u03b3n in Tfrom vn to vn+1. (If this line segment is not tangent to any of the circles of C, then the set ofcircles intersected by the segment is exactly a path in T .) The path \u03b3 is defined by concatenatingthe paths \u03b3n. For every \u03b5 > 0, there are at most finitely many v for which the radius of the circlecorresponding to v is greater than \u03b5, since the sum of the squared radii of all the circles in thepacking is at most 1. Thus, for large n, all the circles corresponding to vertices used by the path \u03b3nare small. The circles that are intersected by the line segment between z(vn) and z(vn+1) thereforenecessarily have centers close to \u03be0 for large n. We deduce that z(\u03b3i)\u2192 \u03be0 as i\u2192\u221e.Proof of Theorem 8.1.5. Let \u3008Xn\u3009n\u22650 be a random walk on T . Our assumptions guarantee that\u03b8(Xn) and z(Xn) both converge almost surely as n\u2192\u221e and that the laws of these limits are bothnon-atomic and have support R\/\u03b7Z and \u2202D respectively.Suppose that \u03b3 is a path in T that visits each vertex at most finitely often. We claim that \u03b8(\u03b3i)converges if and only if z(\u03b3i) converges if and only if Xn almost surely does not hit \u03b3i infinitelyoften. We prove that \u03b8(\u03b3i) converges if and only if Xn almost surely does not hit \u03b3i infinitely often.The proof for z(\u03b3i) is similar.If \u03b8(\u03b3i) converges, then Xn almost surely does not hit \u03b3i infinitely often, since otherwiselimi\u2192\u221e \u03b8(\u03b3i) would be an atom in the law of limn\u2192\u221e \u03b8(Xn). Conversely, if \u03b8(\u03b3i) does not con-verge, then there exist at least two distinct points \u03b81, \u03b82 \u2208 R\/\u03b7Z such that \u03b81\u00d7{1} and \u03b82\u00d7{1} arein the closure of {(\u03b8(\u03b3i), y(\u03b3i)) : n \u2265 0}. Let \u3008Yn\u3009n\u22650 be a random walks started at \u03c1 independentof \u3008Xn\u3009n\u22650. Since the law of limn\u2192\u221e \u03b8(Xn) has full support, we have with positive probabilitythat limn\u2192\u221e \u03b8(Xn) \u2208 (\u03b81, \u03b82) and limn\u2192\u221e \u03b8(Yn) \u2208 (\u03b82, \u03b81). On this event, the union of the traces{Xn : n \u2265 0} \u222a {Yn : n \u2265 0} disconnects \u03b81\u00d7{1} from \u03b82\u00d7{1}, and consequently the path \u3008\u03b3i\u3009n\u22650must hit {Xn : n \u2265 0} \u222a {Yn : n \u2265 0} infinitely often. By symmetry, there is a positive probabilitythat \u3008Xn\u3009n\u22650 hits the trace {\u03b3i : n \u2265 0} infinitely often as claimed.We deduce that for every path \u03b3 in T that visits each vertex of T at most finitely often, \u03b8(\u03b3i)converges if and only if z(\u03b3i) converges. It follows from this and Lemmas 8.4.3 and 8.4.4 that for anysequence of vertices \u3008vi\u3009i\u22650 in T that includes each vertex of T at most finitely often, the sequence\u03b8(vi) converges if and only if z(vi) converges, and hence that the map \u03be 7\u2192 \u03b8(\u03be) = limz(v)\u2192\u03be \u03b8(v)is well defined. To see that this map is a homeomorphism, suppose that \u03ben is a sequence of pointsin \u2202D converging to \u03be. For every n, there exists a vertex vn \u2208 V such that |\u03ben \u2212 z(vn)| \u2264 1\/n and2038.4. Identification of the Martin boundary|\u03b8(\u03ben)\u2212 \u03b8(vn)| \u2264 1\/n. Thus, z(vn)\u2192 \u03be and we have|\u03b8(\u03be)\u2212 \u03b8(\u03ben)| \u2264 |\u03b8(\u03be)\u2212 \u03b8(vn)|+ |\u03b8(vn)\u2212 \u03b8(\u03ben)| \u2212\u2212\u2212\u2192n\u2192\u221e 0.The proof of the continuity of the inverse is similar.Proof of Theorem 8.1.4. By [17, Theorem 1.2], a sequence of vertices \u3008vi\u3009i\u22650 converges to a pointin the Martin boundary of T if and only if z(vi) converges to a point in \u2202D, and the map\u03be 7\u2192M\u03be := lim\u0131\u2192\u221eMvi where z(vi)\u2192 \u03beis a homeomorphism from \u2202D to \u2202M(T ). Combining this with Theorem 8.1.5 completes theproof.204Chapter 9Hyperbolic and parabolic unimodularrandom mapsSummary. We show that for infinite planar unimodular random rooted maps, many global geo-metric and probabilistic properties are equivalent, and are determined by a natural, local notionof average curvature. This dichotomy includes properties relating to amenability, conformal ge-ometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation.We also prove that every simply connected unimodular random rooted map is sofic, that is, aBenjamini-Schramm limit of finite maps.9.1 IntroductionIn the classical theory of Riemann surfaces, the Uniformization Theorem states that every simplyconnected, non-compact Riemann surface is conformally equivalent to either the plane or the disc,which are inequivalent to each other by Liouville\u2019s theorem. The dichotomy provided by thistheorem manifests itself in several different ways, relating to analytic, geometric and probabilisticproperties of surfaces. In particular, if S is a simply connected, non-compact Riemann surface,then eitherS is parabolic: it is conformally equivalent to the plane, admits a compatible Rieman-nian metric of constant curvature 0, does not admit non-constant bounded harmonicfunctions, and is recurrent for Brownian motion,or elseS is hyperbolic: it is conformally equivalent to the disc, admits a compatible Rieman-nian metric of constant curvature \u22121, admits non-constant bounded harmonic functions,and is transient for Brownian motion.In the 1990\u2019s, a discrete counterpart to this dichotomy began to develop in the setting of boundeddegree planar graphs [35, 47, 48, 121, 122]. A milestone in this theory was the work of He andSchramm [121, 122], who studied circle packings of infinite triangulations of the plane. They provedthat every infinite triangulation of the plane can be circle packed, either in the unit disc or in theplane, but not both. A triangulation is called CP hyperbolic or CP parabolic accordingly. Heand Schramm also connected the circle packing type to isoperimetric and probabilistic properties2059.1. Introductionof the triangulation, showing in particular that, in the bounded degree case, CP parabolicity isequivalent to the recurrence of simple random walk. Later, Benjamini and Schramm [47, 48]provided an analytic aspect to this dichotomy, showing that every bounded degree, infinite planargraph admits non-constant bounded harmonic functions if and only if it is transient for simplerandom walk, and that in this case the graph also admits non-constant bounded harmonic functionsof finite Dirichlet energy. Most of this theory fails without the assumption of bounded degrees, asone can easily construct pathological counterexamples to the theorems above.The goal of this paper is to develop a similar theory for unimodular random rooted maps,without the assumption of bounded degree. In our earlier work [21], we studied circle packingsof, and random walks on, random plane triangulations of unbounded degree. In this paper, westudy many further properties of unimodular random planar maps, which we do not assume tobe triangulations. Our main result may be stated informally as follows; see Theorem 9.1.1 for acomplete and precise statement.Theorem (The Dichotomy Theorem). Every infinite, planar, unimodular random rooted planarmap is either hyperbolic or parabolic. The map is hyperbolic if and only if its average curvatureis negative and is parabolic if and only if its average curvature is zero. The type of a unimodularrandom rooted map determines many of its properties.The many properties we show to be determined by the type of the map are far-reaching, relatingto aspects of the map including amenability, random walks, harmonic functions, spanning forests,Bernoulli bond percolation, and the conformal type of associated Riemann surfaces. The seeds ofsuch a dichotomy were already apparent in [21], in which we proved that a unimodular randomrooted plane triangulation is CP parabolic almost surely if and only if the expected degree of theroot is six (which is equivalent to the average curvature being zero), if and only if the triangu-lation is invariantly amenable \u2013 a notion of amenability due to Aldous and Lyons [7] that isparticularly suitable to unimodular random rooted graphs. A notable property that is not a part ofTheorem 9.1.1 is recurrence of the random walk: while every hyperbolic unimodular random rootedmap is transient, not every parabolic unimodular random rooted map is recurrent. (Theorem 9.1.1can be combined with the work of Gurel-Gurevich and the third author [107] to deduce that aparabolic unimodular random planar map is recurrent under the additional assumption that thedegree of the root has an exponential tail.)A map is a proper (i.e., locally finite) embedding of a graph into an oriented surface viewedup to orientation preserving homeomorphisms of the surface. (Other definitions extend to non-orientable maps, but we shall not be concerned with those here.) A rooted map is a map togetherwith a distinguished root vertex. The map is called planar if the surface is homeomorphic to anopen subset of the sphere, and is simply connected if the surface is homeomorphic to the sphereor the plane. (In particular, every simply connected map is planar.) A random rooted map issaid to be unimodular if it satisfies the mass-transport principle, which can be interpreted asmeaning that \u2018every vertex of the map is equally likely to be the root\u2019. See Section 10.2 for precisedefinitions of each of these terms.2069.1. IntroductionThe curvature of a map is a local geometric property, closely related to the Gaussian curvatureof manifolds that may be constructed from the map; see Section 9.4 for a precise definition. Forone natural manifold constructed from the map by gluing together regular polygons, the curvatureat each vertex v is\u03ba(v) = 2pi \u2212\u2211f\u22a5vdeg(f)\u2212 2deg(f)pi,where the sum is taken over faces of the map incident to v, and a face is counted with multiplicityif more than one of the corners of the face are located at v. We can define the average curvature of(M,\u03c1) to be the expectation E[\u03ba(\u03c1)]. Theorem 9.4.9 states that the average curvature is a canoni-cal quantity associated to the random map, in the sense that any unimodular way of associating amanifold to the map will result in the same average curvature. Observe that the average curvatureof a unimodular random triangulation is equal to (6 \u2212 E[deg(\u03c1)])pi\/3, so that a unimodular trian-gulation has expected degree greater than six if and only if it has negative average curvature. Thisrelates the dichotomy described in [21] to that of Theorem 9.1.1.Classical examples of unimodular random maps are provided by Voronoi diagrams of stationarypoint processes [37] and (slightly modified) Galton-Watson trees [7, Example 1.1], as well as latticesin the Euclidean and hyperbolic planes, and arbitrary local limits of finite maps. Many localmodifications of maps, such as taking Bernoulli percolation or uniform or minimal spanning trees,preserve unimodularity, giving rise to many additional examples. Unimodular random maps, mostnotably the uniform infinite planar triangulation (UIPT) [24] and quadrangulation (UIPQ) [73, 157],have also been studied in the context of 2-dimensional quantum gravity; see the survey [98] andreferences therein. More recently, hyperbolic variants of the UIPT have been constructed [23, 75].Many of these examples do not have uniformly bounded degrees, so that the deterministic theoryis not applicable to them.Our results also have consequences for unimodular random maps that are not planar, which wedevelop in Section 9.7.9.1.1 The Dichotomy TheoremSince many of the notions tied together in the following theorem are well known, we first state thetheorem, and defer detailed definitions to individual sections dealing with each of the properties.Theorem 9.1.1 (The Dichotomy Theorem). Let (M,\u03c1) be an infinite, ergodic, unimodular randomrooted planar map and suppose that E[deg(\u03c1)] <\u221e. Then the average curvature of (M,\u03c1) is non-positive and the following are equivalent:1. (M,\u03c1) has average curvature zero.2. (M,\u03c1) is invariantly amenable.3. Every bounded degree subgraph of M is amenable almost surely.4. Every subtree of M is amenable almost surely.5. Every bounded degree subgraph of M is recurrent almost surely.6. Every subtree of M is recurrent almost surely.2079.1. Introduction7. (M,\u03c1) is a Benjamini-Schramm limit of finite planar maps.8. (M,\u03c1) is a Benjamini-Schramm limit of a sequence \u3008Mn\u3009n\u22650 of finite maps such thatgenus(Mn)#{vertices of Mn} \u2212\u2212\u2212\u2192n\u2192\u221e 0.9. The Riemann surface associated to M is conformally equivalent to either the plane C or thecylinder C\/Z almost surely.10. M does not admit any non-constant bounded harmonic functions almost surely.11. M does not admit any non-constant harmonic functions of finite Dirichlet energy almostsurely.12. The laws of the free and wired uniform spanning forests of M coincide almost surely.13. The wired uniform spanning forest of M is connected almost surely.14. Two independent random walks on M intersect infinitely often almost surely.15. The laws of the free and wired minimal spanning forests of M coincide almost surely.16. Bernoulli(p) bond percolation on M has at most one infinite connected component for everyp \u2208 [0, 1] almost surely (in particular, pc = pu).17. M is vertex extremal length parabolic almost surely.In light of this theorem, we call a unimodular random rooted map (M,\u03c1) with E[deg(\u03c1)] < \u221eparabolic if its average curvature is zero (and, in the planar case, clauses (1)\u2013(17) all hold), andhyperbolic if its average curvature is negative (and, in the planar case, the clauses all fail).As one might guess from the structure of Theorem 9.1.1, the proof consists of many separatearguments for the different implications. Some of the implications are already present in theliterature, and part of this paper is spent surveying the earlier works that form the individualimplications between the long list of equivalent items in Theorem 9.1.1. For the sake of completenesswe also include proofs of several standard technical results from the ergodic theory literature usingprobabilistic terminology.Some of the implications in Theorem 9.1.1 hold in any graph. For example, (10) implies (11)for any graph, and (14) and (13) are always equivalent (see [44]). Other implications hold for anyplanar graph. For example (10) is equivalent to (14), see [40]. We do not provide a comprehensivelist of the assumptions needed for each implication, but some of this information is encoded inFigure 9.1.Most of the paper is dedicated to proving the theorem under the additional assumption that Mis simply connected; the multiply-connected case is easier and is handled separately in Section 9.7.The logical structure of the proof in the simply connected case is summarized in Figures 9.1 and 9.2.Figure 9.1 also shows which implications were already known and which are proved in the presentpaper.2089.1. IntroductionConnectivityoftheFUSFLipton\u2013TarjanBLPSBLPSAldous\u2013LyonsMasstransport+dualityBenjamini,Curien,GeorgakopoulosAldous\u2013LyonsInvariantly amenableAverage curvature zeroConformally parabolicWMSF = FMSF pc = puHarmonicDirichletfunctions areconstantLiouvilleIntersectionPropertyWUSF connectedBenjamini-SchrammLPSBLS,Aldous-LyonsAldous\u2013LyonsBenjamini\u2013SchrammBenjamini-Schrammlimit of finitelow-genus mapsBenjamini-Schrammlimit of finiteplanar mapsWUSF = FUSFVELparabolicSubgraphconditionsFigure 9.1: The logical structure for the proof of Theorem 9.1.1 in the simply connected case.Implications new to this paper are in red. Blue implications hold for arbitrary graphs; the orangeimplication holds for arbitrary planar graphs, and green implications hold for unimodular randomrooted graphs even without planarity. A few implications between items that are known but notused in the proof are omitted.2099.1. Introduction18172 1311101415 169Theorem 3.19Theorem3.18Lemma3.17Lemma3.16Corollary3.5Theorem3.6Corollary 4.7Proposition5.5Theorem5.6Proposition5.7Theorem 5.8Proposition 5.10Theorem 6.1Theorem 5.22Theorem5.21Theorem5.1Corollary5.12Corollary5.14Theorem 5.33 45 6712Theorem 5.23Corollary5.25Proposition5.2Figure 9.2: The numbers of the theorems, propositions, lemmas and corollaries forming the individ-ual implications used to prove Theorem 9.1.1 in the simply connected case. Unlabelled implicationsare trivial.2109.2. Unimodular maps9.1.2 Unimodular planar maps are soficLet \u3008Gn\u3009n\u22650 be a sequence of (possibly random) finite graphs. We say that a random rootedgraph (G, \u03c1) is the Benjamini-Schramm limit of the sequence \u3008Gn\u3009n\u22650 if the random rootedgraphs (Gn, \u03c1n) converge in distribution to (G, \u03c1) with respect to the local topology on rootedgraphs (see Section 9.2.2), where \u03c1n is a uniform vertex of Gn. Benjamini-Schramm limits of finitemaps and networks are defined similarly, except that the local topology takes into account theadditional structure. When Gn is a uniformly chosen map of size n, this construction gives rise tothe aforementioned UIPT and UIPQ.It is easy to see that every (possibly random) finite graph with conditionally uniform root isunimodular. Moreover, unimodularity is preserved under distributional limits in the local topology.It follows that every Benjamini-Schramm limit of finite random graphs is unimodular. A randomrooted graph that can be obtained in this way is called sofic. It is a major open problem todetermine whether the converse holds, that is, whether every unimodular random rooted graphis sofic [7, Section 10]. The next theorem answers this question positively for simply connectedunimodular random maps.Theorem 9.1.2. Every simply connected unimodular random rooted map is sofic.The proof of Theorem 9.1.2 relies on the corresponding result for trees, which is due to Bowen[56], Elek [85], and Benjamini, Lyons and Schramm [46]. As was observed by Elek and Lippner [86],it follows that treeable unimodular graphs (that is, unimodular graphs that exhibit an invariantspanning tree) are sofic, and so the key new step is proving the connectivity of the free uniformspanning forest.Note that the finite maps converging to a given infinite unimodular random rooted map (M,\u03c1)need not be planar. Indeed, Theorem 9.1.1 characterises the Benjamini-Schramm limits of finiteplanar maps exactly as the parabolic unimodular random rooted maps. Moreover, if \u3008Mn\u3009n\u22650 is asequence of finite maps converging to an infinite hyperbolic unimodular random rooted map, thenthe approximating maps Mn must have genus comparable to their number of vertices as n tendsto infinity.9.2 Unimodular maps9.2.1 MapsWe provide here a brief background to the concept of maps, and refer the reader to [160, Chapter1.3] for a comprehensive treatment. Let G = (V,E) be a connected graph, which may containself-loops and multiple edges. An embedding of a graph in a surface S is a drawing of the graphin the surface with non-crossing edges. Given an embedding, the connected components of thecomplement of the image of G are called faces. An embedding is said to be proper if the followingconditions hold:2119.2. Unimodular maps1. it is locally finite (every compact set in S intersects finitely many edges),2. every face is homeomorphic to an open disc, and3. for every face f , if we consider the oriented edges of G that have their right hand side incidentto f , and consider the permutation that maps each such oriented edge to the oriented edgefollowing it in the clockwise order around the face, then this permutation has a single orbit.(Note that an edge can have the same face both to its right and its left.)For example, the complete graph on three vertices can be properly embedded in the sphere butnot in the plane. If S is simply connected or compact, then any embedding that satisfies (1) and(2) must also satisfy (3). For this reason, the condition (3) is not included in many references thatdeal primarily with finite maps. An example of an embedding that satisfies (1) and (2) but not (3)is given by drawing Z along a straight line in an infinite cylinder.We define a (locally finite) map M to be a connected, locally finite graph G together withan equivalence class of proper embeddings of G into oriented surfaces, where two embeddings areequivalent if there is an orientation preserving homeomorphism between the two surfaces thatsends one embedding to the other. If M is a map with underlying graph G, we refer to any properembedding of G that falls into the equivalence class of embeddings corresponding to M as anembedding of M . A map is said to be planar if it is embedded into a surface homeomorphic toan open subset of the sphere, and is said to be simply connected if it is embedded into a simplyconnected surface (which is necessarily homeomorphic to either the sphere or the plane).Let M be a map with underlying graph G and let z be a proper embedding of M into a surfaceS. If every face of M has finite degree, the dual map of M , denoted M \u2020 is defined as follows.The underlying graph of M \u2020, denoted G\u2020, has the faces of M as vertices, and has an edge drawnbetween two faces of M for each edge in M that is incident to both of the faces. We define anembedding z\u2020 of G\u2020 into S by placing each vertex of G\u2020 in the interior of the corresponding face ofM and each edge of G\u2020 so that it crosses the corresponding edge of G but no others. We define M \u2020to be the map with underlying graph G represented by the pair (S, z\u2020): Although the embedding z\u2020is not uniquely defined, every choice of z\u2020 defines the same map. The construction gives a canonicalbijection between edges of G and edges of G\u2020. We write e\u2020 for the edge of G\u2020 corresponding to e.If e is an oriented edge, we let e\u2020 be oriented so that it crosses e from right to left as viewed fromthe orientation of e.Despite their topological definitions, maps and their duals can in fact be defined entirely com-binatorially. Given any graph, we consider each edge as two oriented edges in opposite directions.We write E\u2192 = E\u2192(G) for the set of oriented edges of a graph. For each directed edge e, we havea head e+ and tail e\u2212, and write \u2212e for the reversal of e. Given a map M and a vertex v of M , let\u03c3v = \u03c3v(M) be the cyclic permutation of the set {e \u2208 E\u2192 : e\u2212 = v} of oriented edges emanatingfrom v corresponding to counter-clockwise rotation in S. This procedure defines a bijection betweenmaps and graphs labelled by cyclic permutations.2129.2. Unimodular mapsFigure 9.3: Different maps with the same underlying graph. The two maps both have K4 as theirunderlying graph, but the left is a sphere map while the right is a torus map. Both maps arerepresented both abstractly as a graph together with a cyclic permutation of the edges emanatingfrom each vertex (left) and as a graph embedded in a surface (right).Theorem 9.2.1 ([160]). Given a connected, locally finite graph G and a collection of cyclic per-mutations \u03c3v of the sets {e \u2208 E\u2192 : e\u2212 = v}, there exists a unique map M with underlying graph Gsuch that \u03c3(M) = \u03c3.In light of Theorem 9.2.1, we identify a map M with the pair (G, \u03c3). Given such a combinatorialspecification of a map M as a pair (G, \u03c3), we may form an embedding of the map into a surfaceS(M) by gluing topological polygons according to the combinatorics of the map (see Figures 9.3and 9.4). The faces of a map M = (G, \u03c3) can be defined abstractly as orbits of the permutation\u03c3\u2020 : E\u2192 \u2192 E\u2192 defined by \u03c3\u2020(e) = \u03c3\u22121(\u2212e) for each e \u2208 E\u2192. The dual e\u2020 of a directed edge eis defined to have the orbit of e as its tail and the orbit of \u2212e as its head, so that we again havea bijection between directed edges of M and their duals. Using this bijection, we can consider\u03c3\u2020 to acts on dual edges, and the dual map M \u2020 is then constructed abstractly as M \u2020 = (G\u2020, \u03c3\u2020).We define maps with infinite degree vertices, and duals of maps with infinite degree faces, directlythrough this abstract formalism.If e is an oriented edge in a map, we write e` for the face of M to the left of e and er for theface of M to the right of e, so that e` = (e\u2020)\u2212 and er = (e\u2020)+. Given a map M , we write f \u22a5 vif the face f is incident to the vertex v, that is, if there exists an oriented edge e of M such thate\u2212 = v and e` = f . When writing a sum of the form\u2211f\u22a5v, we use the convention that a face f iscounted with multiplicity according to the number of oriented edges e of M such that e\u2212 = v ande` = f . Similarly, when writing a sum of the form\u2211u\u223cv, we count each vertex u with multiplicityaccording to the number of oriented edges e of M such that e\u2212 = v and e+ = u. The degree of aface is defined to be the number of oriented edges with er = f , i.e., the degree of the face in thedual.9.2.2 Unimodularity and the mass transport principleAs noted, a rooted graph (G, \u03c1) is a connected, locally finite (multi)graph G = (V,E) togetherwith a distinguished vertex \u03c1, called the root. A graph isomorphism \u03c6 : G\u2192 G\u2032 is an isomorphismof rooted graphs if it maps the root to the root.The local topology (see [50]) is the topology on the set G\u2022 of isomorphism classes of rooted2139.2. Unimodular mapsgraphs induced by the metricdloc((G, \u03c1), (G\u2032, \u03c1\u2032))= e\u2212R,whereR = R((G, \u03c1), (G\u2032, \u03c1\u2032))= sup{R \u2265 0 : BR(G, \u03c1) \u223c= BR(G\u2032, \u03c1\u2032)},i.e., the maximal radius such that the balls BR(G, \u03c1) and BR(G\u2032, \u03c1\u2032) are isomorphic as rootedgraphs.A random rooted graph is a random variable taking values in the space G\u2022 endowed withthe local topology. Similarly, a doubly-rooted graph is a graph together with an ordered pairof distinguished (not necessarily distinct) vertices. Denote the space of isomorphism classes ofdoubly-rooted graphs equipped with this topology by G\u2022\u2022.The spaces of rooted and doubly-rooted maps are defined similarly and are denoted M\u2022 andM\u2022\u2022 respectively. For this, an isomorphism \u03c6 of rooted maps is preserves the roots and the mapstructure, so that \u03c6 \u25e6 \u03c3 = \u03c3\u2032 \u25e6 \u03c6 for vertices inside the balls.A further generalisation of these spaces will also be useful. A marked graph (referred to byAldous and Lyons [7] as a network) is defined to be a locally finite, connected graph together witha function m : E \u222a V \u2192 X assigning each vertex and edge of G a mark in some Polish space X,referred to as the mark space. The local topology on the set of isomorphism classes of rootedgraphs with marks in X is the topology induced by the metricdloc((G, \u03c1,m), (G\u2032, \u03c1\u2032,m\u2032)) = e\u2212Rwhere R((G, \u03c1,m), (G\u2032, \u03c1\u2032,m\u2032)) is the largest R such that there exists an isomorphism of rootedgraphs \u03c6 : (BG(\u03c1, b),m, \u03c1) \u2192 (BG\u2032(\u03c1\u2032, b), \u03c1\u2032) such that dX(m\u2032(\u03c6(x)),m(x)) \u2264 1\/R for every vertexor edge x of BG(\u03c1, n), and dX is a metric compatible with the topology of X. The space ofisomorphism classes of rooted marked graphs with marks in X is denoted GX\u2022 . The space of rootedmarked maps is defined similarly.It is possible to consider rooted maps and rooted marked maps as rooted marked graphs byencoding the permutations \u03c3v in the marks \u2013 in particular, this means that any statement thatholds for all unimodular random rooted marked graphs also holds for all unimodular random rootedmarked maps. See [7, Example 9.6].A mass transport is a Borel function f : G\u2022\u2022 \u2192 [0,\u221e]. A random rooted graph (G, \u03c1) is saidto be unimodular if it satisfies the Mass Transport Principle: for every mass transport f ,E[\u2211v\u2208Vf(G, \u03c1, v)]= E[\u2211u\u2208Vf(G, u, \u03c1)]. (MTP)That is,Expected mass out equals expected mass in.A probability measure P on G\u2022 is said to be unimodular if a random rooted map with law P is2149.2. Unimodular mapsunimodular. Unimodular probability measures on rooted maps, marked graphs and marked mapsare defined similarly.The Mass Transport Principle was first introduced by Ha\u00a8ggstro\u00a8m [110] to study dependentpercolation on Cayley graphs. The formulation of the Mass Transport Principle presented here wassuggested by Benjamini and Schramm [50] and developed systematically by Aldous and Lyons [7].The unimodular probability measures on G\u2022 form a weakly closed, convex subset of the space ofprobability measures on G\u2022, so that weak limits of unimodular random graphs are unimodular. Inparticular, a weak limit of finite graphs with uniformly chosen roots is unimodular: such a limitof finite graphs is referred to as a Benjamini-Schramm limit. It is a major open problem todetermine whether all unimodular random rooted graphs arise as Benjamini-Schramm limits offinite graphs [7, \u00a710].We will make frequent use of the fact that unimodularity is stable under most reasonable waysof modifying a graph locally without changing its vertex set. The following lemma is a formalizationof this fact.Lemma 9.2.2. Let X1 and X2 be polish spaces. Let (G, \u03c1,m) be a unimodular random rootedX1-marked graph, and suppose that (G\u2032, \u03c1,m\u2032) is a random rooted marked graph with the samevertex set as G, such that, for every pair of vertices u, v in G, the conditional distribution of(G\u2032, u, v,m\u2032) given (G, \u03c1,m) coincides a.s. with some measurable function of the isomorphism classof (G, u, v,m). Then (G\u2032, \u03c1,m\u2032) is unimodular.Proof. Let f : GX2\u2022\u2022 \u2192 [0,\u221e] be a mass transport. For each pair of vertices u, v in G, the conditionalexpectationF (G, u, v,m) = E[f(G\u2032, u, v,m\u2032) | (G, \u03c1,m)]coincides a.s. with a measurable function of (G, u, v,m), and so is itself a mass transport. Since(G, \u03c1,m) is unimodular we deduce thatE\u2211vf(G\u2032, \u03c1, v,m\u2032) = E\u2211vF (G, \u03c1, v,m)= E\u2211vF (G, v, \u03c1,m) = E\u2211uf(G\u2032, u, \u03c1,m\u2032), (9.2.1)verifying that (G\u2032, \u03c1,m\u2032) satisfies the Mass Transport Principle.Later in the paper, we will often claim that various random rooted graphs obtained fromunimodular random rooted graphs in various ways are unimodular. These claims can always bejustified either as a direct consequence of Lemma 9.2.2, or otherwise by applying the Mass-TransportPrinciple to the conditional expectation as in eq. (9.2.1).2159.2. Unimodular maps9.2.3 ReversibilityRecall that the simple random walk on a locally finite graph G = (V,E) is the Markov process\u3008Xn\u3009n\u22650 on the state space V with transition probabilities p(u, v) defined to be the fraction of edgesemanating from u that end in v. A probability measure P on G\u2022 that is supported on rooted graphswith more than one vertex is said to be stationary if, when (G, \u03c1) is a random rooted graph withlaw P and \u3008Xn\u3009n\u22650 is a simple random walk on G started at the root, (G, \u03c1) and (G,Xn) have thesame distribution for all n. The measure P is said to be reversible if furthermore(G, \u03c1,Xn)d= (G,Xn, \u03c1)for all n. A random rooted graph is said to be reversible if its law is reversible. Reversible randomrooted maps, marked graphs and marked maps are defined similarly. Every reversible randomrooted graph is clearly stationary, but the converse need not hold in general [36, Examples 3.1 and3.2].Let G\u2194 be the space of isomorphism classes of connected, locally finite graphs equipped witha distinguished (not necessarily simple) bi-infinite path, which we endow with a natural variant ofthe local topology. If (G, \u03c1) is a reversible random rooted graph and \u3008Xn\u3009n\u22650 and \u3008X\u2212n\u3009n\u22650 areindependent simple random walks started from \u03c1, then the sequence\u3008(G, \u3008Xn+k\u3009n\u2208Z)\u3009k\u2208Zof G\u2194-valued random variables is stationary.The following correspondence between unimodular and reversible random rooted graphs is im-plicit in [7] and was proven explicitly in [38]. Similar correspondences also hold between unimodularand reversible random rooted maps, marked graphs and marked maps.Proposition 9.2.3 ([7, 38]). Let P be a unimodular probability measure on G\u2022 that is supported onrooted graphs with more than one vertex and has P[deg(\u03c1)] < \u221e, and let Prev denote the deg(\u03c1)-biasing of P, defined byPrev((G, \u03c1) \u2208 A ) := P[deg(\u03c1)1((G, \u03c1) \u2208 A )]P[deg(\u03c1)]for every Borel set A \u2286 G\u2022. Then Prev is reversible. Conversely, if P is a reversible probabilitymeasure on G\u2022, then biasing P by deg(\u03c1)\u22121 yields a unimodular probability measure on G\u2022.9.2.4 ErgodicityA probability measure P on G\u2022 is said to be ergodic if P(A ) \u2208 {0, 1} for every event A \u2286 G\u2022 thatis invariant to changing the root in the sense that(G, \u03c1) \u2208 A \u21d0\u21d2 (G, v) \u2208 A for all v \u2208 V.2169.2. Unimodular mapsA random rooted graph is said to be ergodic if its law is ergodic. Aldous and Lyons [7, \u00a74] provedthat the ergodic unimodular probability measures on G\u2022 are exactly the extreme points of theweakly closed, convex set of unimodular probability measures on. It follows by Choquet theorythat every unimodular random rooted graph is a mixture of ergodic unimodular random rootedgraphs, meaning that the graph may be sampled by first sampling a random ergodic unimodularprobability measure on G\u2022 from some distribution, and then sampling from this randomly chosenmeasure. In particular, to prove almost sure statements about unimodular random rooted graphs,it suffices to consider the ergodic case. Analogous statements hold for unimodular random rootedmaps, marked graphs, and marked maps.9.2.5 DualityLet P be a unimodular probability measure on M\u2022 such that M \u2020 is locally finite P-a.s., and let(M,\u03c1) be a random rooted map with law Prev. Conditional on (M,\u03c1), let \u03b7 be an oriented edge ofM sampled uniformly at random from the set E\u2192\u03c1 , and let \u03c1\u2020 = \u03b7r. We define P\u2020rev to the law ofthe random rooted map (M \u2020, \u03c1\u2020).Proposition 9.2.4 (Aldous-Lyons [7, Example 9.6]). Let P be a unimodular probability measureonM\u2022 such that M \u2020 is locally finite P-a.s. and P[deg(\u03c1)] <\u221e. Then P\u2020rev is a reversible probabilitymeasure on M\u2022.Intuitively, under the measure Prev, the edge \u03b7 is \u2018uniformly distributed\u2019 among the edges ofM . Since the map sending each edge to its dual is a bijection, it should follow that \u03b7\u2020 is uniformlydistributed among the edges of M \u2020, making the measure P\u2020rev reversible also.We define P\u2020 to be the unimodular measure on M\u2022 obtained as the deg(\u03c1)\u22121-biasing of P\u2020rev.We refer to P\u2020 as the dual of P and say that P is self-dual if P\u2020 = P. We write Erev, E\u2020rev, and E\u2020for the associated expectation operators. We may express P\u2020 directly in terms of P byP\u2020((M \u2020, \u03c1\u2020) \u2208 A ) = E[\u2211f\u22a5\u03c1deg(f)\u22121]\u22121E[\u2211f\u22a5\u03c1deg(f)\u221211((M \u2020, f) \u2208 A)](9.2.2)for every Borel set A \u2286M\u2022. In particular, we can calculateE\u2020[deg(\u03c1\u2020)] = E[\u2211f\u22a5\u03c1deg(f)\u22121]\u22121E[\u2211f\u22a5\u03c1deg(f)\u22121 deg(f)]= E[\u2211f\u22a5\u03c1deg(f)\u22121]\u22121E[deg(\u03c1)] <\u221e. (9.2.3)The factor E[\u2211f\u22a5\u03c1 deg(f)\u22121]\u22121 may be thought of as the ratio of the number of vertices of M tothe number of faces of M .Example 9.2.5. Let M be a finite map and let P be the law of the unimodular random rooted map(M,\u03c1) where \u03c1 is a vertex of M chosen uniformly at random. Then P\u2020 is the law of the unimodular2179.3. Percolations and invariant amenabilityrandom rooted map (M \u2020, \u03c1\u2020) obtained by rooting the dual M \u2020 of M at a uniformly chosen face \u03c1\u2020of M .9.3 Percolations and invariant amenability9.3.1 Markings and percolationsLet (G, \u03c1) be a unimodular random rooted graph with law P. Given a random rooted graph (G, \u03c1)and a polish space X, an X-marking of (G, \u03c1) is a random assignment m : E \u222aV \u2192 X of marks tothe edges and vertices of (G, \u03c1) (possibly defined on some larger probability space) such that therandom rooted marked graph (G, \u03c1,m) is unimodular. A percolation on (G, \u03c1) is a {0, 1}-markingof (G, \u03c1), which we think of as a random subgraph of G consisting of the open edges satisfying\u03c9(e) = 1, and open vertices satisfying \u03c9(v) = 1. We assume without loss of generality that if anedge is open then so are both of its endpoints. We call \u03c9 connected if the subgraph of \u03c9 spannedby the vertices {v \u2208 V : \u03c9(v) = 1} is connected. We call \u03c9 a bond percolation if \u03c9(v) = 1for every vertex v \u2208 V almost surely. The cluster K\u03c9(v) of \u03c9 at the vertex v is the connectedcomponent of v in \u03c9. A percolation is said to be finitary if all of its clusters are finite almostsurely.Let us gather here the following simple and useful technical lemmas concerning markings andpercolations.Lemma 9.3.1. Let (G, \u03c1,m) be a unimodular random marked graph. Let \u03c9 be a percolationon (G, \u03c1,m) and let m|K\u03c9(\u03c1) denote the restriction of m to K\u03c9(\u03c1). Then the conditional law of(K\u03c9(\u03c1), \u03c1,m|K\u03c9)\u03c1)) given that \u03c9(\u03c1) = 1 is also unimodular. In particular, if \u03c9 is finitary, then \u03c1 isuniformly distributed on its cluster.Proof. Let f be a mass transport, and let g be the mass transportg(G, u, v,m, \u03c9) := 1(\u03c9(u) = \u03c9(v) = 1, v \u2208 K\u03c9(u))f(K\u03c9(u), u, v,m|K\u03c9(u)).Then, applying the Mass Transport principle for (G, \u03c1, \u03c9), we haveE\u2211v\u2208K\u03c9(\u03c1)f(K\u03c9(\u03c1), \u03c1, v,m|K\u03c9(\u03c1)) = E\u2211v\u2208V (G)g(G, \u03c1, v,m, \u03c9) = E\u2211v\u2208V (G)g(G, v, \u03c1,m, \u03c9)= E\u2211v\u2208K\u03c9(\u03c1)f(K\u03c9(\u03c1), v, \u03c1,m|K\u03c9(\u03c1)).Lemma 9.3.2 (Coupling markings I). Let (G, \u03c1) be a unimodular random rooted graph and let{mi : i \u2208 I} be a set of markings of (G, \u03c1) indexed by a countable set I and with mark spaces{Xi : i \u2208 I}. Then there exists a\u220fi\u2208I Xi-marking m of (G, \u03c1) such that (G, \u03c1, pii \u25e6m) and (G, \u03c1,mi)have the same distribution for all i \u2208 I, where pii denotes the projection of\u220fi\u2208I Xi onto Xi for eachi \u2208 I.2189.3. Percolations and invariant amenabilityProof. One such marking is given by first sampling (G, \u03c1) and then sampling the markings miindependently from their conditional distributions given (G, \u03c1). The details of this construction areare omitted.Lemma 9.3.3 (Coupling markings II). Let (G, \u03c1) be a unimodular random rooted graph, let \u03c9 be aconnected percolation on (G, \u03c1), and let m be an X-marking of the unimodular random rooted graphgiven by sampling (K\u03c9(\u03c1), \u03c1) conditional on \u03c9(\u03c1) = 1. Then there exists a X-marking m\u02c6 of (G, \u03c1)such that the laws of (K\u03c9(\u03c1), \u03c1, m\u02c6|K\u03c9(\u03c1)) and (K\u03c9(\u03c1), \u03c1,m) coincide.Proof. One such marking is given as follows: First sample (K\u03c9(\u03c1), \u03c1) from its conditional distri-bution given \u03c9(\u03c1) = 1. Then, independently, sample (G, \u03c1) and m independently conditional on(K\u03c9(\u03c1), \u03c1). Extend m to the vertices and edges of G not present in \u03c9 by setting m to be someconstant x0 \u2208 X on those vertices. This yields the law of a random rooted network (G, \u03c1, \u03c9,m)in which the root always satisfies \u03c9(\u03c1) = 1. Now, for each vertex u of G, let v(u) be chosen uni-formly from the set of vertices of \u03c9 closest to u, independently from everything else. (In particular,v(u) = u if \u03c9(u) = 1.) Let \u03c1\u2032 be chosen uniformly from the set {u : v(u) = \u03c1}. Sampling thenetwork (G, \u03c1\u2032, \u03c9,m) biased by |{u : v(u) = \u03c1}| yields the desired coupling. The details of thisconstruction are omitted.9.3.2 AmenabilityRecall that the (edge) Cheeger constant of an infinite graph G = (V,E) is defined to beiE(G) = inf{ |\u2202EW ||W | : W \u2282 V finite}where \u2202EW denotes the set of edges with exactly one end in W . The graph is said to be amenableif its Cheeger constant is zero and nonamenable if it is positive.Non-amenability is often too strong a condition to hold for random graphs. For example, it iseasily seen that every infinite cluster of a Bernoulli-(1 \u2212 \u03b5) percolation on a nonamenable Cayleygraph is almost surely amenable, since the cluster will contain \u2018bad regions\u2019 that contain sets ofsmall expansion. In light of this, Aldous and Lyons introduced the following weakened notionof non-amenability for unimodular random graphs. (Another way to overcome this issue is touse anchored expansion, which is less relevant in our setting. See [7] for a comparison of the twonotions.) The invariant Cheeger constant of an ergodic unimodular random rooted graph (G, \u03c1)is defined to beiinv(G, \u03c1) = inf{E[ |\u2202EK\u03c9(\u03c1)||K\u03c9(\u03c1)|]: \u03c9 a finitary percolation on G}. (9.3.1)(This is a slight abuse of notation: iinv(G, \u03c1) is really a functions of the law of (G, \u03c1).) An ergodicunimodular random rooted graph (G, \u03c1) is said to be invariantly amenable if iinv(G, \u03c1) = 0 andinvariantly nonamenable otherwise. More generally, we say that a unimodular random rooted2199.3. Percolations and invariant amenabilitygraph is invariantly amenable if its ergodic decomposition is supported on invariantly amenableunimodular random rooted graphs, and invariantly nonamenable if its ergodic decomposition issupported on invariantly nonamenable unimodular random rooted graphs. This is a property ofthe law of (G, \u03c1) and not of an individual graph. Invariant amenability and non-amenability isdefined similarly for ergodic unimodular random rooted maps, marked graphs and marked maps.Let (G, \u03c1) be a unimodular random rooted graph and let \u03c9 be a finitary percolation on G. Letdeg\u03c9(\u03c1) denote the degree of \u03c1 in \u03c9 and let\u03b1(G, \u03c1) = sup{E[deg\u03c9(\u03c1)]: \u03c9 a finitary percolation on G}. (9.3.2)An easy application of the mass transport principle [7, Lemma 8.2] shows thatE[deg\u03c9(\u03c1)] = E[\u2211v\u2208K\u03c9(\u03c1) deg\u03c9(v)|K\u03c9(\u03c1)|]. (9.3.3)It follows that, if E[deg(\u03c1)] <\u221e,iinv(G, \u03c1) = E[deg(\u03c1)]\u2212 \u03b1(G, \u03c1). (9.3.4)Similarly, if (G, \u03c1) is an ergodic unimodular random rooted graph with E[deg(\u03c1)] = \u221e, then\u03b1(G, \u03c1) <\u221e is a sufficient condition for iinv(G, \u03c1) to be positive.9.3.3 HyperfinitenessA unimodular random rooted graph (G, \u03c1) is said to be hyperfinite if there exists a {0, 1}N-marking \u3008\u03c9i\u3009i\u22651 of (G, \u03c1) such that each of the percolations \u03c9i is finitary, \u03c9i \u2286 \u03c9i+1 almost surely,and\u22c3i\u22651 \u03c9i = G almost surely. We call such an \u3008\u03c9i\u3009i\u22650 a finitary exhaustion for (G, \u03c1). Thefollowing is standard in the measured equivalence relations literature and was noted to carry throughto the unimodular random rooted graph setting by Aldous and Lyons [7].Theorem 9.3.4 ([7, Theorem 8.5]). Let (G, \u03c1) be a unimodular random rooted graph with E[deg(\u03c1)] <\u221e. Then (G, \u03c1) is invariantly amenable if and only if it hyperfinite.Since a complete proof is not provided in [7], we provide a proof below for completeness.Proof. First suppose that (G, \u03c1) is hyperfinite, and let \u3008\u03c9i\u3009i\u22650 be a finitary exhaustion for (G, \u03c1).By the monotone convergence theorem, E[deg\u03c9n(\u03c1)]\u2192 E[deg(\u03c1)], so that\u03b1(G, \u03c1) \u2265 lim supi\u2192\u221eE[deg\u03c9i(\u03c1)] = E[deg(\u03c1)]and hence (G, \u03c1) is invariantly amenable. Suppose conversely that (G, \u03c1) is invariantly amenable.For each i \u2265 1, there exists a finitary percolation \u03c9i on G such that E[deg(\u03c1)\u2212deg\u03c9i(\u03c1)] \u2264 2\u2212i. ByLemma 9.3.2, there exists {0, 1}N-marking \u3008\u03c9i\u3009i\u22651 of (G, \u03c1) such that E[deg(\u03c1) \u2212 deg\u03c9i(\u03c1)] \u2264 2\u2212i2209.3. Percolations and invariant amenabilityfor each i \u2265 1. For each i \u2265 1, let \u03c9\u02c6i =\u22c2j\u2265i \u03c9j . Clearly \u3008\u03c9\u02c6i\u3009i\u22651 is a {0, 1}N-marking of (G, \u03c1), and\u03c9\u02c6i \u2286 \u03c9\u02c6i+1 for every i \u2265 1. Furthermore, by construction,E[deg(\u03c1)\u2212 deg\u03c9\u02c6i(\u03c1)] \u2264\u2211j\u2265iE[deg(\u03c1)\u2212 deg\u03c9i(\u03c1)] \u2264 2\u2212i+1and hence, by Borel-Cantelli, \u03c1 is in the interior of its cluster K\u03c9i(\u03c1) for all sufficiently large ialmost surely. It follows by unimodularity that\u22c3\u03c9i = G, so that \u3008\u03c9\u02c6i\u3009i\u22650 is a finitary exhaustionfor (G, \u03c1).Remark 9.3.5. Invariantly amenable unimodular random rooted graphs are always hyperfinitewhether or not E[deg(\u03c1)] <\u221e. However, the graph obtained by replacing each edge of the canopytree (i.e., the Benjamini-Schramm limit of the balls in a 3-regular tree) at height n by 2n paralleledges is hyperfinite but nonamenable.Corollary 9.3.6 (Theorem 9.1.1, (2) implies (7).). Let (M,\u03c1) be a hyperfinite ergodic randomrooted map. Then (M,\u03c1) is a Benjamini-Schramm limit of finite planar maps.Proof. Let \u3008\u03c9i\u3009i\u22651 be a finitary exhaustion for (M,\u03c1). Let Mi be the finite map with underlyinggraph K\u03c9i(\u03c1) and map structure inherited from M . Then (Mi, \u03c1i) is a finite unimodular randomrooted map, and converges to (M,\u03c1) almost surely as i\u2192\u221e.Theorem 9.3.7 (Theorem 9.1.1: (7) implies (2)). Let (M,\u03c1) be an ergodic unimodular randomplanar map that is a Benjamini-Schramm limit of finite planar maps. Then (M,\u03c1) is hyperfinite.This proposition follows as a corollary to the Lipton-Tarjan planar separator theorem [167]; see[21] for details. Similarly, it is possible to deduce that item (8) of Theorem 9.1.1 implies item (2)as an application of the low-genus separator theorem of Gilbert, Hutchinson, and Tarjan [102].9.3.4 EndsRecall that an infinite connected graph G = (V,E) is said to be k-ended if k is minimal such thatfor every finite set of vertices W in G, the graph induced by the complement V \\W has at mostk distinct infinite connected components. In particular, an infinite tree is one-ended if it does notcontain a simple bi-infinite path, and is two-ended if it contains a unique bi-infinite path.The following proposition, due primarily to Aldous and Lyons [7], connects the number of endsof a unimodular random rooted graph to invariant amenability.Proposition 9.3.8 (Ends and Amenability [7, 21]). Let (G, \u03c1) be an infinite ergodic unimodularrandom rooted graph with E[deg(\u03c1)] < \u221e. Then G has either one, two, or infinitely many endsalmost surely. If G has infinitely many ends, then (G, \u03c1) is invariantly nonamenable. If G has twoends almost surely, then it is almost surely recurrent and (G, \u03c1) is invariantly amenable.Similarly, a connected topological space X is said to be k-ended if over all compact subsets Kof X, the complement X \\K has a maximum of k connected components that are not precompact.2219.3. Percolations and invariant amenabilityLemma 9.3.9. Let M be a map. Then the underlying graph of M has at least as many ends asthe associated surface S = S(M).Proof. For each compact subset K of S, let WK be the set of vertices of M that are adjacent toan edge intersecting K. Since every face of M in S is a topological disc, each non-precompactconnected component of S \\K contains infinitely many edges of M , and there are no connectionsin M \\WK between these components, so that M \\WK has at least as many infinite connectedcomponents as S \\K has non-precompact connected components.Note however that the 3-regular tree is infinitely-ended while its associated surface is homeo-morphic to the plane and hence one-ended.Finally, let us note the following simple topological fact.Lemma 9.3.10. Let M be a map. Then the underlying graph of M has at least as many ends asM has faces of infinite degree.9.3.5 Unimodular couplings and soficityLet (G1, \u03c11) and (G2, \u03c12) be unimodular random rooted graphs. A unimodular coupling of(G1, \u03c11) and (G2, \u03c12) is a unimodular random rooted {0, 1}2-marked graph (G, \u03c1, \u03c91, \u03c92) such that\u03c91 and \u03c92 are both connected almost surely and the law of the subgraph (\u03c9i, \u03c1) conditioned onthe event that \u03c9i(\u03c1) = 1 is equal to the law of (Gi, \u03c1i) for each i. We say that two unimodularrandom rooted graphs (G1, \u03c11) and (G2, \u03c12) are coupling equivalent if they admit a unimodularcoupling. (It is not difficult to show that this is an equivalence relation, arguing along similar linesto Lemmas 9.3.2 and 9.3.3.) Unimodular couplings and coupling equivalence are defined similarlyfor unimodular random rooted maps, marked graphs and marked maps. Coupling equivalence isclosely related to the notion of two graphings generating the same equivalence relation in the theoryof measured equivalence relations (see [7, Example 9.9]).Example 9.3.11. Let (G, \u03c1) be a unimodular random rooted graph and let \u03c9 be a bond percolationon G that is connected almost surely. Then (G, \u03c1) and (\u03c9, \u03c1) are coupling equivalent.Example 9.3.12. Let (M,\u03c1) be a unimodular random rooted map with locally finite dual. Then(M,\u03c1) is coupling equivalent to its unimodular dual: both (M,\u03c1) and its dual can be representedas percolations of the graph formed by combining M and its dual by adding a vertex wherevera primal edge crosses a dual edge, as well as keeping the original primal and dual edges (see [7,Example 9.6] for a similar construction).The main use of unimodular coupling will be that hyperfiniteness and strong soficity are bothpreserved under unimodular coupling. The corresponding statements for measured equivalencerelations are due to Elek and Lippner [86].Proposition 9.3.13. Let (G1, \u03c11) and (G2, \u03c12) be coupling equivalent unimodular random rootedgraphs. Then (G2, \u03c12) is hyperfinite if and only if (G1, \u03c11) is.2229.3. Percolations and invariant amenabilityProof. It suffices to prove that if (G, \u03c1) is a unimodular random rooted graph and \u03c9 is a perco-lation on (G, \u03c1) that is almost surely connected, then (G, \u03c1) is hyperfinite if and only if (\u03c9, \u03c1) ishyperfinite. Suppose first that (G, \u03c1) is hyperfinite and let \u3008\u03c9\u02dci\u3009i\u22651 be a finitary exhaustion for G.By Lemma 9.3.2, we may couple \u03c9 and \u3008\u03c9i\u3009i\u22651 so that (G, \u03c1, (\u03c9, \u3008\u03c9\u02dci\u3009i\u22651)) is unimodular. Undersuch a coupling, \u3008\u03c9\u2229 \u03c9\u02dci\u3009i\u22651 is a finitary exhaustion for (\u03c9, \u03c1), and consequently (\u03c9, \u03c1) is hyperfinite.Suppose conversely that (\u03c9, \u03c1) is hyperfinite, and let \u3008\u03c9\u02dci\u3009i\u22651 be a finitary exhaustion for (\u03c9, \u03c1). ByLemma 9.3.3, we may assume that (G, \u03c1, \u03c9, \u3008\u03c9\u02dci\u3009i\u22651) is unimodular. For each vertex v of G, let w(v)be a chosen uniformly from the set of vertices of \u03c9 minimising the graph distance to v in G. Foreach i, define a subgraph \u03c9\u02c6i of G by\u03c9\u02c6i(e) = 1 \u21d0\u21d2 w(e\u2212) and w(e+) are connected in \u03c9\u02dci. (9.3.5)Then \u3008\u03c9\u02c6i\u3009i\u22651 is a finitary exhaustion for (G, \u03c1).Remark 9.3.14. It can be deduced from Proposition 9.3.13 and [7, Theorem 8.9] that a unimodularrandom rooted graph with finite expected degree is hyperfinite if and only if it is coupling equivalentto Z.One useful application of Proposition 9.3.13 is the following.Lemma 9.3.15. Let (M,\u03c1) be an ergodic unimodular random rooted map. The number of infinitedegree faces is either 0, 1, 2, or \u221e, and if it is 1 or 2 then (M,\u03c1) is hyperfinite.Proof. By applying Proposition 9.3.8 and Lemma 9.3.10, it suffices to prove that if M has a non-zero but finite number of infinite degree faces almost surely, then (M,\u03c1) is hyperfinite. Supposethat M has finitely many infinite degree faces almost surely. Conditional on (M,\u03c1), choose one suchface, f , uniformly at random. If for every edge e of M incident to f there exists a vertex v of Msuch that e is contained in a finite component of M \\ {v}, then inductively there exists a sequenceof vertices \u3008vn\u3009n\u22651 of M such that vn is contained in a finite connected component of M \\ {vn+1}for every n \u2265 1. We deduce in this case that M is hyperfinite by taking as a finitary exhaustion thesequence of random subgraphs \u3008\u03c9\u02dcm\u3009m\u22651 induced by the standard monotone coupling of Bernoulli1 \u2212 1\/m site percolations on M , all of the clusters of which are almost surely finite due to theexistence of the cutpoints \u3008vn\u3009n\u22651.Otherwise, define a percolation \u03c9 on (M,\u03c1) by setting an edge e of M to be open if and only ifexactly one side of e is incident to f and there does not exist a vertex v of M such that e is containedin a finite connected component of M \\ {v}, and setting a vertex v of M to be open if and only if itis incident to an open edge. Then \u03c9 is connected and is isomorphic to the bi-infinite line graph Z. Itfollows that (G, \u03c1) is coupling equivalent to (Z, 0) and hence hyperfinite by Proposition 9.3.13Remark 9.3.16. An alternate proof in the case that E[deg(\u03c1)] < \u221e is as follows. Suppose that Mhas finitely many infinite degree faces almost surely. Conditional on (M,\u03c1), choose one such face,f , uniformly at random. We will use the boundary of this face to construct a unimodular couplingbetween M and Z. let \u3008ei(f)\u3009i\u2208Z be the set of oriented edges of M with eli(f) = f , enumerated so2239.3. Percolations and invariant amenabilitythat ei(f)+ = e\u2212i+1(f) for all i \u2208 Z. (this enumeration is unique up to translations of the indices,and this choice will not matter). Let G = (V,E) be the underlying graph of M , and consider thegraph G\u2032 with vertex set V \u2032 = V \u222a Z and edges given by the edges of G, the edges of Z, and anedge connecting v \u2208 V to i \u2208 Z if and only if v = ei(f)\u2212. Let \u03c1\u2032 be chosen uniformly from the set{\u03c1}\u222a{i : ei(f)\u2212 = \u03c1}. It is easy to verify that the graph (G\u2032, \u03c1\u2032) is unimodular when sampled biasedby 1 + |{i : ei(f)\u2212 = \u03c1}|, which has finite expectation by assumption. Thus, we have constructeda unimodular coupling between (G, \u03c1) and Z, so that the claim follows by Proposition 9.3.13.We call a sofic unimodular random rooted graph (G, \u03c1) strongly sofic if (G, \u03c1,m) is sofic forevery marking m of (G, \u03c1). The main input to Theorem 9.1.2 is the following, which was provenfor Cayley graphs of free groups by [56].Theorem 9.3.17 (Bowen [56]; Elek [85]; Elek and Lippner [86]; Benjamini, Lyons and Schramm[46]). Every unimodular random rooted tree is strongly sofic.The following theorem is an adaptation of a related theorem of Elek and Lippner [86] in thesetting of group actions. Even in our setting, it is well-known to experts that treeable unimodularrandom graphs (i.e., unimodular random graphs admitting a unimodular random spanning tree)are sofic.Theorem 9.3.18. Let (G1, \u03c11) and (G2, \u03c12) be coupling equivalent unimodular random rootedgraphs. Then (G1, \u03c11) is strongly sofic if and only if (G2, \u03c12) is strongly sofic.The proof can be summarised as follows: Suppose (G1, \u03c11) is strongly sofic, and let m be amarking of (G2, \u03c12). We can encode both the structure of G2 and the marks m as a marking m\u02c6of (G1, \u03c11). The strong soficity of (G1, \u03c11) allows us to approximate (G1, \u03c11, m\u02c6) by a sequence offinite graphs. We then use this sequence to define an approximating sequence for (G2, \u03c12,m). Thisargument takes some care to make rigorous.Proof. Since (G1, \u03c11) and (G2, \u03c12) can both be considered as percolations on some unimodularrandom graph (G, \u03c1), it suffices to prove that if (G, \u03c1) is a unimodular random rooted graph and\u03c9 is an almost surely connected percolation on G, then (G, \u03c1) is strongly sofic if and only if theunimodular random rooted graph (H, \u03c1\u2032) obtained from (\u03c9, \u03c1) by conditioning on \u03c9(\u03c1) = 1 isstrongly sofic.First suppose that (G, \u03c1) is strongly sofic and let m be a marking of (H, \u03c1\u2032). By Lemma 9.3.3,there exists a marking m of G such that (G, \u03c1, \u03c9,m) is unimodular and such that the law of(H, \u03c1\u2032,m) coincides with the law of (\u03c9, \u03c1,m) conditional on \u03c9(\u03c1) = 1. Since (G, \u03c1) is strongly sofic,there exists a sequence of finite unimodular random marked graphs (Gn, \u03c1n, \u03c9n,mn) converging to(G, \u03c1, \u03c9,m) in distribution. Let (Hn, \u03c1\u2032n,mn) be the unimodular random rooted graph obtainedfrom (\u03c9n, \u03c1n,mn) by conditioning on \u03c9(\u03c1) = 1. The sequence (Hn, \u03c1\u2032n,mn) converges to (H, \u03c1\u2032,m)and, since m was arbitrary, (H, \u03c1\u2032) is strongly sofic.Suppose conversely that (H, \u03c1\u2032) is strongly sofic and let m be an X-marking of (G, \u03c1). ByLemma 9.3.2, we may assume that (G, \u03c1, \u03c9,m) is unimodular. For each vertex u of G, let v(u)2249.3. Percolations and invariant amenabilitybe chosen uniformly from the set of vertices in \u03c9 that minimize the graph distance to u, and foreach vertex v of \u03c9 let Uv = {v(u) = v}. Transporting mass 1 from u to v(u) for every vertex uof G shows that E|U\u03c1| < \u221e, and in particular Uv is finite for every vertex v of \u03c9 almost surley.Conditional on (G, \u03c1, \u03c9,m), let {U(v) : v \u2208 V } \u222a {U(e) : e \u2208 E} be a collection i.i.d. uniform [0, 1]random variables indexed by the vertices and edges of G, and for each vertex v of G such that\u03c9(v) = 1, definem\u02c6(v) ={(U(u),m(u), {(U(e),m(e)) : e is an edge incident to u in G}) : u \u2208 Uv} .If we denote the space of finite subsets of the metric space X by X\u2217, then m\u02c6 takes values in themetric space ([0, 1]\u00d7 X\u00d7 ([0, 1]\u00d7 X)\u2217)\u2217 .By conditioning on \u03c9(\u03c1) = 1, we obtain a unimodular random rooted marked graph (H, \u03c1, m\u02c6). Since(H, \u03c1) is strongly sofic, there exists a sequence of finite unimodular random rooted marked graphs(Hn, \u03c1n, m\u02c6n) converging in distribution to (H, \u03c1, m\u02c6), and we may assume that E|m\u02c6n(\u03c1n)| < \u221efor each n \u2265 1. For each \u03b5 > 0 and R \u2208 N, bias (Hn, \u03c1n, m\u02c6n) by |m\u02c6(\u03c1)| and construct a finiteunimodular random rooted marked graph (GR,\u03b5n , \u03c1n,mR,\u03b5n ) from (Hn, \u03c1\u2032n, m\u02c6n) as follows.1. Let the vertex set of GR,\u03b5n be the union\u22c3v\u2208Hn m\u02c6(v), and let \u03c1n be chosen uniformly fromm\u02c6(\u03c1\u2032n). For each vertex u of GR,\u03b5n , let u = (u1, u2, u3) be the three coordinates of u, and letm(u) = u2.2. Draw an edge between two vertices u1 and u2 of GR,\u03b5n if and only if(a) u1 \u2208 m\u02c6n(v1) and u2 \u2208 m\u02c6n(v2) for some vertices v1 and v2 of Hn that are at distance atmost R in Hn, and(b) there exists a pair of points(t, x) \u2208 u3 \u2282 [0, 1]\u00d7 X and (s, y) \u2208 u3 \u2282 [0, 1]\u00d7 Xsuch that the distance between (t, x) and (s, y) is less than \u03b5 in the product metric. Ifthere are multiple such pairs, we draw multiple edges as appropriate.3. Let {Zn(e)} be a collection of i.i.d. Bernoulli-1\/2 random variables indexed by the edges ofGR,\u03b5n . For each edge e of GR,\u03b5n , let (t, x) and (s, y) be the matching pair of points in [0, 1]\u00d7Xthat led us to draw e in step (2), and let mR,\u03b5n (e) = x if Zn(e) = 0 and mR,\u03b5n (e) = y ifZn(e) = 1.This construction is continuous for the local topology, and hence for each fixed \u03b5 and R, the finitenetworks (GR,\u03b5n , \u03c1n,mR,\u03b5n ) converge to the network (GR,\u03b5, \u03c1,mR,\u03b5) defined by applying the sameprocedure to (H, \u03c1\u2032, m\u02c6). Taking \u000f\u2192 0, we obtain the network (GR, \u03c1,mR) which consists of thoseedges of (G, \u03c1) whose endpoints are of distance at most R in \u03c9. Finally, taking R\u2192\u221e we recover2259.3. Percolations and invariant amenability(G, \u03c1,m). Thus, (G, \u03c1,m) is a weak limit of sofic unimodular random rooted marked graphs, andit follows that (G, \u03c1,m) is sofic.Proof of Theorem 9.1.2. Let (M,\u03c1) be a simply connected unimodular random rooted map, let(G, \u03c1) be the underlying graph of M , and let F be a sample of FUSFM . By Theorem 9.5.13, F isconnected almost surely, and so (G, \u03c1) is coupling equivalent to the unimodular random rooted tree(F, \u03c1). It follows from Theorems 9.3.17 and 9.3.18 that (G, \u03c1) is strongly sofic. By encoding themap (M,\u03c1) as a marking of (G, \u03c1) as in Section 9.2.2 and [7, Example 9.6], we conclude that theunimodular random rooted map (M,\u03c1) is also sofic.9.3.6 Vertex extremal length and recurrence of subgraphsLet G be an infinite graph. For each vertex v of G, the vertex extremal length from v to infinityis defined to beVELG(v,\u221e) = supminf\u03b3:v\u2192\u221em(\u03b3)2\u2016m\u20162 , (9.3.6)where the supremum is over measures m on the vertex set of G such that \u2016m\u20162 = \u2211m(u)2 <\u221e,and the infimum is over paths \u03b3 from v to\u221e in G. A connected graph is said to be VEL parabolicif VEL(v \u2192 \u221e) = \u221e for some vertex v of G (and hence for every vertex), and VEL hyperbolicotherwise. The VEL type is easily seen to be monotone in the sense that subgraphs of VELparabolic graphs are also VEL parabolic.Lemma 9.3.19 (He and Schramm [121]: Theorem 9.1.1, (17) implies (5)). Let G be a locally finite,connected graph. If G is VEL hyperbolic, then it is transient. If G has bouneded degrees then theconverse also holds, so that G is transient if and only if it is VEL hyperbolic.Lemma 9.3.20 (Theorem 9.1.1, (17) implies (6)). Let T be a tree. Then T is transient if and onlyif it is VEL hyperbolic.Proof. Let G be an infinite connected graph. Recall that the effective resistance from a v toinfinity in a graph G is defined to beReff(v \u2192\u221e) = supminf\u03b3:v\u2192\u221em(\u03b3)2\u2016m\u20162 ,where the supremum is over measures m on the edge set of G such that \u2016m\u20162 = \u2211m(u)2 < \u221e,and the infimum is over paths \u03b3 from v to \u221e in G. Recall also that G is transient if and only ifthe effective resistance from v to infinity is finite for some vertex (and hence every vertex) v of G.Let T be a VEL parabolic tree, and let v be a vertex of T . For every M \u2265 1, there exists be ameasure m on the vertex set of T such thatinf\u03b3:v\u2192\u221em(\u03b3)2\u2016m\u20162 \u2265M.2269.4. CurvatureFor each edge e of T , let u(e) be the endpoint of e farthest to v, and define a measure m\u02c6 on theedge set of T by setting m\u02c6(e) = m(u(e)) for every edge e of T . Then\u2016m\u02c6\u20162 = \u2016m\u20162 \u2212m(v)2 and m\u02c6(\u03b3) = m(\u03b3)\u2212m(v)for every path \u03b3 from v to \u221e. Since \u2016m\u20162 \u2265 m(v)2, it follows thatReff(v \u2192\u221e) \u2265 inf\u03b3:v\u2192\u221e m\u02c6(\u03b3)2\u2016m\u02c6\u20162 \u2265 inf\u03b3:v\u2192\u221e(m(\u03b3)\u2212m(v))2\u2016m\u20162 \u2265 inf\u03b3:v\u2192\u221e\uf8eb\uf8edm(\u03b3)2\u2016m\u20162 \u2212 2\u221am(\u03b3)2\u2016m\u20162\uf8f6\uf8f8= inf\u03b3:v\u2192\u221em(\u03b3)2\u2016m\u20162 \u2212 2\u221ainf\u03b3:v\u2192\u221em(\u03b3)2\u2016m\u20162 \u2265M \u2212 2\u221aM.Since M was arbitrary, we deduce that T is recurrent.Vertex extremal length was introduced by He and Schramm [121] and is closely connected tocircle packing. Benjamini and Schramm [50] used circle packing to prove the following remarkabletheorem. See [21] for an alternative proof.Theorem 9.3.21 (Benjamini-Schramm [50]: Theorem 9.1.1, (7) implies (17)). Let (M,\u03c1) be aBenjamini-Schramm limit of finite planar maps. Then M is VEL parabolic almost surely. Inparticular, if M has bounded degrees, then it is almost surely recurrent for simple random walk.The converse is provided by the following theorem.Theorem 9.3.22 (Benjamini, Lyons and Schramm [45]; Aldous and Lyons [7]: Theorem 9.1.1,(3) implies (2)). Let (G, \u03c1) be an invariantly nonamenable unimodular random rooted graph. Thenthere exists a percolation \u03c9 on G such that every connected component of \u03c9 are nonamenablealmost surely, and there exists a constant M such that deg(v) \u2264 M for every vertex v of G suchthat \u03c9(v) = 1. Furthermore, the percolation \u03c9 can be taken to be a forest.This theorem is also very useful for studying random walks on invariantly nonamenable uni-modular random rooted graphs; see [21, Section 5.1]. See [21, Section 3.3.1] for a complete proofof the first part of the theorem (in which \u03c9 is not taken to be a forest).9.4 CurvatureIn this section, we introduce the average curvature of a unimodular random rooted map, andestablish some of its basic properties. We begin by giving a combinatorial definition of the averagecurvature; in Section 9.4, we show that show that the average curvature is a canonical quantityassociated to the random map, in the sense that any unimodular way of embedding the map ina Riemannian manifold (satisfying certain integrability conditions) will result in the same averagecurvature.2279.4. CurvatureRecall that the internal angles of a regular k-gon are given by (k\u2212 2)pi\/k. We define the anglesum at a vertex v of a map M to be\u03b8(v) = \u03b8M (v) =\u2211f\u22a5vdeg(f)\u2212 2deg(f)piThis definition extends to maps with infinite faces, with the convention that (\u221e\u22122)\/\u221e = 1. In thecase that every face of M has degree at least 3, we interpret \u03b8(v) as the total angle of the cornersat v if we form M by gluing together regular polygons, where we consider the upper half-space{x + iy \u2208 C : y > 0} with edges {[n, n + 1] : n \u2208 Z} to be a regular \u221e-gon. Of course, M cannotnecessarily be drawn in the plane with regular polygons, and the angle sum at a vertex of M neednot be 2pi. We define the curvature of M at the vertex v to be the angle sum deficit\u03ba(v) = \u03baM (v) = 2pi \u2212 \u03b8(v)and define the average curvature of a unimodular random rooted map (M,\u03c1), denoted K(M,\u03c1),to be the expected curvature at the rootK(M,\u03c1) = E[\u03ba(\u03c1)] = 2pi \u2212 E\uf8ee\uf8f0\u2211f\u22a5\u03c1deg(f)\u2212 2deg(f)pi\uf8f9\uf8fb .Note that if E[deg(\u03c1)] is finite then K(M,\u03c1) is also finite.Example 9.4.1 (Finite Maps). Let M be a finite map and let \u03c1 be a vertex of M chosen uniformlyat random. ThenK(M,\u03c1) =1|V |\u2211v\u2208V\u03ba(v) =1|V |(2pi|V | \u2212\u2211v\u2208V\u2211f\u22a5vdeg(f)\u2212 2deg(f)pi)=1|V |(2pi|V | \u2212\u2211f\u2208F(deg(f)\u2212 2)pi)=1|V |2pi(|V | \u2212 |E|+ |F |) = 1|V |2pi (2\u2212 2 genus(M)) . (9.4.1)where the final equality follows from Euler\u2019s formula. If g is a Riemannian metric on S(M) and\u03ba(x) is the Gaussian curvature of (S, g) at x, then the Gauss-Bonnet formula implies thatK(M,\u03c1) =1|V |\u222bS\u03ba(x) dx,so that the two notions of average curvature coincide. This is a special case of Theorem 9.4.9 below.2289.4. CurvatureFigure 9.4: Left: A vertex with positive curvature pi\/15. Centre: A vertex with zero curvature.Right: A vertex with negative curvature \u2212pi\/3.Example 9.4.2 (k-angulations). If M is a k-angulation (i.e., every face of M has degree k), then\u03b8(v) =k \u2212 2kdeg(v)pifor every vertex v of M . In particular, if M is an infinite plane tree, the angle sum at a vertex v ofM is simply \u03b8(v) = pi deg(v). Consequently, if (M,\u03c1) is a unimodular random k-angulation, thenK(M,\u03c1) =(2\u2212 k \u2212 2kE[deg(\u03c1)])pi.Example 9.4.3 (Curvature of the dual measure). Let P be a unimodular probability measure onM\u2022 such that P[deg(\u03c1)] <\u221e and M has locally finite dual P-a.s., and let P\u2020 be the dual measure.Then, by (9.2.2),E\u2020[\u03ba(\u03c1\u2020)]= 2pi \u2212 E\u2020[ \u2211f\u22a5\u03c1\u2020deg(f)\u2212 2deg(f)]pi= 2pi \u2212 E[\u2211f\u22a5\u03c11deg(f)]\u22121E[\u2211f\u22a5\u03c11deg(f)\u2211v\u22a5fdeg(v)\u2212 2deg(v)]pi.Applying the Mass Transport Principle yields thatE\u2020[\u03ba(\u03c1\u2020)]= 2pi \u2212 E[\u2211f\u22a5\u03c11deg(f)]\u22121E[\u2211f\u22a5\u03c11deg(f)\u2211v\u22a5fdeg(\u03c1)\u2212 2deg(\u03c1)]piwhich can be rearranged to giveE\u2020[\u03ba(\u03c1\u2020)] = E[\u2211f\u22a5\u03c11deg(f)]\u22121E[\u03ba(\u03c1)]In particular, the average curvature of (M,\u03c1) under P has the same sign as that of (M \u2020, \u03c1\u2020) underP\u2020.Example 9.4.4 (Self-dual maps). Let (M,\u03c1) be a self-dual unimodular random rooted map. Then2299.4. Curvature(9.2.3) implies that E[\u2211f\u22a5\u03c1 deg(f)\u22121] = 1 and soK(M,\u03c1) = 2pi \u2212 E\uf8ee\uf8f0\u2211f\u22a5\u03c1deg(f)\u2212 2deg(f)\uf8f9\uf8fbpi= 2pi \u2212 E[deg(\u03c1)]pi + 2E[\u2211f\u22a5\u03c1deg(f)\u22121]pi = 4pi \u2212 E[deg(\u03c1)]pi.9.4.1 Curvature of submapsLet M be a map with underlying graph G, and let z be a proper embedding of M into an orientablesurface S. A submap of M is a map represented by a triple (H,S, z), where H is a connectedsubgraph of G such that the restriction of z to H is a proper embedding of H into S. If M is simplyconnected, then every connected subgraph of G is also a submap of M . This no longer holds if M isnot simply connected. For example, if M is the product Z\u00d7K3 of the integers Z with the trianglegraph (a.k.a. the complete graph on three vertices) properly embedded into the infinite cylinder,then the subgraph Z\u00d7 {v}, where v is one of the vertices of K3, does not correspond to a submapof M .Proposition 9.4.5 (Curvature of random submaps). Let (M,\u03c1) be a unimodular random rootedmap with E[deg(\u03c1)] < \u221e and let \u03c9 be a connected bond percolation on M that is almost surely asubmap of M . ThenK(\u03c9, \u03c1) = K(M,\u03c1).Proof. First suppose that M and \u03c9 both have a locally finite duals a.s. In this case, every face of\u03c9 is a union of finitely many faces of M . Define a mass transport as follows. For each vertex v ofM and every face f of M incident to v, let f\u02c6 be the face of \u03c9 containing f , and transport a massof2pideg(f)\u2212 2deg(f) deg(f\u02c6)to each vertex u in the boundary of f\u02c6 , where as usual we count f with multiplicity if it has multiplecorners incident to v, and count u with multiplicity if it has multiple corners incident to f . Thetotal mass sent out by each vertex is \u03b8M (v). For each face f\u02c6 of \u03c9, consider the subgraph G(f\u02c6) of Mspanned by all the edges of M that are incident to a face of M that are contained in f\u02c6 . This graphcan be drawn in the plane so that its faces correspond to the faces of M contained in f\u02c6 , along with2309.4. Curvaturean outside face. Applying Euler\u2019s formula to this graph, we have that\u2211f\u2208F (M),f\u2286f\u02c6(deg(f)\u2212 2) = 2|{edges of G(f\u02c6)}| \u2212 2|{faces of G(f\u02c6)}|+ 2\u2212 |{edges with exactly one side incident to f\u02c6}|= 2|{vertices of G(f\u02c6)}| \u2212 2\u2212 |{edges with exactly one side incident to f\u02c6}|.= deg(f\u02c6)\u2212 2,(This is similar to [9, Lemma 18].) It follows that the total mass received by each vertex v is givenby\u2211f\u02c6\u2208F (\u03c9),f\u02c6\u22a5v\u2211f\u2208F (M),f\u2286f\u02c6\u2211u\u22a5f2pideg(f)\u2212 2deg(f) deg(f\u02c6)=\u2211f\u02c6\u2208F (\u03c9),f\u02c6\u22a5v\u2211f\u2208F (M),f\u2286f\u02c62pideg(f)\u2212 2deg(f\u02c6)=\u2211f\u02c6\u2208F (\u03c9),f\u02c6\u22a5v2pideg(f\u02c6)\u2212 2deg(f\u02c6)= \u03b8\u03c9(v).The claim now follows by applying the mass-transport principle.Next, suppose that M has locally finite dual but that \u03c9 does not. For each infinite face f\u02c6 of \u03c9, let\u3008ei(f\u02c6)\u3009i\u2208Z be the set of oriented edges of \u03c9 that have their left hand side incident to f\u02c6 , enumeratedin counterclockwise order around the boundary of f . For each n \u2265 1, define a percolation \u03c9n byletting e \u2208 \u03c9n if and only if either e \u2208 \u03c9 or if e\u2212 and e+ are both in the boundary of some infiniteface f\u02c6 of \u03c9 and e\u2212 = e\u2212i (f\u02c6) and e+ = e+j (f\u02c6) for some i, j \u2208 Z with |i\u2212 j| \u2265 n. Observe that, sinceM has locally finite dual, \u03c9n has locally finite dual for every n \u2265 1. Thus, it follows as above thatK(\u03c9n, \u03c1) = K(M,\u03c1) for every n \u2265 1. The sequence of random variables |\u03b8\u03c9n(\u03c1)| are bounded by2pi deg(\u03c1), and so it follows from the dominated convergence theorem that K(\u03c9n, \u03c1) \u2192 K(\u03c9, \u03c1) asn\u2192\u221e, completing the proof in this case.Now suppose that the dual M \u2020 is not locally finite. Let F\u221e be the set of infinite degree facesof M . For each n \u2265 1, let the map Mn be constructed from M as follows. For each infiniteface f of (M,\u03c1), let \u3008ei(f)\u3009i\u2208Z be the set of oriented edges of M with eli(f) = f , enumerated incounterclockwise order around the boundary of f . Conditional on (M,\u03c1), let {Ui(f) : f \u2208 F\u221e, i \u2208N} be a collection of i.i.d. Bernoulli-1\/2 random variables. For each infinite face f of G and eachn \u2265 1, let Kn(f) =\u2211nj=1 Uj(f)2j . For each n \u2265 1, we form Mn by drawing an edge betweene(f)\u2212Km+i2m and e(f)\u2212Km+(i+1)2mfor each i \u2208 Z and m \u2265 n. (Only M1 is required for the currentproof, we include the definition of Mn here for later use.) See Figure 4 for an illustration. It isclear that Mn is almost surely a locally finite map with a locally finite dual, and it follows fromLemma 9.2.2 that (Mn, \u03c1) is unimodular.For each oriented edge e emanating from \u03c1 in M that is incident to an infinite face of M , the2319.4. CurvatureFigure 9.5: The map M1 is defined by filling in each infinite face of M with a system of arcs. Thisfigure demonstrates this procedure applied to one of the infinite faces of Z.expected number of additional arcs in M1 drawn into the corner of this face at e is 2, so thatE[degMn(\u03c1)] \u2264 E[degM1(\u03c1)] = E[deg(\u03c1)] + 2E\u2223\u2223\u2223\u2223{e \u2208 E\u2192 : e\u2212 = \u03c1 and deg(e`) =\u221e}\u2223\u2223\u2223\u2223\u2264 3E[deg(\u03c1)] <\u221e.By Lemma 9.3.2, we can consider both \u03c9 and M to be percolations on M1 that are almost surelysubmaps, and we deduce from the above argument that K(\u03c9, \u03c1) = K(M1, \u03c1) = K(M,\u03c1).Remark 9.4.6. A straightforward extension of Proposition 9.4.5 is that, if \u03c9 is a connected perco-lation on (M,\u03c1) that is almost surely a submap, but might not include every vertex, then, letting(N, \u03c1\u2032) be the unimodular random map obtained from (\u03c9, \u03c1) by conditioning on the event that \u03c1 isin \u03c9, we have thatK(N, \u03c1\u2032) = P(\u03c1 \u2208 \u03c9)\u22121K(M,\u03c1).Proposition 9.4.7 (Upper semicontinuity of the average curvature). Let (Mn, \u03c1n) be a sequence ofunimodular random rooted maps with E[deg(\u03c1n)] < \u221e converging weakly to a unimodular randomrooted map (M,\u03c1), and suppose that E[deg(\u03c1)] <\u221e. ThenK(M,\u03c1) \u2265 limn\u2192\u221eK(Mn, \u03c1n).Proof. Let \u03c1n be a uniformly chosen root vertex of Mn for each n \u2265 1. First suppose that none of themaps Mn have any faces of degree 1. In this case, \u03b8Mn(v) is positive for every n \u2265 1, and the claimfollows from Fatou\u2019s lemma. Otherwise, let (M\u02c6, \u03c1) and (M\u02c6n, \u03c1) be the unimodular random rootedmaps obtained by removing all self-loops from M and Mn respectively. Clearly (M\u02c6n, \u03c1n) convergesweakly to (M,\u03c1), and by Fatou we have that K(M\u02c6, \u03c1) \u2265 limn\u2192\u221eK(M\u02c6n, \u03c1n) as above. We conclude2329.4. Curvatureby applying Proposition 9.4.5 to deduce that K(M,\u03c1) = K(M\u02c6, \u03c1) and K(Mn, \u03c1n) = K(M\u02c6n, \u03c1n) forevery n \u2265 1.Combining Proposition 9.4.7 with the equation (9.4.1) has the following immediate corollary.Corollary 9.4.8 (Theorem 9.1.1, (8) implies (1)). Let (M,\u03c1) be a unimodular random rooted mapwith E[deg(\u03c1)] < \u221e that is obtained as a Benjamini-Schramm limit of a sequence of finite maps\u3008Mn\u3009n\u22651 such thatgenus(Mn)|V (Mn)| \u2212\u2212\u2212\u2192n\u2192\u221e 0.Then K(M,\u03c1) \u2265 0.We will later prove that K(M,\u03c1) \u2264 0 for any infinite, simply connected, unimodular randomrooted planar map with finite expected degree in Section 9.5.9.4.2 Invariance of the curvatureIn this section, we consider unimodular embeddings of unimodular random rooted maps. We define anotion of average curvature associated to the embedding and show that, under certain integrabilityconditions, the average curvature associated to the embedding agrees with the average curvaturethat we defined combinatorially. This shows that the average curvature is a canonical quantity.Since this section is somewhat tangential to the rest of the paper, we will skip over some of thetechnical details.We define a metric surface embedded map (MSEM) to be a locally finite map M togetherwith a proper embedding z of M into an oriented metric surface S. A rooted MSEM is a MSEMtogether with a distinguished root vertex. Two rooted MSEMs are isomorphic if they are isomorphicas rooted maps, and there is an orientation preserving isometry between the two surfaces sendingone embedding to the other.We define the local topology on the set of isomorphism classes of rooted MSEMs by a variationon the local Gromov-Haussdorf topology: Namely, we set the distance between two rooted MSEMs(M1, \u03c11, S1, z1) and (M2, \u03c12, S2, z2) to be e\u2212r, where r is maximal such that there is a rooted graphisomorphism \u03c6 from the ball of radius r around \u03c11 in M1 to the (graph distance) ball of radiusr about \u03c12 in M2 such that \u03c6 preserved the cycling of the edges emanating from each vertex inthe interior of the ball, and there exist functions \u03c81 and \u03c82 from the balls of (metric) radius raround z1(\u03c11) and z2(\u03c12) in S1 and S2 respectively, denoted B1(r) and B2(r), into some commonmetric space X so that the Haussdorf distance between \u03c81(B1) and \u03c82(B2) is at most 1\/r, and theHaussdorf distance between \u03c81(z1(e) \u2229 B1(r)) and \u03c82(z2(\u03c6(e)) \u2229 B2(r)) is at most 1\/r for everyedge e contained in the graph distance ball of radius r around \u03c11.We define doubly rooted MSEMs, the local topology on the set of isomorphism classes ofdoubly rooted MSEMs, and unimodular random rooted MSEMs similarly to the graph case. Itis straightforward (but rather tedious) to encode the structure of a MSEM as a marking of theunderlying map of the MSEM, so that all of the usual machinery of unimodularity transfers to this2339.4. Curvaturesetting. If (M,\u03c1) is a unimodular random rooted map and (M,S, z, \u03c1) is a unimodular randomrooted MSEM with underlying map (M,\u03c1), we call (S, z) a unimodular embedding of M .In practice, unimodular embeddings often arise as measurable, automorphism equivariant func-tions of the map, such as the embeddings given by circle packing and the conformal embedding.These are easily seen to be unimodular since every mass-transport on the associated MSEM isinduced by a mass-transport on the map.Let us give an example of a unimodular embedding. Given a map M such that every face ofM has degree at least three, a natural way to embed it on a surface is as follows. We associateto each face f of degree d a regular d-gon with sides of length 1. If two faces share an edge, weidentify the corresponding edges of the polygons. For infinite faces, the corresponding polygon is ahalf-plane {z \u2208 C : Im(z) \u2265 0} with edges {[n, n+ 1] : n \u2208 Z} along the boundary, which we thinkof as a regular \u221e-gon. In this surface, there is no curvature on any of the faces, as they are allflat pieces of R2. The surface is also smooth along the edges, as two faces are glued along straightsegments of unit length. Thus, all of the curvature of the surface is concentrated on the vertices,and in fact the atom of curvature \u03bas(v) at v (see below) is exactly the value \u03ba(v) that we definedcombinatorially at the beginning of this section.One can more generally obtain a unimodular embedding by gluing together (possibly random)shapes that are not necessarily polygons, and, in particular, can also extend the construction toallow for faces of degree 1 and 2.In what follows, we restrict ourselves to surfaces that have a smooth Riemannian metric, exceptpossibly for cone-like singularities at vertices of the map, and assume that the edges of the map areembedded as smooth curves in the surface. We call a unimodular embedding of a map satisfyingthese conditions a smooth embedding. Given a smooth embedding of a map M , for each orientededge e of M , we let ang(e) be the angle between e and the edge following e in the clockwiseordering of the oriented edges emanating from e\u2212. Every smooth Riemannian surface with cone-like singularities has a Gaussian curvature \u03ba associated with its metric, which is a signed measureon the surface (see e.g. [9]). The curvature measure \u03ba is absolutely continuous with respect to thearea measure on the surface except possibly at the cone-like singularities at the vertices. We let\u03bas(v) be the atom of curvature at a vertex v, and let \u03ba(f) be the total curvature of the face f ,which is well-defined if f has finite degree. Moreover, every oriented edge e of the map has a welldefined total geodesic curvature in the embedding, which we denote \u03bag(e). In a smooth embeddingof a map, the mass of the atom of curvature at a vertex is given by\u03bas(v) = 2pi \u2212\u2211e: e\u2212=vang(e).Given a smooth embedding of a map with a locally finite dual, we define the total curvature ata vertex v by\u03ba(v) = \u03bas(v) +\u2211f\u22a5v\u03ba(f)deg(f).2349.4. CurvatureWe are now ready to state our \u201cinvariance of the average curvature\u201d theorem.Theorem 9.4.9. Let (M,\u03c1) be a unimodular random rooted map with locally finite dual and let(S, z) be a smooth unimodular embedding of M . Suppose further that either1. \u03bag(e) = 0 for every oriented edge e of M , and \u03ba(f) \u2264 0 for every face f of M , or2.\u2211e:e\u2212=\u03c1 |\u03bag(e)| and |\u03ba(\u03c1)| both have finite expectation.ThenE[\u03ba(\u03c1)] = K(M,\u03c1).We shall require the following signed version of the mass-transport principle: If (G, \u03c1) is aunimodular random rooted graph and \u03c6 is a measurable function from G\u2022\u2022 to R such thatE\uf8ee\uf8f0\u2211v\u2208V\u2223\u2223\u03c6(G, \u03c1, v)\u2223\u2223\uf8f9\uf8fb <\u221e, then E\uf8ee\uf8f0\u2211v\u2208V\u03c6(G, \u03c1, v)\uf8f9\uf8fb = E\uf8ee\uf8f0\u2211v\u2208V\u03c6(G, v, \u03c1)\uf8f9\uf8fb .This follows easily from the usual mass-transport principle for \u03c6 non-negative. A similar signedmass-transport principle holds for unimodular random rooted maps and MSEMs.Proof. For each oriented edge e of M , let ang(e) be the angle between e and the edge \u03c3(e) followinge in the clockwise ordering of the oriented edges emanating from e\u2212. The Gauss-Bonnet Theorem(see e.g. [9, Chapter VI, Section 7]) implies that for any face f of M ,\u2211e:er=f[pi \u2212 ang(e)] +\u2211e:er=f\u03bag(e) + \u03ba(f) = 2pi.We rewrite this identity as(deg(f)\u2212 2)pi =\u2211e:er=f[ang(e)\u2212 \u03ba(f)deg(f)\u2212 12\u03bag(e)\u2212 12\u03bag(\u2212\u03c3(e))]. (9.4.2)We now define a mass transport on M . Given a vertex u, for each face f that u belongs to, usends to each vertex v in the boundary of f (including u itself) the mass\u03c8(u, f) =1deg(f)\u2211e:er=f, e\u2212=u[ang(e)\u2212 \u03ba(f)deg(f)\u2212 12\u03bag(e)\u2212 12\u03bag(\u2212\u03c3(e))],where, as usual, v receives a multiple of this mass if it is multiply incident to the face. Note that ifu and v share more than a single face, there is a term for each face containing the two, and thereis also mass sent from u to itself, with a term for each face incident to u.Let \u03c6(u, v) be the total mass sent from u to v. If condition (1) holds, then \u03c6 is positive.2359.4. CurvatureOtherwise, condition (2) holds, and we have that\u2211v\u2208V|\u03c6(u, v)| \u2264\u2211f\u22a5udeg(f)|\u03c8(u, f)| \u2264 |\u03ba(u)|+\u2211e:e\u2212=u|\u03bag(e)|and henceE\u2211v\u2208V|\u03c6(\u03c1, v)| <\u221e.We can apply the Mass-Transport Principle to \u03c6 in either case.Let us consider the total mass sent out from v. The quantity corresponding to a face f \u22a5 vis sent to each of the deg(f) corners of f , which cancels the 1\/deg(f) factor. The sum over facesadjacent to v of ang(e) is precisely 2pi \u2212 \u03bas(v). Together with the sum over f of \u03ba(f)\/ deg(f) thiscomes to 2pi\u2212 \u03ba(v). Meanwhile, since \u03bag(e) = \u2212\u03bag(\u2212e), the terms involving geodesic curvatures ofedges cancel when we sum over all faces incident to v. Thus, we have that the total mass sent outby v is exactly \u03ba(v). On the other hand, the total mass received by a vertex v from vertices of aface f \u22a5 v is (deg(f)\u2212 2)pi\/deg(f) by (9.4.2), and so the result follows.Example 9.4.10 (Integrability conditions are necessary). Consider the map obtained by triangu-lating each of the two infite faces of Z as in Figure 9.5. This map admits a unimodular embedding inthe hyperbolic plane, defined by drawing the edges of Z as segments of length one along a bi-infinitegeodesic, and drawing the other edges of the map as circular arcs. This embedding is a measurablefunction of the map, and is therefore unimodular. However, it does not satisfy the integrabilityrequirements of Theorem 9.4.9. Indeed, the conclusion of Theorem 9.4.9 fails for this embedding:the curvature \u03bas(v) +\u2211f\u22a5v \u03ba(f)\/ deg(f) is negative for every vertex of the map, but it is easilyseen that the map is invariantly amenable and has finite expected degree, so that K(M,\u03c1) \u2265 0 byCorollary 9.4.8 and Corollary 9.3.6.Example 9.4.11 (Voronoi diagrams and Delaunay triangulations of point processes). We say thata set of points in the plane is in general position if no more than three points in the set lie onany given circle or line. Given a such a set of points Z in general position in either the Euclideanplane or the hyperbolic plane, the Delaunay triangulation of Z is the simple triangulation thathas the points of Z as its vertices, and, for each triple of distinct points u, v and w in Z, contains ageodesic triangle with corners u, v and w if and only if the unique disc containing u, v and w in itsboundary does not contain any other points of Z. Suppose that Z is an isometry-invariant, locallyfinite point process in either the Euclidean plane or the hyperbolic plane, and let Z\u02c6 be the Palmversion of Z that is conditioned to have a point at the origin. Then the Delaunay triangulationof Z\u02c6 is unimodular when rooted at the point at the origin; see [37] for a study of the Poissoncase. This embedding satisfies condition (1) of Theorem 9.4.9, and we deduce that, unsurprisingly,Delaunay triangulations of hyperbolic point processes are hyperbolic while Delaunay triangulationsof Euclidean point processes are parabolic. (This also follows from the methods of [21].) The dualof the Delaunay triangulation is the Voronoi diagram of the point process, which can also be madeunimodular as in Section 9.2.5.2369.5. Spanning forestsWe remark that it is possible to use Theorem 9.4.9 to prove Proposition 9.4.5 by embedding Min the polygonal manifold associated to \u03c9.9.5 Spanning forests9.5.1 Uniform spanning forestsOur primary way to relate the average curvature of a map M with the various probabilistic prop-erties listed in Theorem 9.1.1 is a formula relating the average degree of the free uniform spanningforest of M to its average curvature. To this aim, we begin by succinctly discussing the topic ofuniform spanning forest, referring the reader to [44, 173] for a comprehensive treatment.For each finite graph G, let USTG be the uniform measure on spanning trees of G (i.e. connectedsubgraphs of G containing every vertex and no cycles), which is the law of a percolation on G. Thereare two natural ways to define infinite volume limits of the uniform spanning tree. Let G = (V,E)be an infinite, locally finite, connected graph. An exhaustion of G is an increasing sequence\u3008Vn\u3009n\u22651 of finite connected subsets of V such that\u22c3n\u22651 Vn = V . Given an exhaustion \u3008Vn\u3009n\u22651of G, we define Gn to be the subgraph of G induced by Vn for each n \u2265 1. The free uniformspanning forest measure of G is defined as the weak limit of the uniform spanning tree measureson the graphs Gn. That is, for each finite set S \u2282 E,FUSFG(S \u2282 F) := limn\u2192\u221eUSTGn(S \u2282 T ),where F is a sample of the FUSF of G and T is a sample of the UST of Gn. For each n \u2265 1, wealso construct a graph G\u2217n from G by identifying every vertex in V \\ Vn into a single vertex \u2202n,and deleting all of the resulting self loops from \u2202n to itself. We then define the wired uniformspanning forest measure of G to be the weak limit of the uniform spanning tree measures on thegraphs G\u2217n. That is, for each finite set S \u2282 E,WUSFG(S \u2282 F) := limn\u2192\u221eUSTG\u2217n(S \u2282 T ),where F is a sample of the WUSF of G and T is a sample of the UST of Gn. The study of uniformspanning forests was pioneered by Pemantle [190], who showed that both limits exist for any graphG and in particular are independent of the choice of exhaustion.The link between the USFs and amenability is the following.Theorem 9.5.1 ([7, Proposition 18.14]: Theorem 9.1.1, (2) implies (12)). If (G, \u03c1) is an invariantlyamenable unimodular random rooted graph, then FUSFG = WUSFG almost surely.The USFs enjoy the following properties:1. (Free dominates wired.) The measure FUSFG stochastically dominates the measureWUSFG for every graph G.2379.5. Spanning forests2. (Domination and subgraphs) let H be a connected subgraph of G. Then the FUSF of Hstochastically dominates the restriction of the FUSF of G to H.3. (Expected degree of the WUSF.) The expected degree in the WUSF of the root of anyunimodular random rooted graph is 2 [7, Proposition 7.3].Note that, since a connected spanning forest cannot be strictly contained in another spanningforest, the stochastic domination (1) above has the following immediate consequence.Proposition 9.5.2 (Theorem 9.1.1, (13) implies (12)). Let G be a graph. If the wired uniformspanning forest of G is connected almost surely, then the wired and free spanning forests of Gcoincide.If (G, \u03c1) is a unimodular random rooted graph and F is a sample of either WUSFG or FUSFG,then the marked graph (G, \u03c1,F) is also unimodular (i.e., F is a percolation on (G, \u03c1)): Since thedefinitions of FUSFG andWUSFG do not depend on the choice of exhaustion, for each mass transportf : G{0,1}\u2022\u2022 \u2192 [0,\u221e], the expectationsfF (G, u, v) = FUSFG[f(G, u, v,F)]and fW (G, u, v) = WUSFG[f(G, u, v,F)]are also mass transports. Using this observation, we deduce the mass-transport principle for(G, \u03c1,F) from that of (G, \u03c1).Connections to random walk and potential theory.Although the uniform spanning tree of each Gn or G\u2217n is connected, the limiting random subgraphcan be disconnected. Indeed, Pemantle [190] proved that WUSF and FUSF of Zd coincide for alld \u2265 1, and are connected if and only if d \u2264 4. A complete characterisation of the connectivity of theWUSF was given by Benjamini, Lyons, Peres and Schramm [44]. A connected, locally finite graphis said to have the intersection property if the traces of two independent simple random walksstarted from any two vertices of the graph have infinite intersection almost surely (or, equivalently,if the two traces have non-empty intersection almost surely). A graph is said to have the non-intersection property if the traces of two independent simple random walks on the graph havefinite intersection almost surely.Theorem 9.5.3 (Benjamini, Lyons, Peres and Schramm [44]: Theorem 9.1.1, equivalence of (14)and (13)). Let G be an infinite, locally finite, connected graph and let F be a sample of WUSFG.Then F is connected almost surely if and only if G has the intersection property. If G has thenon-intersection property, then F has infinitely many connected components almost surely.In general, a graph need not have either of the intersection or non-intersection properties. Forexample, the graph formed by connecting two disjoint copies of Z3 by a single edge between theirorigins does not have either property. However, it is easily seen that this is not the case for reversiblerandom rooted graphs.2389.5. Spanning forestsLemma 9.5.4. Let (G, \u03c1) be a unimodular random rooted graph with E[deg(\u03c1)] < \u221e. Then Geither has the intersection property or the non-intersection property almost surely.Proof. By biasing by the degree we may assume that (G, \u03c1) is reversible. We may assume also that(G, \u03c1) is ergodic, otherwise taking an ergodic decomposition. Let \u3008Xn\u3009n\u22650 and \u3008X\u2212n\u3009n\u22650 be inde-pendent random walks on G started at \u03c1. Then the event that the traces of \u3008Xn\u3009n\u22650 and \u3008X\u2212n\u3009n\u22650have infinite intersection is an invariant event for the stationary sequence \u3008(G, \u3008Xn+k\u3009n\u2208Z\u3009k\u2208Z andtherefore has probability either zero or one by ergodicity.Thus, the WUSF of a unimodular random rooted graph with finite expected degree is eitherconnected or has infinitely many connected components almost surely.Recall that a function h : V \u2192 R defined on the vertex set of a graph G = (V,E) is said to beharmonic ifh(v) =1deg(v)\u2211u\u223cvh(u)for every vertex v of G, or equivalently if \u3008h(Xn)\u3009n\u22650 is a martingale when \u3008Xn\u3009n\u22650 is a randomwalk on G. A graph is said to be Liouville if it does not admit any non-constant boundedharmonic functions, and non-Liouville otherwise. The following proposition, which follows fromthe martingale convergence theorem, is well-known (see [173, Exercise 14.28]).Proposition 9.5.5 (Theorem 9.1.1, (14) implies (10)). Let G be a connected graph. If G has theintersection property, then G is Liouville.The converse of Proposition 9.5.5 does not hold for general graphs. For example, Zd is Liouvillefor all d \u2265 1 but has the intersection property only for d \u2264 4. However, Benjamini, Curien andGeorgakopoulos [40] proved that the converse does hold for planar graphs.Theorem 9.5.6 ([40] Theorem 9.1.1, (10) implies (14)). ] Let G be a planar graph. Then G isLiouville if and only if it has the intersection property.The Dirichlet energy of a function f : V \u2192 R defined on the vertex set of G is defined to beE(f) = 12\u2211e\u2208E\u2192(f(e\u2212)\u2212 f(e+))2.The following is classical; see [173, Exercise 9.43].Proposition 9.5.7 (Theorem 9.1.1, (10) implies (11)). Let G be a connected graph. Then thebounded harmonic functions of finite Dirichlet energy are dense in the space of harmonic func-tions of finite Dirichlet energy. In particular, if G admits a non-constant harmonic function offinite Dirichlet energy, then G admits a bounded non-constant harmonic function of finite Dirichletenergy.Benjamini, Lyons, Peres and Schramm [44] related analytic properties of G to the WUSF andFUSF of G.2399.5. Spanning forestsTheorem 9.5.8 (Benjamini, Lyons, Peres and Schramm [44]: Theorem 9.1.1, equivalence of items(12) and (11)). Let G be an infinite connected graph. Then the measures FUSFG and WUSFG aredistinct if and only if G admits harmonic functions of finite Dirichlet energy.In general, the FUSF is not nearly as well understood as the WUSF: no criterion for its connec-tivity is known, nor is it known whether the number of components of the FUSF is non-random inevery graph. However, for simply connected maps, the FUSF is relatively well understood thanksto the following duality: Given a map M and a set W \u2282 E, let W \u2020 := {e\u2020 \u2208 E\u2020 : e \/\u2208 W} be theset of dual edges whose corresponding primal edges are not contained in W . Observe that if t is aspanning tree of a finite planar map M , then the dual t\u2020 is a spanning tree of M \u2020 \u2013 it is connectedbecause t has no cycles, and has no cycles because t is connected. This observation leads to thefollowing.Proposition 9.5.9 (USF Duality [44, Theorem 12.2]). Let M be a simply connected map withlocally finite dual M \u2020 and let F be a random variable with law FUSFM . Then F\u2020 has the lawof WUSFM\u2020.In general, if M is an infinite simply connected map and F is an essential spanning forest of M(that is, a spanning forest such that every component is infinite), then F\u2020 is an essential spanningforest of M \u2020 \u2013 it is a forest because every component of F is infinite, and is essential because F hasno cycles. Moreover, the forest F is connected if and only if every component of F\u2020 is one-ended.Furthermore, we have the following.Proposition 9.5.10 (Aldous-Lyons [7, Theorem 8.9]: Theorem 9.1.1, (13) implies (2)). Let (G, \u03c1)be a unimodular random rooted graph with E[deg(\u03c1)] < \u221e and let F be a sample of WUSFG. If Fis connected almost surely, then (G, \u03c1) is invariantly amenable.Proof. Suppose that F is connected almost surely. Since F has at most two ends almost surely by[7, Theorem 6.2], [7, Theorem 8.9] implies that (G, \u03c1) is invariantly amenable.9.5.2 ResultsThe first main result of this section relates the average curvature and the expected degree of theFUSF in a simply connected unimodular random rooted map.Theorem 9.5.11. Let (M,\u03c1) be an infinite, simply connected unimodular random map, and supposethat E[deg(\u03c1)] <\u221e, and let F be a sample of FUSFM . ThenE[degF(\u03c1)] =1piE[\u03b8(\u03c1)] = 2\u2212 1piK(M,\u03c1). (9.5.1)As an easy consequence we get the following component of Theorem 9.1.1.Corollary 9.5.12 (Theorem 9.1.1, non-positivity of average curvature and equivalence of (1) and(12)). Let (M,\u03c1) be an ergodic, unimodular random graph with E[deg(\u03c1)] < \u221e. Then the averagecurvature P[\u03ba(\u03c1)] is non-positive and is zero if and only if FUSFM = WUSFM almost surely.2409.5. Spanning forestsProof. Let F be a sample of FUSFM . Since the expected degree of the WUSF in any unimod-ular random rooted graph is two, and FUSFM stochastically dominates WUSFM , we have thatE[degF(\u03c1)] \u2265 2 and that the measures FUSFM and WUSFM differ almost surely if and only if thisinequality is strict. We conclude by applying Theorem 9.5.11.Theorem 9.5.11 follows as an immediate corollary of Proposition 9.4.5 and the following theorem,which is the second main result of this section. In Section 9.5.6, we give an alternative, duality-based proof of Theorem 9.5.11 that does not rely on Theorem 9.5.13 or Proposition 9.4.5, and alsoapplies to the free minimal spanning forest.Theorem 9.5.13 (Connectivity of the FUSF). Let (M,\u03c1) be a simply connected unimodular ran-dom rooted map with E[deg(\u03c1)] < \u221e. Then the free uniform spanning forest of M is connectedalmost surely.Since the measure FUSFG stochastically dominates WUSFG for every graph G, we deduce thefollowing immediate corollary.Corollary 9.5.14 (Theorem 9.1.1, equivalence of (12) and (13)). Let (M,\u03c1) be a simply connectedunimodular random map with E[deg(\u03c1)] < \u221e. Then FUSFM = WUSFM if and only if the wireduniform spanning forest of M is connected almost surely.If M has locally finite dual almost surely, then, by Proposition 9.5.9, Theorem 9.5.13 is equiv-alent to the statement that every component of the WUSF of M \u2020 is one-ended almost surely.Fortunately, it is known that every component of the WUSF is one-ended almost surely in severallarge classes of graphs: The following was proven by the second author [127, 128] and followed ear-lier works by Pemantle [190], Benjamini, Lyons, Peres, and Schramm [44], and Aldous and Lyons[7].Theorem 9.5.15 ([127, 128]). Let (G, \u03c1) be a transient unimodular random rooted graph. Thenevery component of the wired uniform spanning forest of G is one-ended almost surely.In the case that the dual M \u2020 is locally finite, Theorem 9.5.13 follows as a corollary to Theo-rem 9.5.15. The case that dual is not locally finite is contained in Proposition 9.5.18 and is handledby a separate argument. Theorem 9.5.13 is complemented by concurrent work by the second andthird authors [130], who prove the corresponding theorem for deterministic simply connected mapswith bounded degrees. See [170] for further one-endedness results in the deterministic setting.It is an open question whether the WUSF is one-ended almost surely in every one-ended uni-modular random rooted graph (G, \u03c1) with E[deg(\u03c1)] < \u221e, without the assumption of transience.The final result of this section is to prove that this is holds in the planar case.Theorem 9.5.16. Let (M,\u03c1) be a recurrent unimodular random rooted planar map with E[deg(\u03c1)] <\u221e. Then the WUSF of M has the same number of ends as M almost surely (which is either oneor two since M is recurrent).2419.5. Spanning forestsWhen M has locally finite dual, Theorem 9.5.16 follows immediately by duality and the factthat the WUSF of a reversible random rooted graph is either connected or has infinitely manyconnected components almost surely; our contribution is to handle the case that M has infinitefaces.9.5.3 Proof of Theorem 9.5.13 and Theorem 9.5.16Proof of Theorem 9.5.13 when the dual is locally finite. Let (M,\u03c1) be a unimodular random rootedmap with P[deg(\u03c1)] < \u221e, and suppose that the dual M \u2020 is locally finite almost surely. We mayassume that (M,\u03c1) is ergodic, otherwise taking an ergodic decomposition. Let F be a sample ofFUSFM and let F\u2020 be the dual forest. Since E[deg(\u03c1)] < \u221e, the law of (M \u2020, \u03c1\u2020) is equivalent tothe law of a reversible random rooted map by Proposition 9.2.4. If M \u2020 is almost surely transient,Theorem 9.5.15 implies that every component of F\u2020 is one-ended almost surely, and we deduce thatF is connected almost surely. If M \u2020 is almost surely recurrent, then F\u2020 is connected almost surely,and consists of a single tree with at most two ends. It follows that the measures FUSFM\u2020 andWUSFM\u2020 coincide and hence the measures FUSFM coincide WUSFM by duality. If the single treeof F\u2020 were two-ended then F would have exactly two components, contradicting Lemma 9.5.4.The remainder of this section is dedicated to the proof of Theorems 9.5.13 and 9.5.16 in thepresence of infinite faces. We begin by developing a variant of Wilson\u2019s algorithm that allows us tosample the dual of the FUSF using random walks when the dual is not locally finite.Given a graph G and a path \u03b3 in G that is either finite or transient, i.e. visits each vertex of G atmost finitely many times, the loop-erasure LE(\u03b3) is formed by erasing cycles from \u03b3 chronologicallyas they are created. Formally, LE(\u03b3)i = \u03b3ti where the times ti are defined recursively by t0 = 0and ti = 1 + max{t \u2265 ti\u22121 : \u03b3t = \u03b3ti\u22121}. The loop-erasure of a simple random walk is known asloop-erased random walk and was first studied by Lawler [161].Wilson\u2019s algorithm [228] is a method of sampling a uniform spanning tree of a finite graph byjoining together loop-erased random walk paths. Benjamini, Lyons, Peres and Schramm [44] laterintroduced a variant of Wilson\u2019s algorithm for sampling the WUSF of an infinite transient graph,known as Wilson\u2019s algorithm rooted at infinity. Let G = (V,E) be a connected, locally finitegraph and let \u3008vi : i \u2265 1\u3009 be an enumeration of the vertex set V . We sample a sequence of forests\u3008Fi\u3009i\u22650 in G recursively as follows:1. If G is finite or recurrent, fix a vertex v0 of G and let F0 = {v}. Otherwise G is transient andwe set F0 = \u2205.2. Given Fi, start a simple random walk from fi in M independently of everything we havealready sampled, stopped if and when it first hits a vertex already included in Fi.3. Take the loop-erasure of this random walk path, and let Fi+1 be the union of Fi and thisloop-erased path.2429.5. Spanning forests4. Let F =\u22c3i\u22650 Fi.This procedure is referred to as Wilson\u2019s algorithm rooted at v0 when G is finite or recurrent,and Wilson\u2019s algorithm rooted at infinity when G is transient: The resulting random foresthas law USTG when G is finite [228] and WUSFG when G is infinite [44].Now suppose that M is a simply connected map with at least one infinite degree face. LetFfin = {f \u2208 F (M) : deg(f) < \u221e} be the set of finite degree faces of M and let F\u221e = F \\ Ffin bethe set of infinite degree faces of M . Let \u3008fi : i \u2265 1\u3009 be an enumeration of Ffin. We sample anincreasing sequence of forests \u3008F\u2020i : i \u2265 1\u3009 in M \u2020 recursively as follows:1. Let F\u20200 = F\u221e.2. Given F\u2020i , start a simple random walk from fi in M\u2020 independently of everything we havealready sampled, stopped if and when it first hits a vertex already included in F\u2020i .3. Take the loop-erasure of this random walk path, and let F\u2020i+1 be the union of F\u2020i and thisloop-erased path.4. Let F\u2020 =\u22c3i\u22650 F\u2020i .We call this procedure Wilson\u2019s algorithm rooted at {\u221e} \u222a F\u221e.Proposition 9.5.17. Let M = (V,E, \u03c3) be a simply connected map with dual M \u2020 and let F\u2020 bea random subset of E\u2020 sampled by Wilson\u2019s algorithm rooted at {\u221e} \u222a F\u221e. Then F = (F\u2020)\u2020 is asample of FUSFM .Proof. Let \u3008Vn\u3009n\u22650 be an exhaustion of V such that the submap of M induced by V \\ Vn, denotedMn, does not have any finite connected components for any n. The dual of Mn may be constructedfrom M \u2020 by identifying every face f of M that does not have all of its constituent vertices includedin Mn into a single vertex \u2202n, and deleting all the self-loops that are created. In particular, allinfinite faces of M are identified with \u2202n for every n \u2265 1. Note also that if we start a simple randomwalk \u3008Xn\u3009n\u22650 on M \u2020, started at some face f \u2208 Ffin and stopped the first time it hits F\u221e, then,since every time the walk revisits f it has a constant, positive probability of hitting F\u221e beforerevisiting f , the stopped random walk path is either finite or transient almost surely. Given theseobservations, the rest of the proof proceeds similarly to the usual proof of the veracity of Wilson\u2019salgorithm rooted at infinity [44, Theorem 5.1].Let H be a finite set of edges of M \u2020, and let f1, . . . , fl be an enumeration of the set of faces f ofM that are endpoints of at least one of the edges in H. Let \u3008\u3008Xij\u3009j\u22650 : i = 1, . . . , l\u3009 be a collectionof independent random walks in M \u2020, where the walk \u3008Xij\u3009j\u22650 is started at fi and stopped the firsttime that it hits an infinite face of M . Run Wilson\u2019s algorithm in M \u2020n, rooted at \u2202n and startingwith the faces f1, . . . , fl in that order, using the random walks \u3008Xij\u3009j\u22650: For each i \u2208 [l], let \u03c4ni bethe first time that the random walk \u3008Xij\u3009j\u22650 visits the portion of the spanning tree generated up2439.5. Spanning foreststo time i\u2212 1, so thatUSTM\u2020n(H \u2282 T ) = P(H \u2286l\u22c3i=1LE(\u3008Xij\u3009\u03c4nij=0)).Now, similarly, run Wilson\u2019s algorithm on M \u2020 rooted at {\u221e}\u222aF\u221e, starting with the faces f1, . . . , flin that order and using the random walks \u3008Xij\u3009j\u22650, and let \u03c4i be the first time that the randomwalk \u3008Xij\u3009j\u22650 visits the portion of the spanning tree generated up to time i\u2212 1 (which might nowbe infinite). Since the walks \u3008Xij\u3009j\u22650 are finite or transient almost surely, we have that\u03c4ni \u2192 \u03c4i and LE(\u3008Xij\u3009\u03c4nij=0)\u2192 LE(\u3008Xij\u3009\u03c4ij=0)almost surely as n\u2192\u221e. It follows thatFUSFM (H \u2282 F\u2020) = limn\u2192\u221eUSTM\u2020(H \u2282 T ) = limn\u2192\u221eP(H \u2286l\u22c3i=1LE(\u3008Xij\u3009\u03c4nij=0))= P(H \u2286l\u22c3i=1LE(\u3008Xij\u3009\u03c4ij=0))completing the proof.In the case that M has faces of infinite degree, Theorems 9.5.13 and 9.5.16 follow immediatelyfrom the following proposition. (The case of Theorem 9.5.16 in which M is not simply connectedis trivial, since in this case M and F must both have two ends.)Proposition 9.5.18. Let (M,\u03c1) be a simply connected, unimodular random rooted map withP[deg(\u03c1)] < \u221e and suppose that the dual M \u2020 contains a vertex of infinite degree almost surely.Let F be a sample of FUSFM . Then every connected component of F\u2020 \\ F\u221e is finite almost surely,and consequently F is connected almost surely.The main ingredient of Proposition 9.5.18 is the following simple lemma.Lemma 9.5.19. Let (M,\u03b7) be a simply connected reversible random edge-rooted map and supposethat the dual M \u2020 contains a vertex of infinite degree almost surely. Let f \u2208 Ffin and let \u3008Xn\u3009n\u22650 bea random walk on M \u2020 started at f and stopped at \u03c4F\u221e, the first time it visits F\u221e. Then \u03c4F\u221e <\u221ealmost surely.Proof. If there are almost surely no edges e\u2020 \u2208 E\u2020 with both endpoints in Ffin then the result istrivial, so suppose not. Form a (possibly disconnected) graph G\u2021 with edge set E\u2020 by, for eachedge e\u2020 \u2208 E\u2020 with an endpoint in F\u221e, replacing this endpoint with a vertex of degree one. Foreach edge e\u2020 \u2208 E\u2020, let G\u2021e\u2020 denote the connected component of G\u2020 containing e\u2020. The vertex setof G\u2021 may be written as Ffin \u222a L, where L is the set of degree one vertices of G\u2021 correspondingto edges of M incident to an infinite degree face. It is clear that the random edge-rooted graph(G\u2021\u03b7, \u03b7\u2020) is reversible. Every connected component of G\u2021 contains a vertex in L and consequently,2449.5. Spanning forestsFigure 9.6: The maps T4 (left) and M4 (right).by stationarity, a random walk on G\u2021 visits L infinitely often almost surely. We conclude by notingthat the random walk on M \u2020 started at a vertex f \u2208 Ffin and stopped when it first hits F\u221e can becoupled with the random walk on G\u2021 started at the same f \u2208 Ffin and stopped when it first hits Lso that the two hitting times agree.Proof of Proposition 9.5.18. Generate F\u2020 with Wilson\u2019s algorithm rooted at {\u221e} \u222a F\u221e. It followsfrom Lemma 9.5.19 that for every face f of M , there exists a path in F\u2020 connecting f to F\u221e, andthis path is easily seen to be unique. For each oriented edge e of M with e` \u2208 Ffin, transportmass one from e\u2212 to d\u2212, where d\u2020 is the last edge of the unique path connecting e` to F\u221e in F\u2020.Then each vertex u sends a mass of at most deg(u), so that the expected mass sent by the root isfinite. For each connected component of F\u2020 \\ F\u221e, there exists an edge e\u2020 \u2208 F\u2020 that connects thisinfinite connected component to F\u221e, and the vertex e\u2212 receives mass equal to at least the numberof edges in the component. Thus, the existence of an infinite connected component of F\u2020 \\ F\u221ewould contradict the mass transport principle.9.5.4 Finite expected degree is neededExample 9.5.20. Let Tn be a binary tree of height n drawn in the plane. The Benjamini-Schrammlimit of Tn as n tends to infinity is known as the canopy tree, and can be thought of as an \u2018infinitebinary tree viewed from a leaf\u2019. Let Mn be the finite map obtained by drawing two copies of Tnso that one is the reflection of the other, and attaching these two copies together at their leaves(see Figure 5). Then the Benjamini-Schramm limit of Mn exists and is formed of two canopy treesattached together at their leaves. Let T \u2032n be the map formed from Tn by replace each edge atdistance k from the leaves of Tn by 3k parallel edges, and let M \u2032n be the map obtained by drawingtwo copies of T \u2032n so that one is the reflection of the other, and attaching these two copies togetherat their leaves. These strong drifts ensure that, in the Benjamini-Schramm limit M \u2032 of the mapsM \u2032n, the distance between a simple random walk \u3008Xk\u3009k\u22650 and the set of former leaves of the canopytree will increase linearly in k almost surely, and the walk therefore gets stuck in one of the twotrees almost surely. Since the walk can get stuck in either of the two trees, we obtain a boundedharmonic function on M defined by letting h(v) be the probability starting from v that the random2459.5. Spanning forestswalk gets stuck in the first tree. Moreover, Wilson\u2019s algorithm shows that the WUSF of M \u2032 hasexactly two components, corresponding to the two trees of M \u2032.9.5.5 Percolation and minimal spanning forestsWhile the uniform spanning forests are related to random walks, the minimal spanning forests arerelated to bernoulli bond percolation. Recall that a Bernoulli-p bond percolation on G, denoted\u03c9p, is a random subgraph of G defined by keeping each edge of G independently with probabilityp and deleting the rest. The Bernoulli bond percolations {\u03c9p}p\u2208[0,1] on a graph G may be coupledmonotonically by letting {U(e)}e\u2208E be a collection of i.i.d. Uniform([0, 1]) random variables indexedby the edge set of G and setting \u03c9p(e) = 1(U(e) \u2264 p) for every e \u2208 E and p \u2208 [0, 1]. The criticalprobability of G is defined bypc(G) := inf{p : P(\u03c9p has an infinite connected component) = 1}.It is well-known [7, 187] that if (G, \u03c1) is an ergodic unimodular random rooted graph, then for eachp \u2208 [0, 1] the number of infinite connected components of \u03c9p for any is in {0, 1,\u221e} almost surelyand is non-random. Moreover, if (G, \u03c1) is a unimodular random rooted graph and p is such that\u03c9p has a unique infinite cluster almost surely, then \u03c9p\u2032 has a unique infinite connected componentalmost surely for every p\u2032 \u2265 p [7, 113, 176]. In light of this, the uniqueness threshold of a graphG is defined to bepu(G) = inf{p : P(\u03c9p has a unique infinite connected component) = 1} \u2265 pc(G).Note that if (G, \u03c1) is an ergodic, infinite unimodular random rooted graph, then the quantities pc(G)and pu(G) are non-random. It is of interest to determine which graphs have a non-uniqueness phasefor Bernoulli bond percolation. For example, pc = pu for every amenable transitive graph, while along standing conjecture of Benjamini and Schramm [51] asserts conversely that every nonamenabletransitive graph has pc < pu.Lyons, Peres and Schramm [175] related the non-uniqueness phase to minimal spanning forests.Given a finite graph G and an injective function U : E(G)\u2192 R assigning weights to the edges ofG, the minimal spanning tree of G with respect to U is defined to be the spanning tree T of Gminimising the total weight\u2211e\u2208T U(e). Equivalently, an edge e of G is contained in T if and onlyif there does not exist a simple cycle in G containing e such that e maximises U(e) among the edgesin this cycle. We write MSTG for the distribution on spanning trees of G obtained by letting T bethe minimal spanning tree of G with respect to weights {U(e)}e\u2208E given by i.i.d. Uniform([0, 1])random variables. As for the uniform spanning trees, given an exhaustion \u3008Vn\u3009n\u22650 of an infinitegraph G, we the free and wired minimal spanning forests as the weak limitsFMSFG(S \u2282 T ) := limn\u2192\u221eMSTGn(S \u2282 T )2469.5. Spanning forestsandWMSFG(S \u2282 T ) := limn\u2192\u221eMSTG\u2217n(S \u2282 T ).The limits do not depend on the choice of exhaustion and, if (G, \u03c1) is a unimodular random rootedgraph and F is a sample of either WMSFG or FMSFG, then F is a percolation on (G, \u03c1). Unlike inthe uniform case, both of the minimal spanning forests may also be defined directly on the infinitegraph G as follows. Let {U(e) : e \u2208 E} be a collection of i.i.d. Uniform([0, 1]) random variablesindexed by the edge set of G. An edge e of G is included in free minimal spanning forest of G ifand only if it is not the heaviest edge in any simple cycle in G. An edge e of G is included in thewired minimal spanning forest of G if and only if it is not the heaviest edge in any simple cycle inG or in any bi-infinite simple path (or \u2018cycle through infinity\u2019) in G.Lyons, Peres and Schramm [175] proved that an infinite connected graph G has FMSFG =WMSFG if and only if for \u03c9p has a unique infinite cluster for Lebesgue-a.e. p \u2208 [0, 1]. Combiningthis with uniqueness monotonicity [7, 113, Theorem 6.7] yields the following.Theorem 9.5.21 ([7, 113, 175]: Theorem 9.1.1, equivalence of (15) and (16)). Let (G, \u03c1) be aninfinite unimodular random rooted graph with E[deg(\u03c1)] < \u221e. Then pc(G) < pu(G) if and only ifFMSFG 6= WMSFG.The minimal spanning forests share several properties with their uniform cousins:1. (Free dominates wired.) The measure FMSFG stochastically dominates the measureWMSFG for every graph G.2. (Domination and subgraphs) let H be a connected subgraph of G. Then the FMSF of Hstochastically dominates the restriction of the FMSF of G to H.3. (Expected degree of the WMSF.) The expected degree in the WMSF of root of anyunimodular random rooted graph is two [7, Proposition 7.3].4. (Amenability and boundary conditions.) If (G, \u03c1) is an invariantly amenable randomrooted graph, then FMSFG = WMSFG almost surely [7, Proposition 18.14].5. (Planar duality.) If M is a simply connected map with locally finite dual M \u2020 and F is asample of FMSFM , then F\u2020 has law WMSFM\u2020 [173, \u00a711.5].Combining items 1 and 2 above, we deduce that if (G, \u03c1) a unimodular random rooted graph, themeasures FMSFG and WMSFG coincide almost surely if and only if the expected degree of \u03c1 in theFMSF of G is two.By using the properties above, the proof of Theorem 9.5.11 also yields a formula relating theexpected degree of the FMSF to the average curvature.Theorem 9.5.22 (Theorem 9.1.1, equivalence of (1) and (15)). Let (M,\u03c1) be an infinite simplyconnected unimodular random rooted map with E[deg(\u03c1)] < \u221e and let F be a sample of FMSFM .2479.5. Spanning forestsThenE[degF(\u03c1)] = 2\u22121piK(M,\u03c1).In particular, FMSFM = WMSFM almost surely if and only if K(M,\u03c1) = 0.The equivalence of (16) and (2) in Theorem 9.1.1 can also be proven directly as follows. Let(M,\u03c1) be a hyperbolic simply connected unimodular random rooted map. If (M,\u03c1) has locallyfinite dual, we deduce that pc(M) < pu(M) by applying the following two results.Theorem 9.5.23 (Benjamini, Lyons, Peres and Schramm [36, 43]; Aldous and Lyons [7]). Let(G, \u03c1) be an invariantly nonamenable unimodular random rooted graph with E[deg(\u03c1)] <\u221e. Then\u03c9pc does not contain any infinite connected components almost surely.Theorem 9.5.24 (Benjamini and Schramm [49, Theorem 3.1]). Let (M,\u03c1) be an invariantly non-amenable, simply connected unimodular random rooted map with locally finite dual M \u2020 and supposethat E[deg(\u03c1)] < \u221e. Then \u03c9p has a unique infinite connected component if and only if everycomponent of \u03c9\u2020p is finite. It follows thatpu(M) = 1\u2212 pc(M \u2020)almost surely and that \u03c9pu contains a unique infinite connected component almost surely.Benjamini and Schramm proved their theorem for transitive planar graphs, but their proofextends immediately to our setting.If M does not have locally finite dual, then it must have infinitely many infinite faces byLemma 9.3.15, so that the underlying graph of M is infinitely ended. In this case, we have thatpu(M) = 1 almost surely (see Proposition 9.7.7), while pc(M) < 1 by Theorem 9.5.23.Thus, we have the following.Corollary 9.5.25 (Theorem 9.1.1, (2) implies (16)). Let (M,\u03c1) be a simply connected, invariantlynonamenable, unimodular random rooted map with E[deg(\u03c1)] < \u221e. Then pc(M) < pu(M) almostsurely.9.5.6 Expected degree formulaWe prove Theorem 9.5.22. Every property of the minimal spanning forests that we use also holdsfor the uniform spanning forests, so that we also obtain an alternative proof of Theorem 9.5.11 thatdoes not rely on Theorem 9.5.13.Proof of Theorem 9.5.11.Locally finite dual case. Let \u03c9 be a percolation on (M,\u03c1), and let \u03c9\u2020 = {e\u2020 \u2208 E\u2020 : e \/\u2208 \u03c9}.As in Section 9.2.5, let \u03b7 be chosen uniformly at random from the set E\u2192\u03c1 of oriented edges of Memanating from \u03c1, let \u03c1\u2020 = \u03b7r, and let Prev be the deg(\u03c1)-biasing of P and let P\u2020 be the deg(\u03c1\u2020)\u221212489.5. Spanning forestsbiasing of Prev, so that (M \u2020, \u03c1\u2020) is a unimodular random rooted map under Prev. We write E\u2020 forthe expectation operator associated to P\u2020.Lemma 9.5.26. The following equation holds.E[deg\u03c9(\u03c1)] = E[deg(\u03c1)]\u2212 E[\u2211f\u22a5\u03c1deg(f)\u22121]E\u2020[deg\u03c9\u2020(\u03c1\u2020)]. (9.5.2)Proof. Observe that, since \u03b7 is uniformly distributed on E\u2192\u03c1 conditional on (M,\u03c1), we haveE[deg\u03c9(\u03c1)] = E[deg(\u03c1)1(\u03b7 \u2208 \u03c9)] = E[deg(\u03c1)(1\u2212 1(\u03b7\u2020 \u2208 \u03c9\u2020))],and soE[deg\u03c9(\u03c1)] = E[deg(\u03c1)](1\u2212 Prev(\u03b7\u2020 \u2208 \u03c9\u2020)).Similarly, since under the measure P\u2020 and conditional on (M \u2020, \u03c1\u2020), \u03b7\u2020 is uniformly distributed onE\u2192\u03c1\u2020 ,E\u2020[deg\u03c9\u2020(\u03c1\u2020)] = E\u2020[deg(\u03c1\u2020)]Prev(\u03b7\u2020 \u2208 \u03c9\u2020).It follows thatE[deg\u03c9(\u03c1)] = E[deg(\u03c1)](1\u2212 E\u2020[deg(\u03c1\u2020)]E[deg(\u03c1)]E\u2020[deg\u03c9\u2020(\u03c1\u2020)]).Applying the expected degree formula (9.2.3), we deduce thatE[deg\u03c9(\u03c1)] = E[deg(\u03c1)](1\u2212 E[\u2211f\u22a5\u03c1 deg(f)\u22121]E[deg(\u03c1)]E\u2020[deg\u03c9\u2020(\u03c1\u2020)]),which rearranges to give the desired expression.Let F have law FMSFM . By Proposition 9.5.9, the dual forest F\u2020 is distributed according toWMSFM\u2020 . Since the expected degree at the root of the WMSF in any unimodular random rootedgraph is 2, we have E\u2020[degF\u2020(\u03c1\u2020)] = 2 and consequently, by Lemma 9.5.26,E[degF(\u03c1)] = E[deg(\u03c1)]\u2212 2E[\u2211f\u22a5\u03c1deg(f)\u22121]= E\uf8ee\uf8f0\u2211f\u22a5\u03c1deg(f)\u2212 2deg(f)\uf8f9\uf8fb = 2\u2212 1piK(M,\u03c1).This completes the proof in the case that the dual of M is locally finite.Non-locally finite dual case. Observe that, if \u03c9 is a percolation on a unimodular random map(M,\u03c1) that is almost surely a spanning forest of M , then every component of \u03c9\u2020 is infinite, and it2499.6. The conformal typefollows from [7, Theorem 6.1] that E\u2020[deg\u03c9\u2020(\u03c1\u2020)] \u2265 2. Thus, we deduce from Lemma 9.5.26 thatE[degF(\u03c1)] \u2264 2\u22121piK(M,\u03c1). (9.5.3)For each n \u2265 1, let Mn be the locally finite map containing M as a submap that was defined inthe proof of Proposition 9.4.5, and let Fn be a sample of FMSFMn . Since (Mn, \u03c1) has locally finitedual, we have thatE[degFn(\u03c1)] = 2\u22121piK(Mn, \u03c1) = 2\u2212 1piK(M,\u03c1),where the second equality follows from Proposition 9.4.5.Since the underlying graph of M is a subgraph of the underlying graph of Mn, the forest Fstochastically dominates the restriction Fn \u2229 E for every n \u2265 1, and soE[degF(\u03c1)] \u2265 E[degFn(\u03c1)]\u2212 E[degMn(\u03c1)\u2212 degM (\u03c1)]= 2\u2212 1piK(M,\u03c1)\u2212 E[degMn(\u03c1)\u2212 degM (\u03c1)] \u2212\u2212\u2212\u2192n\u2192\u221e 2\u22121piK(M,\u03c1).(The limit can be deduced either by direct calculation or by invoking the dominated convergencetheorem.) To obtain a corresponding upper bound, consider F as a subgraph of M1. Since F is aspanning forest of M1, we can apply (9.5.3) to deduce thatE[degF(\u03c1)] \u2264 2\u22121piK(M1, \u03c1) = 2\u2212 1piK(M,\u03c1)completing the proof.9.6 The conformal typeGiven a map M such that every face of M has degree at least three, we may form a surface S(M)by gluing regular unit polygons together according to the combinatorics of M , the boundaries ofthese polygons becoming the edges of M embedded in S(M). As before, we consider the upperhalf-space {x+ iy \u2208 C : y > 0} with edges {[n, n+ 1] : n \u2208 Z} to be a regular \u221e-gon. The surfaceS(M) can be endowed naturally with a conformal structure by defining an atlas as follows.\u2022 For each face f of M , we take as a chart the identity map from the interior of the regularpolygon corresponding to f to itself.\u2022 For each edge e of M , we define an open neighbourhood of the interior of e in S by takingthe two triangles formed by the endpoints of e and the centres of the two faces adjacentto e (if either face is infinite, we interpret this triangle to be the infinite strip starting ate and perpendicular to the boundary of the face). To define a coordinate chart on thisneighbourhood, we simply place the two triangles next to each other in the plane. This chart2509.6. The conformal typeis well defined even if both sides of e are incident to the same face.\u2022 For each vertex v of M , we define an open neighbourhood of v in S similarly by intersectingthe corners of the faces adjacent to v with open discs of radius 1\/2 centred at v. We define achart on this neighbourhood by first laying the corners out in a (possibly overlapping) spiralaround the origin, and then applying the function z 7\u2192 z2pi\/\u03b8(v), suitably interpreted to getan injective map into the plane.z2pi\/\u03b8(v)The coordinate changes are easily seen to be analytic, so that this atlas does indeed define aRiemann surface structure on S(M). We denote this Riemann surface by R(M).The definition of R(M) can be extended (somewhat arbitrarily) to maps containing faces ofdegree 1 and 2 as follows: Given a map M , let M\u02c6 be obtained from M by adding a vertex insideeach face of M that has degree 1 or 2, and connecting this vertex to each of the corners of theface. Every face of M\u02c6 has degree at least three, and we define R(M) = R(M\u02c6). The map M canbe embedded in R(M) by restricting the natural embedding of M\u02c6 into R(M\u02c6).If M is simply connected, the uniformization theorem implies that R(M) is conformally equiv-alent to the sphere, the plane or the disc, and we call M conformally elliptic, parabolic, orhyperbolic accordingly. We refer to the embedding of M into the sphere, the plane, or the discgiven by uniformizing R(M), which is unique up to Mo\u00a8bius transformations of the sphere, plane ordisc as appropriate, as the conformal embedding of M . Conformal embeddings of unimodularrandom planar maps are conjectured to play a key role in the theory of two-dimensional quantumgravity, see [74] and references therein.Gill and Rohde [103] proved that every Benjamini-Schramm limit of finite planar maps withuniformly bounded codegrees is conformally parabolic almost surely. The main result of this sectiongeneralises and, together with Corollary 9.3.6, provides a converse to their result.2519.6. The conformal typeTheorem 9.6.1 (Theorem 9.1.1, equivalence of (9) and (2)). Let (M,\u03c1) be an infinite, ergodic,simply connected unimodular random rooted map with E[deg(\u03c1)] < \u221e. Then M is conformallyparabolic almost surely if and only if (M,\u03c1) is hyperfinite.Proof of Theorem 9.6.1: Conformally parabolic implies hyperfinite. Let (M,\u03c1) be a unimodular ran-dom rooted map such that M is conformally parabolic a.s., and let z be the conformal embedding ofM into the plane, which is unique up to translation and scaling. We claim that (M,\u03c1) is hyperfinite.We claim that if H and H \u2032 are two open half-planes that are disjoint from z(V ), then either His contained in H \u2032, H \u2032 is contained in H, or H and H \u2032 are disjoint. Indeed, if this is not the case,then the set z(V ) is contained in some cone. It follows that there exists a unique, not necessarilydistinct, pair of angles \u03b81, \u03b82 \u2208 [0, 2pi) with |\u03b82 \u2212 \u03b81| mod 2pi < pi such that for every \u03b5 > 0 andevery point z0 \u2208 C, the cone{z \u2208 C : arg(z \u2212 z0) \u2208 [\u03b81 \u2212 \u03b5, \u03b82 + \u03b5]}contains all but finitely many points of z(V ), but, if \u03b81 6= \u03b82, then infinitely many points of z(V )are not in the cone{z \u2208 C : arg(z \u2212 z0) \u2208 [\u03b81 + \u03b5, \u03b82 \u2212 \u03b5]}.In particular, there exists a finite, nonempty set of vertices of z(V ) that have maximally negativeinner product with ei\u03b81 . This set of vertices does not depend on the choice of conformal embedding,and so we obtain a contradiction by transporting a mass of one from each vertex of M to a uniformlychosen element of this set.If there are a.s. two disjoint open half-planes that are both disjoint from z(V ), let H1 and H2 bethe two disjoint open half-planes disjoint from z(V ) that are maximal in the sense that any open-half-plane strictly containing either H1 or H2 must intersect z(V ). By translating, rotating, andscaling, we may assume that H1 is the lower half-plane {z \u2208 C : I(z) < 0} and H2 is the half-plane{z \u2208 C : I(z) > 1}. Let U be a uniform [0, 1] random variable, and consider the randomly shiftedreal integers Z+U \u2282 C. For each n \u2208 Z, let v(n) \u2208 V be chosen uniformly at random from amongthose v \u2208 V such that z(v) is of minimal distance to n + U . Drawing an edge between v(n) andv(n+ 1) for every n \u2208 Z defines a unimodular coupling between (M,\u03c1) and Z, and it follows fromProposition 9.3.13 that (M,\u03c1) is hyperfinite in this case.Now suppose that there is a unique open half-plane H disjoint from z(V ) that is maximal inthe sense that any open-half-plane strictly containing H must intersect z(V ). By rotating andtranslating, we may assume that H is the right half-plane, H = {x + iy \u2208 C : x > 0}. We willuse the linear structure of the boundary of H to show that (M,\u03c1) is coupling equivalent to Z, sothat it will follow from Proposition 9.3.13 that (M,\u03c1) is hyperfinite. Since z(V ) is locally finite,there exists a vertex v such that the straight line {x+ z(v) : x \u2265 0} is disjoint from z(V ) \\ {z(v)}.For any such vertex, again by local finiteness of z(V ), there exists a positive angle \u03b8 such thatthe cone C\u03b8(v) = {z \u2208 C : arg(z \u2212 z(v)) \u2208 [\u2212\u03b8, \u03b8]} is disjoint from C\u03b8(u) for every u \u2208 V withRz(u) \u2265 Rz(v). We say that v is \u03b8-exposed if this condition holds. By the above discussion, we2529.6. The conformal typemay choose \u03b8 > 0 such that \u03c1 is \u03b8-exposed with positive probability, in which case it follows fromergodicity and unimodularity that there are infinitely many \u03b8-exposed vertices a.s. Observe that,again using the fact that z(V ) is locally finite, for every bounded interval [a, b], there exist at mostfinitely many \u03b8-exposed vertices v of M that have Iz(v) \u2208 [a, b]. Define a graph whose vertices arethe \u03b8-exposed vertices of M , and where we draw an edge between two distinct \u03b8-exposed vertices uand v if and and only if there is no \u03b8-exposed vertex w, distinct from u and v, such that Iz(w) liesbetween Iz(u) and Iz(v). The above discussion implies that this graph is isomorphic to Z, and itfollows that (M,\u03c1) and Z are coupling equivalent as claimed.Now suppose that z(V ) intersects every open half-plane. We define D to be the Delaunaytessellation with vertex set given by the set of points z(V ). That is, D is the map that hasembedded vertex set z(V ), and has a straight line between two points z(u) and z(v) if and only ifthere exists either a disc or a half-plane in the plane such that1. the disc or half-plane does not have any points of z(V ) in its interior,2. z(u), z(v), and at least one other point of z(V ) lie on the boundary of the disc or half-plane,and3. z(u) and z(v) are adjacent in the cyclic ordering of the set of points of z(V ) that intersectthe boundary of the disc or half-plane.Note that the isomorphism class of (D, \u03c1) is independent of the choice of the conformal embedding.If the dual of D is not locally finite, observe that the union of the finite degree faces of D is equalto the convex hull of the set z(V ), and consequently any infinite face of D must contain a half-spacedisjoint from z(V ). On the other hand, if D is not locally finite, it follows from the definition ofD that for every infinite degree vertex v of D, there exists an infinite sequence of vertices ui andclosed discs or half-planes Ci such that Ci contains both z(v) and z(ui) in its boundary and nopoints of z(V ) in its interior. By taking a subsequential limit, it follows from the fact that z(V )is locally finite that there must exist a half-space containing z(v) in its boundary and no points ofz(V ) in its interior.Thus, we may assume that D is finite and has locally finite dual a.s. In this case, it follows fromthe measurability of conformal embedding that (D, \u03c1) is a unimodular random rooted map, andthat (G, \u03c1) and (D, \u03c1) are coupling equivalent. Thus, by Proposition 9.3.13, it suffices to prove that(D, \u03c1) is hyperfinite. Define a mass transport as follows. For each vertex u and face f of D incidentto u, let ang(f, u) be the angle of the corner of f at u. Transport a mass of ang(f, u)\/ deg(f) fromu to each of the vertices incident to f , including u itself. The mass sent out by each vertex u is 2pi,while, since each face is a polygon in the plane, the mass received is\u2211f\u22a5usum of internal angles of fdeg(f)=\u2211f\u22a5udeg(f)\u2212 2deg(f)pi.Applying the mass-transport principle yields that the average curvature of (D, \u03c1) is zero, and it2539.6. The conformal typefollows from the equivalence of items (1) and (2) of Theorem 9.1.1 that (D, \u03c1) is hyperfinite asclaimed.Before proving the converse, our immediate goal is to prove the following lemma. Suppose Mis a conformally hyperbolic map. For each oriented edge e of M , let Ue be the subset of R(M)defined as follows. If the face er to the right of e has finite degree, let Ue be the quadrilateral inthe polygon corresponding to f that is formed from the first half of e, the first half of \u03c3(e) (i.e., thenext edge clockwise from e in the cyclic ordering of the edges emanating from e\u2212), and the straightlines between the center of f and the midpoints of these two edges. If deg(f) is infinite, let Ue bethe infinite strip in the half-plane corresponding to f whose boundary is given by the first half ofe, the first half of \u03c3(e), and the two half-infinite straight lines that start at the midpoints of theseedges and are perpendicular to the boundary of f . For each vertex v of M , we defineUv =\u22c3e\u2212=vUe.Given a subset K of a Riemann surface that is conformally equivalent to the hyperbolic plane,we write areaH(K) of the hyperbolic area of the image of K under a uniformizing map from theRiemann surface to the hyperbolic plane.Lemma 9.6.2. There exists a constant C such that the following holds. Let M be a conformallyhyperbolic, simply connected map such that every face of M has degree at least three. Then forevery vertex v of M ,areaH(Uv) \u2264 C deg(v).Proof. We shall require the following classical facts about hyperbolic area.1. Schwarz-Pick. If R is a Riemann surface that is conformally equivalent to the disc, D is asimply connected open subset of R, and K is a Borel subset of D, then the hyperbolic area ofK considered as a subset of D is greater than or equal to the hyperbolic area of K consideredas a subset of R. (Schwarz-Pick is usually stated in terms of the metrics, but immediatelyimplies this statement.)2. For every 0 < \u03b5 < 1, the hyperbolic area of the set {z \u2208 C : |z| < 1 \u2212 \u03b5}, considered as asubset of the open unit disc, is given by4pi(1\u2212 \u03b5)21\u2212 (1\u2212 \u03b5)2 \u22642pi\u03b5. (9.6.1)For each oriented edge e, let U1e , . . . , U4e be the following subsets of the polygon associated tothe face er. See Figure 9.7 for an illustration.1. U1e is the 1\/8 neighbourhood of the midpoint of e.2549.6. The conformal typeFigure 9.7: The covering of Ue (grey) by discs (dashed boundaries) used in the proof of Lemma 9.6.2.Left: the case that er has degree three. Right: the case that er has infinite degree.2. U2e is the 1\/8 neighbourhood of the midpoint of the edge following e in the clockwise orderof the edges adjacent to e\u2212.3. U3e is the 7\/16 neighbourhood of e\u2212.4. If er has finite degree, we define U4e is the intersection of Ue with the disc that is centred atthe centre of the polygon corresponding to er and that has distance 1\/16 from the boundaryof the polygon corresponding to er. If er has infinite degree, we let U4e be the intersectionof Ue with the upper half-plane that has distance 1\/16 from the boundary of the half-planecorresponding to er.It is easily verified by elementary trigonometry that the Ue is contained in the union\u22c34i=1 Uie.We first claim that the hyperbolic areas of U1e and U2e are bounded above by a constant. Recallthat in the chart at e, we simply place the two triangles formed by the endpoints of e and the centresof the two faces adjacent to e next to each other (where the triangles become infinite strips if aface has infinite degree). It is easily verified that the open ball of radius 1\/4 around the midpointof e is always contained in the domain formed by placing the two triangles together. Since U1e iscontained in the ball of radius 1\/8 around the midpoint of e, it follows that the hyperbolic area ofU1e is bounded above by a constant (namely, 4pi\/3) by (9.6.1) and Schwarz-Pick. The correspondingclaim for U2e follows similarly.We next claim that the hyperbolic area of U4e is bounded above by a constant. By Schwarz-Pick,it suffices to prove that the hyperbolic area of U4e is bounded above by a constant when consideredas a subset of the polygon corresponding to er. If er has infinite degree, this area is the same as thehyperbolic area of {x+ iy \u2208 C : x \u2208 [0, 1], y \u2265 1\/16} considered as a subset of the upper half-plane{x+ iy \u2208 C : y > 0}, which is finite (in fact, it is equal to 16). On the other hand, if er has finitedegree, then we can consider the disc D centred at the centre of the polygon corresponding to erthat has distance 1\/16 to the boundary of the polygon. The radius of the disc D\u2032 that is centredat the centre of the polygon and just touches the boundary of the polygon is1 + cos(2pi\/deg(er))4 sin(pi\/deg(er))\u2264 deg(er)2pi.2559.6. The conformal typeThus, it follows that the hyperbolic area of D considered as a subset of D\u2032 is at most 16 deg(er).We deduce that, by symmetry, the area of U4e considered at a subset of D\u2032 is at most 16. The claimnow follows by applying Schwarz-Pick.Finally, we claim that for every vertex v, the hyperbolic area of U3v :=\u22c3e\u2212=v U3e is at mostC deg(v), where C is a universal constant. Observe that U3v is simply the set of points that are atdistance at most 7\/16 from v in one of the polygons corresponding to a face incident to v. Underthe chart associated to v, this set gets mapped to a ball of radius (7\/16)2pi\/\u03b8(v), while the set of allpoints at distance at most 1\/2 from v gets mapped to the ball of radius (1\/2)2pi\/\u03b8(v) with the samecentre. It follows that the hyperbolic area of U3v considered as a subset of this larger ball is at most2pi1\u2212 (7\/8)2pi\/\u03b8(v) \u2264 C\u03b8(v) \u2264 piC deg(v),where C is a constant. Verifying that such a constant exists is a simple calculus exercise. Theclaim now follows from Schwarz-Pick.The lemma follows by combining the three estimates given.Proof of Theorem 9.6.1: Conformally hyperbolic implies invariantly nonamenable. We may assumethat every face of M has degree at least three, since otherwise the map M\u02c6 used to define R(M)can easily be rerooted and biased to be unimodular, and is coupling equivalent to M . Let \u03c6 be aconformal equivalence between R(M) and the hyperbolic plane, let Z be a Poisson point processof intensity 1 on the hyperbolic plane, and let D be the Delaunay triangulation associated to Z.We form a graph G whose vertex set is V \u222a Z, and has as edges the edges of M , the edges ofD, and an edge connecting each v \u2208 V to every point z \u2208 Z \u2229 \u03c6(Uv). We also mark the edges ofG according to which of these three types they come from. Note that the law of (G, \u03c1) does notdepend on the choice of \u03c6. It follows from Lemma 9.6.2 that the expected number of points inZ \u2229\u03c6(U\u03c1) is finite. It is easily verified, using the measurability of the conformal embedding, that ifwe sample (G, \u03c1) biased by 1 + |Z \u2229 \u03c6(U\u03c1)| and then let \u03c1\u02c6 be uniform on the set {\u03c1} \u222a (Z \u2229 \u03c6(U\u03c1)),then the resulting random rooted network (G, \u03c1\u02c6) is unimodular. Similarly, if we sample (M,\u03c1) andZ biased by |Z\u2229\u03c6(U\u03c1)|, and let \u03c1\u2032 be uniform on the set Z\u2229\u03c6(U\u03c1), then the resulting graph (D, \u03c1\u2032)is unimodular. Thus, we have defined a unimodular coupling between (M,\u03c1) and (D, \u03c1\u2032). SinceD is a Poisson-Delaunay triangulation of the hyperbolic plane, (D, \u03c1\u2032) is invariantly nonamenable(see [37] and [21, Section 2]), and we conclude by applying Proposition 9.3.13.We remark that there are many other natural (and inequivalent) ways to associate Riemannsurfaces to maps. For example, we could associate to each face of M a disc of circumference k,with boundary split into k arcs of length one corresponding to the edges, and glue adjacent facesaccording to arc length along their shared edges. The proof of Theorem 9.6.1 extends to many ofthese alternatively defined conformal embeddings with only minor modifications.2569.7. Multiply-connected and non-planar mapsPlanarNot planarCompact One Ended Two Ended Infinitely Endedsphere discg-holed torus prison windowinfinitecylinderJacob\u2019s ladderCantor treeblossomingCantor treeFigure 9.8: Possible topologies of a unimodular random map. The surface S(M) associated to aunimodular random map (M,\u03c1) is almost surely homeomorphic to one of the above.9.7 Multiply-connected and non-planar maps9.7.1 The topology of unimodular random rooted maps.In this section we study multiply-connected unimodular random rooted maps. We begin by classi-fying the possible topologies of the surface associated to a unimodular random rooted map (M,\u03c1).Biringer and Raimbault [54] classified the possible topologies of unimodular random rooted com-plete, orientable, hyperbolic surfaces. Their methods readily generalise to our setting, yielding thefollowing theorem. In fact, the proof is slightly less technical in our setting, and we provide a quicksketch below.Theorem 9.7.1 (Topology of unimodular random rooted maps). Let (M,\u03c1) be an infinite uni-modular random rooted map. Then the surface associated to M is almost surely homeomorphic toone of the following surfaces: the plane, the cylinder, the Cantor tree, the infinite prison window,Jacob\u2019s ladder, or the blossoming Cantor tree.Here, the Cantor tree is a \u2018tree made of tubes\u2019 that is homeomorphic to the complementof the Cantor set in the sphere, the infinite prison window is \u2018the lattice Z2 made of tubes\u2019,Jacob\u2019s ladder is \u2018an infinite ladder made of tubes\u2019, and the blossoming Cantor tree is a Cantortree with a handle attached near each bifurcation. See Figure 4 for illustrations. Be warned thathomeomorphism is an extremely weak notion here. For example, \u2018the lattices Zd made of tubes\u2019are all homeomorphic to each other for all d \u2265 2, as indeed are any two \u2018one-ended infinite graphsmade of tubes\u2019.In [54], Biringer and Raimbault must also allow for the surfaces above to be punctured at alocally finite set of points, corresponding to isolated ends of the surface. This does not occur inour setting, as the surfaces corresponding to unimodular random rooted maps do not have isolated2579.7. Multiply-connected and non-planar mapsends.Sketch of proof. An end \u03be of an infinite graph G is a function that assigns a connected component\u03beK of G\\K to each finite set of vertices K of G, and satifies the consistency condition that \u03beK\u2032 \u2286 \u03beKwhenever K \u2032 \u2287 K. The space of ends of G, denoted \u2202E (G) is the topological space with theset of ends of G as its underlying set and with a basis of open sets given by sets of the form{\u03be an end of G : \u03beK = W}, where K \u2282 V is finite and W \u2282 V is a connected component of G \\K.Note that the basis sets are also closed, so that the space of ends is always zero-dimensional, thatis, its topology is induced by a basis of sets that are both open and closed. The space of ends ofa surface is defined similarly, replacing instances of the word \u2018finite\u2019 by \u2018compact\u2019 above, and arealso zero-dimensional.It is well-known that every unimodular random rooted graph either has one, two, or infinitelymany ends, and, in the last case, the space of ends does not have any isolated points [7, Proposition6.10]. Since the space of ends of any graph is also compact, it follows in the last case that the spaceof ends is homeomorphic to the Cantor set (which, by Brouwer\u2019s Theorem [61], is the only compact,zero-dimensional Hausdorff space with no isolated points). A similar proof applies to show that if(M,\u03c1) is a random rooted map with associated surface S = S(M), then S has either one, two, orinfinitely many ends and in the last case the space of ends of S is homeomorphic to a Cantor set.Next, standard mass transport arguments show that if S contains handles, then the handles of Saccumulate towards every end of S. That is, if S has handles then, for every compact subset K ofS, every non-precompact connected component of S \\K contains a handle.We next apply the classification theorem for non-compact surfaces due to Kere\u00b4kja\u00b4rto\u00b4 andRichards [199], which states that if two non-compact orientable surfaces S1 and S2 have the samenumber of handles (which in our case will be zero or infinity) and there exists a homeomorphism\u03c6 : \u2202E (S1) \u2192 \u2202E (S2) such that the handles of S1 accumulate to \u03be \u2208 \u2202E (S1) if and only if thehandles of S2 accumulate to \u03c6(\u03be), then S1 and S2 are homeomorphic and \u03c6 extends to a homeomor-phism from the ends compactification of S1 to the ends compactification of S2. Thus, by the abovediscussion, the homeomorphism class of S = S(M) is determined almost surely by its number ofends and by the existence or non-existence of handles. This yields the six different possibilitieslisted in the statement of the theorem (see Figure 6).Example 9.7.2. The product T3 \u00d7 H of the 3-regular tree T3 and the graph H consisting of 3parallel edges between two vertices is planar but cannot be drawn in the plane without accumulationpoints. By letting the cyclic ordering of the edges emanating from each vertex of T3 \u00d7H alternatebetween T3-edges and H-edges, and making the cyclic ordering of the edges emanating from (v, 0)and (v, 1) reflections of each other, we obtain a transitive quadrangulation of the Cantor tree.The main result of this section is that the average curvature of a unimodular random rootedmap restricts the possible topologies of the map.Theorem 9.7.3 (Topology from average curvature). Let (M,\u03c1) be an ergodic unimodular randommap. Then the almost sure conformal type of M is determined by its average curvature: Either2589.7. Multiply-connected and non-planar maps1. The average curvature of (M,\u03c1) is positive, in which case M is conformally elliptic and S(M)is homeomorphic to the sphere almost surely,2. the average curvature of (M,\u03c1) is zero, in which case M is conformally parabolic and S(M)is homeomorphic to the plane, the cylinder, or the torus almost surely,or else(3) the average curvature of (M,\u03c1) is negative, in which case M is conformally hyperbolic S(M)is homoemorphic to either the plane, the blossoming Cantor tree, Jacob\u2019s ladder, the infiniteprison window, or a compact surface of genus at least two almost surely.The theorem will follow by combining Theorem 9.6.1, Theorem 9.7.1 and the notion of theuniversal cover of a map.Proof. Recall that a surjective, holomorphic function \u03a0 : S \u2192 S\u2032 between two Riemann surfaces isa holomorphic covering if it is locally a homeomorphism, that is, if for every x \u2208 S there existsan open neighbourhood U of S such that the restriction of \u03a0 to U is a homeomorphism betweenU and its image. Given a Riemann surface S, the universal cover of S is a simply connectedRiemann surface S\u02dc together with a covering \u03a0 : S\u02dc \u2192 S. The universal cover exists for any S, andis unique in the sense that if \u03a0\u2032 : S\u02dc\u2032 \u2192 S is another simply connected Riemann surface covering S,then there exists a conformal equivalence \u03c6 : S\u02dc\u2032 \u2192 S\u02dc such that \u03a0\u2032 = \u03a0 \u25e6 \u03c6.The universal cover of a map is defined analogously. Given a pair of maps M = (G, \u03c3) andM \u2032 = (G\u2032, \u03c3\u2032), we say that a graph homomorphism \u03c6 : G \u2192 G\u2032 is a map homomorphism if\u03c3\u2032 \u25e6 \u03c6 = \u03c6 \u25e6 \u03c3, and say that \u03c6 is a covering if for every vertex v and every face f of M , therestriction of \u03c6 to each of {e \u2208 E\u2192 : e\u2212 = v} and {e \u2208 E\u2192 : er = f} is injective. The universalcover of a map M is a simply connected map M\u02dc together with a covering pi : M\u02dc \u2192 M . Everymap has a universal cover, and the universal cover of a map M is unique in the sense that ifpi\u2032 : M\u02dc \u2032 \u2192M is a covering from a simply connected map M\u02dc \u2032 to M , there exists an isomorphism ofmaps \u03c6 : M\u02dc \u2192 M\u02dc \u2032 such that pi\u2032 \u25e6 \u03c6 = pi.The universal cover pi : M\u02dc \u2192 M of M may be constructed by taking every lift of every edgeof M in the Riemann surface R(M) to the universal cover \u03a0 : R\u02dc(M) \u2192 R(M) of R(M) (see e.g.[119, p. 60] for the topological notion of path lifting). In particular, if R(M\u02dc) is the Riemannsurface associated to M\u02dc , then there exists a conformal equivalence \u03a6 : R(M\u02dc) \u2192 R\u02dc(M) such that\u03a0 \u25e6 \u03a6 \u25e6 z\u02dc = z \u25e6 pi. (See e.g. [215, Section 9.2] for a direct construction.) As for Riemann surfaces,the universal cover of a map is easily seen to be unique in the sense that if pi\u2032 : M\u02dc \u2032 \u2192 M is also auniversal cover of M then there exists an isomorphism of maps f : M\u02dc \u2032 \u2192 M\u02dc such that pi\u2032 = pi \u25e6 f .The following is proven in [21, Section 4.1].Lemma 9.7.4. Let (M,\u03c1) be a unimodular random rooted map. Let (M\u02dc, pi) be the universal coverof M and let \u03c1\u02dc be an arbitrary element of pi\u22121(\u03c1). Then (M\u02dc, \u03c1\u02dc) is a unimodular random rootedmap.2599.7. Multiply-connected and non-planar mapsObserve that \u03baM\u02dc (\u03c1\u02dc) = \u03ba(\u03c1) for any rooted map (M,\u03c1). Thus, we conclude by applying Theo-rem 9.6.1 and the classical theory of Riemann surfaces to obtain that\u2022 K(M,\u03c1) > 0 if and only if M is finite and simply connected and R(M) is conformallyequivalent to the sphere,\u2022 K(M,\u03c1) = 0 if and only if R\u02dc(M) is conformally equivalent to the plane, if and only if R(M)is conformally equivalent to one of the plane C, the cylinder C\/Z or a torus C\/\u039b for somelattice \u039b \u2282 C, and\u2022 K(M,\u03c1) < 0 if and only if R\u02dc(M) is conformally equivalent to the disc.In the last case there are many possibilities for the conformal equivalence class R(M); anythingother than the sphere, the plane, the cylinder or a torus will do. We conclude by applying theadditional topological constraints on R(M) imposed by Theorem 9.7.1.We next connect the topology of S(M) to the number of ends of the underlying graph. Sincethe number of ends of the underlying graph of a map M is at least the number of ends of S(M), itfollows that the underlying graph of M has infinitely many ends if S(M) is homeomorphic to theCantor tree or the blossoming Cantor tree, and at least two ends if S(M) is homeomorphic to thecylinder or Jacob\u2019s ladder.Lemma 9.7.5. Let (M,\u03c1) be a unimodular random rooted map with E[deg(\u03c1)] < \u221e. If S(M) ishomeomorphic to the cylinder or to Jacob\u2019s ladder almost surely, then the underlying graph of Mis two-ended almost surely.Proof. We bias by deg(\u03c1) and prove the equivalent statement for (M,\u03c1) reversible. We may alsoassume that (M,\u03c1) is ergodic. Suppose for contradiction that M has more than two ends. In thiscase, M has infinitely many ends almost surely, is invariantly nonamenable, and hence transientalmost surely. Since S(M) is two-ended almost surely, there exists some r and D such, withpositive probability, the ball Br(M,Xn) of radius r about Xn in M has degree sum at most Dand the complement S(M)\\z(Br(M,Xn)) has two non-precompact connected components, each ofwhich is necessarily one-ended. Denote this event by An. By stationarity, An occurs for infinitelymany n almost surely. Let n0 \u2265 0 be the minimal such n, and denote the components of theunderlying graph of M that are contained in these two components of S \\ z(Br(M,Xn0)) by W1and W2. Since M is transient almost surely, the simple random walk \u3008Xn\u3009n\u22650 eventually stays inone of the Wi, and so the subgraph of the underlying graph of M induced by this set Wi must betransient. However, taking a subsequence \u3008Xnm\u3009m\u22650 of the random walk such that Anm occurs forevery m \u2265 0 and the balls Br(M,Xnm) are all disjoint from each other yields an infinite collectionof disjoint cutsets of degree sum at most D separating \u03c1 from infinity in the subgraph induced byWi. Thus, this graph is recurrent by the Nash-Williams criterion [173], a contradiction.2609.7. Multiply-connected and non-planar maps9.7.2 Theorem 9.1.1 in the multiply-connected planar caseSuppose that (M,\u03c1) is an infinite, multiply-connected unimodular random planar map with E[deg(\u03c1)] <\u221e. If K(M,\u03c1) = 0, then the proof of Theorem 9.7.3 implies that R(M) is conformally equivalent tothe cylinder. Lemma 9.7.5 then implies that the underlying graph of G is two-ended almost surely.It follows that M is recurrent, and we immediately deduce from this that items (2), (10), (11), (14),(17), (5), (6), and (4) of Theorem 9.1.1 hold for (M,\u03c1). The remaining items of Theorem 9.1.1hold for (M,\u03c1) as a consequence of invariant amenability.Now suppose that K(M,\u03c1) < 0. In this case, Theorem 9.7.3 implies that S(M) is almost surelyhomeomorphic to the Cantor tree and consequently that the underlying graph of M is infinitely-ended almost surely by Lemma 9.3.9. The following are well-known to experts.Proposition 9.7.6. Let (G, \u03c1) be a unimodular random rooted graph with E[deg(\u03c1)] < \u221e, andsuppose that G is infinitely ended almost surely. Then G admits non-constant harmonic functionsof finite Dirichlet energy almost surely.Proposition 9.7.7. Let (G, \u03c1) be a unimodular random rooted graph with E[deg(\u03c1)] < \u221e, andsuppose that G is infinitely ended almost surely. Then pu(G) = 1 almost surely.Lemma 9.7.8. Let (G, \u03c1) be a unimodular random rooted graph, and suppose that G is infinitelyended almost surely. Let \u3008Xn\u3009n\u22650 be a random walk on G. Then for every finite set K \u2282 V andevery infinite connected component W of G \\K, there is a positive probability that Xn \u2208W for allsufficiently large n.Proof. Suppose for contradiction that there exists a finite set K \u2282 V and an infinite connectedcomponent W of G\\K such that Xn does not visit W infinitely often almost surely. It follows thatfor every vertex v in W , a random walk started at v must hit the set K almost surely, and hencethat, for each vertex u of K, we haveinfv\u2208WPv(hit u) \u2265 infv\u2208WPv(hit K) \u00b7 infw\u2208KPw(hit u) > 0.On the other hand, we must have that for every \u03b5 > 0, there exist at most finitely many verticesv \u2208 W such that Pu(hit v) \u2265 \u03b5, since otherwise the random walk started at u would hit infinitelymany vertices of W with probability at least \u03b5 by Fatou\u2019s Lemma. The identityPv(hit u) = Pu(hit v)deg(u)\u2211n\u22650 pn(u, u)deg(v)\u2211n\u22650 pn(v, v)therefore implies that for every C < \u221e, there exist at most finitely many vertices v of W withdeg(v)\u2211n\u22650 pn(u, u) \u2264 C.Choose C sufficiently large that deg(\u03c1)\u2211n\u22650 pn(\u03c1, \u03c1) \u2264 C with positive probability, and definea mass transport by, for each vertex v of G, transporting a mass of 1 to the closest vertex to v thathas deg(w)\u2211n\u22650 pn(w,w) \u2264 C. If there are multiple choices of the vertex w, choose one uniformly.2619.7. Multiply-connected and non-planar mapsThen every vertex sends a mass of at most one but, in the situation described, there must be somevertices that recieve an infinite amount of mass. This contradicts the Mass-Transport Principle.Proof of Proposition 9.7.6. It is well-known that if G is a graph and there exists a finite set Ksuch that G \\K has more than one transient connected component, then G admits a non-constantharmonic Dirichlet function: The probability that the walk eventually stays in a particular one ofthe connected components is such a function. See [173, Exercise 9.23].An end \u03be of a graph G is said to be thin if there exists a constant C and a sequence \u3008(Ki,Wi)\u3009i\u22651of sets Ki \u2282 V and connected components Wi of G \\ Ki such that |Ki| \u2264 M for all i \u2265 1,diam(Ki) \u2264M for all i \u2265 1, and\u03be =\u22c2i\u22651{\u03be\u2032 an end of G : \u03beKi = Wi}.Proposition 9.7.9. Let (G, \u03c1) be a unimodular random rooted graph with E[deg(\u03c1)] < \u221e, andsuppose that G is infinitely ended almost surely. Then the random walk on G converges almostsurely in the ends compactification to a thin end of G, and the law of the limiting end has noatoms.Proof. We bias by deg(\u03c1) to work in the reversible setting, and let \u3008Xn\u3009n\u2208Z be a bi-infinite randomwalk started at \u03c1. Convergence in the space of ends follows immediately from transience. To seethat the limiting end is almost surely thin, observe that, by Lemma 9.7.8, there exists R <\u221e suchthat, with positive probability, the random walks \u3008Xn\u3009n\u2265m and \u3008Xn\u3009n\u2264m are eventually in differentconnected components of the complement G \\ BR(G,Xm) of the ball BR(G,Xm). By the ergodictheorem, this event must occur for infinitely many m \u2208 Z. The claim now follows immediately.Proof of Proposition 9.7.7. We bias by deg(\u03c1) to work in the reversible setting, and let \u3008Xn\u3009n\u22650 bea random walk started at \u03c1. Let p \u2208 [0, 1]. If \u03c9p has a unique infinite cluster almost surely, thenHarris\u2019s inequality implies thatP(\u03c1 is connected to Xn in \u03c9p | (G, \u03c1,Xn))\u2265 P(\u03c1 is in the unique infinite cluster of \u03c9p | (G, \u03c1))\u00b7 P(Xn is in the unique infinite cluster of \u03c9p | (G,Xn)),which does not converge to zero by stationarity.However, Proposition 9.7.9 implies that there almost surely exists an increasing sequence inwith in \u2192 \u221e as n \u2192 \u221e, a constant C < \u221e, and sequence of disjoint sets Ki with |Ki| < C suchthat any path connecting \u03c1 to Xm must pass through each of the sets Ki with i \u2264 in. If p < 1,then the probability that every edge incident to Ki is closed is at least (1\u2212 p)C , and it follows thatP(\u03c1 is connected to Xn in \u03c9p | (G, \u03c1,Xn)) \u2264 (1\u2212 (1\u2212 p)C)in ,2629.8. Open problemswhich converges to zero almost surely if p < 1. The claim now follows immediately.The negations of the remaining items of Theorem 9.1.1 follow from Proposition 9.7.6 and Propo-sition 9.7.7 using implications, valid for all unimodular random rooted graphs, that we have alreadyreviewed earlier in the paper; see the green and blue arrows in Figure 1.9.8 Open problemsWe expect that the dichotomy of Theorem 9.1.1 extends to many further properties of planarunimodular random rooted maps. In this section, we discuss several such properties that might beaddressed.9.8.1 Rates of escape of the random walkCan the type of a unimodular random planar map be determined by the rate of escape of the randomwalk? The work of Ding, Lee, and Peres [79] (together with the characterization of parabolicunimodular random planar maps as Benjamini-Schramm limits of finite planar maps) implies thatthe random walk is at most diffusive on any parabolic unimodular random planar map of finiteexpected degree.Theorem 9.8.1 ([79]). There exists a universal constant C such that for every parabolic unimodularrandom rooted map (M,\u03c1) with E[deg(\u03c1)] <\u221e, we haveE[deg(\u03c1)d(\u03c1,Xn)2]\u2264 CnE [deg(\u03c1)]for all n \u2265 0.On the other hand, if (M,\u03c1) is a hyperbolic unimodular random rooted map with E[deg(\u03c1)] <\u221ethat has at most exponential growth, meaning thatlim supn\u2192\u221e1nlog |B(\u03c1, n)| <\u221e,then the random walk on M has positive speed, that is,limn\u2192\u221e1nd(\u03c1,Xn) > 0a.s., where the limit exists a.s. by Kingman\u2019s subadditive ergodic theorem. (Note that the exponen-tial growth condition always holds for graphs of bounded degree.) This can be seen in several ways:it is an easy consequence of a theorem of Benjamini, Lyons, and Schramm [45, Theorem 3.2] (seealso [7, Theorem 8.13] and [21, Theorem 3.2]) that every invariantly nonamenable unimodular ran-dom rooted graph with finite expected degree and at most exponential growth has positive speed.2639.8. Open problemsMeanwhile, it is a result of Benjamini and Curien [38], generalizing the work of Kaimanovich, Ver-shik, and others [141, 144\u2013148], that every non-Liouville unimodular random rooted graph withfinite expected degree and at most exponential growth has positive speed.In general, however, there do exist invariantly nonamenable, non-Liouville, unimodular randomrooted graphs with finite expected degree such that the random walk has zero speed almost surely.An example of such a graph appears in a forthcoming paper by the second author. We do not knowof a planar example, which motivates the following question.Question 9.8.2. Let (M,\u03c1) be a hyperbolic unimodular random rooted planar map with E[deg(\u03c1)] <\u221e. Does the random walk on M have positive speed almost surely?See [21] for a related result concerning the positivity of the speed in the hyperbolic met-ric induced by the circle packing of a hyperbolic unimodular random rooted triangulation withE[deg(\u03c1)2] <\u221e.9.8.2 Positive harmonic functionsTheorem 9.1.1 states that the existence of non-constant bounded harmonic functions and of non-constant harmonic Dirichlet functions are both determined by the type. We conjecture that asimilar result holds for positive harmonic functions.Conjecture 9.8.3. Let (M,\u03c1) be a parabolic unimodular random rooted map and suppose thatE[deg(\u03c1)] < \u221e. Then M does not admit any non-constant positive harmonic functions almostsurely.This conjecture would follow from a positive answer to the following question.Question 9.8.4. Let (M,\u03c1) be a parabolic unimodular random rooted map and suppose thatE[deg(\u03c1)] < \u221e. Let \u3008Xn\u3009n\u22650 be a simple random walk on M . Is every component of the com-plement of the trace of \u3008Xn\u3009n\u22650 finite almost surely?Note that the answer to Question 9.8.4 is trivially positive if M is recurrent.264Chapter 10Uniform spanning forests of planargraphsSummary. We prove that the free uniform spanning forest of any bounded degree, proper planegraph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm[44]. We provide a quantitative form of this result, calculating the critical exponents governing thegeometry of the uniform spanning forests of transient proper plane graphs with bounded degreesand codegrees. We find that these exponents are universal in this class of graphs, provided thatmeasurements are made using the hyperbolic geometry of their circle packings rather than theirusual combinatorial geometry.Lastly, we extend the connectivity result to bounded degree planar graphs that admit locallyfinite drawings in countably connected domains, but show that the result cannot be extended togeneral bounded degree planar graphs.10.1 IntroductionThe uniform spanning forests (USFs) of an infinite, locally finite, connected graph G are definedas weak limits of uniform spanning trees of finite subgraphs of G. These limits can be taken witheither free or wired boundary conditions, yielding the free uniform spanning forest (FUSF)and the wired uniform spanning forest (WUSF) respectively. Although the USFs are definedas limits of random spanning trees, they need not be connected. Indeed, a principal result ofPemantle [190] is that the WUSF and FUSF of Zd coincide, and that they are almost surely (a.s.)a single tree if and only if d \u2264 4. Benjamini, Lyons, Peres and Schramm [44] (henceforth referredto as BLPS) later gave a complete characterisation of connectivity of the WUSF, proving that theWUSF of a graph is connected a.s. if and only if two independent random walks on the graphintersect a.s. [44, Theorem 9.2].The FUSF is much less understood. No characterisation of its connectivity is known, nor hasit been proven that connectivity is a zero-one event. One class of graphs in which the FUSF isrelatively well understood are the proper plane graphs. Recall that a planar graph is a graphthat can be embedded in the plane, while a plane graph is a planar graph G together with aspecified embedding of G in the plane (or some other topological disc). A plane graph is proper ifthe embedding is proper, meaning that every compact subset of the plane (or whatever topologicaldisc G was embedded in) intersects at most finitely many edges and vertices of the drawing (see26510.1. IntroductionSection 10.2.4 for further details). For example, every tree can be drawn in the plane withoutaccumulation points, while the product of Z with a finite cycle is planar but cannot be drawn inthe plane without accumulation points (and therefore has no proper embedding in the plane).BLPS proved that the free and wired uniform spanning forests are distinct whenever G is atransient proper plane graph with bounded degrees, and asked [44, Question 15.2] whether theFUSF is a.s. connected in this class of graphs. They proved that this is indeed the case when G isa self-dual plane Cayley graph that is rough-isometric to the hyperbolic plane [44, Theorem 12.7].These hypotheses were later weakened by Lyons, Morris and Schramm [170, Theorem 7.5], whoproved that the FUSF of any bounded degree proper plane graph that is rough-isometric to thehyperbolic plane is a.s. connected.Our first result provides a complete answer to [44, Question 15.2], obtaining optimal hypothesesunder which the FUSF of a proper plane graph is a.s. connected. The techniques we developed toanswer this question also allow us to prove quantitative versions of this result, which we describein the next section. We state our result in the natural generality of proper plane networks. Recallthat a network (G, c) is a locally finite, connected graph G = (V,E) together with a functionc : E \u2192 (0,\u221e) assigning a positive conductance to each edge of G. The resistance of an edgee in a network (G, c) is defined to be 1\/c(e). Graphs may be considered as networks by settingc \u2261 1. A plane network is a planar graph G together with specified conductances and a specifieddrawing of G in the plane.Theorem 10.1.1. The free uniform spanning forest is almost surely connected in any boundeddegree proper plane network with edge conductances bounded above.In light of the duality between the free and wired uniform spanning forests of proper plane graphs(see Section 10.2.4), the FUSF of a proper plane graph G with locally finite dual G\u2020 is connecteda.s. if and only if every component of the WUSF of G\u2020 is a.s. one-ended. Thus, Theorem 10.1.1follows easily from the dual statement Theorem 10.1.2 below. (The implication is immediate whenthe dual graph is locally finite.) Recall that an infinite graph G = (V,E) is said to be one-endedif, for every finite set K \u2282 V , the subgraph induced by V \\K has exactly one infinite connectedcomponent. In particular, an infinite tree is one-ended if and only if it does not contain a simplebi-infinite path. Components of the WUSF are known to be one-ended a.s. in several other classesof graphs [7, 44, 127, 128, 170, 190], and are recurrent in any graph [185]. Recall that a plane graphG is said to have bounded codegree if its faces have a bounded number of sides, or, equivalently,if its dual G\u2020 has bounded degrees.Theorem 10.1.2. Every component of the wired uniform spanning forest is one-ended almostsurely in any bounded codegree proper plane network with edge resistances bounded above.The uniform spanning trees of Z2 and other two dimensional Euclidean lattices are very wellunderstood due to the deep theory of conformally invariant scaling limits. The study of the USTon Z2 led Schramm, in his seminal paper [208], to introduce the SLE processes, which he conjec-tured to describe the scaling limits of the loop-erased random walk and UST. This conjecture that26610.1. Introductionwas subsequently proven in the celebrated work of Lawler, Werner and Schramm [163]. Overall,Schramm\u2019s introduction of SLE has revolutionised the understanding of statistical physics in twodimensions; see e.g. [98, 162, 202] for guides to the extensive literature in this very active field.Although our own setting is too general to apply this theory, we nevertheless keep conformalinvariance in mind throughout this paper. Indeed, the key to our proofs is circle packing, a canonicalmethod of drawing planar graphs that is closely related to conformal mapping (see e.g. [122, 123,200, 202, 215] and references therein). For many purposes, one can pretend that the randomwalk on the packing is a quasiconformal image of standard planar Brownian motion: Effectiveresistances, heat kernels, and harmonic measures on the graph can each be estimated in terms ofthe corresponding Brownian quantities [17, 69].10.1.1 Universal USF exponents via circle packingA circle packing P is a set of discs in the Riemann sphere C \u222a {\u221e} that have disjoint interiors(i.e., do not overlap) but can be tangent. The tangency graph G = G(P ) of a circle packingP is the plane graph with the centres of the circles in P as its vertices and with edges given bystraight lines between the centres of tangent circles. The Koebe-Andreev-Thurston Circle PackingTheorem [156, 179, 221] states that every finite, simple planar graph arises as the tangency graphof a circle packing, and that if the graph is a triangulation then its circle packing is unique up toMo\u00a8bius transformations and reflections (see [58] for a combinatorial proof). The Circle PackingTheorem was extended to infinite plane triangulations by He and Schramm [121, 122], who provedthat every infinite, proper, simple plane triangulation admits a locally finite circle packing in eitherthe Euclidean plane or the hyperbolic plane (identified with the interior of the unit disc), but notboth. We call an infinite, simple, proper plane triangulation CP parabolic if it admits a circlepacking in the plane and CP hyperbolic otherwise.He and Schramm [121] also initiated the use of circle packing to study probabilistic questions onplane graphs. In particular, they showed that a bounded degree, simple, proper plane triangulationis CP parabolic if and only if simple random walk on the triangulation is recurrent (i.e., visits everyvertex infinitely often a.s.). Circle packing has since proven instrumental in the study of planargraphs, and random walks on planar graphs in particular. Most relevantly to us, circle packing wasused by Benjamini and Schramm [47] to prove that every transient, bounded degree planar graphadmits non-constant harmonic Dirichlet functions; BLPS [44] later applied this result to deducethat the free and wired uniform spanning forest of a bounded degree plane graph coincide if andonly if the graph is recurrent. We refer the reader to [215] and [202] for background on circlepacking, and to [17, 21, 22, 47, 50, 69, 107, 108, 121, 132, 139] for further probabilistic applications.A guiding principle of the works mentioned above is that circle packing endows a triangulationwith a geometry that, for many purposes, is better than the usual graph metric. The resultsdescribed in this section provide a compelling instance of this principle in action: we find that thecritical exponents governing the geometry of the USFs are universal over all transient, boundeddegree, proper plane triangulations, provided that measurements are made using the hyperbolic26710.1. IntroductionFigure 10.1: A simple, 3-connected, finite plane graph (left) and its double circle packing (right).Primal circles are filled and have solid boundaries, dual circles have dashed boundaries.geometry of their circle packings rather than the usual combinatorial geometry of the graphs. It iscrucial here that we use the circle packing to take our measurements: The exponents in the graphdistance are not universal and need not even exist (see Figure 10.2). In Remark 10.6.1 we give anexample to show that no such universal exponents hold in the CP parabolic case at this level ofgenerality.In order to state the quantitative versions of Theorems 10.1.1 and 10.1.2 in their full generality,we first introduce double circle packing. Let G be a plane graph with vertex set V and face setF . A double circle packing of G is a pair of circle packings P = {P (v) : v \u2208 V } and P \u2020 = {P \u2020(f) :f \u2208 F} satisfying the following conditions (see Figure 10.1):1. (G is the tangency graph of P .) For each pair of vertices u and v of G, the discs P (u)and P (v) are tangent if and only if u and v are adjacent in G.2. (G\u2020 is the tangency graph of P \u2020.) For each pair of faces f and g of G, the discs P \u2020(f)and P \u2020(g) are tangent if and only if f and g are adjacent in G\u2020.3. (Primal and dual circles are perpendicular.) For each vertex v and face f of G, thediscs P \u2020(f) and P (v) have non-empty intersection if and only if f is incident to v, and in thiscase the boundary circles of P \u2020(f) and P (v) intersect at right angles.It is easily seen that any finite plane graph admitting a double circle packing must be simple (i.e.,not containing any loops or multiple edges) and 3-connected (meaning that the subgraph inducedby V \\ {u, v} is connected for each u, v \u2208 V ). Conversely, Thurston\u2019s interpretation of Andreev\u2019sTheorem [179, 221] implies that every finite, simple, 3-connected plane graph admits a double circlepacking (see also [58]). The corresponding infinite theory was given developed by He [120], whoproved that every infinite, simple, 3-connected, proper plane graph G with locally finite dual admitsa double circle packing in either the Euclidean plane or the hyperbolic plane (but not both) andthat this packing is unique up to Mo\u00a8bius transformations and reflections. As before, we say thatG is CP parabolic or CP hyperbolic as appropriate. As in the He-Schramm Theorem [121], CPhyperbolicity is equivalent to transience for graphs with bounded degrees and codegrees [120].26810.1. IntroductionFigure 10.2: Two bounded degree, simple, proper plane triangulations for which the graph distanceis not comparable to the hyperbolic distance. Similar examples are given in [215, Figure 17.7]. Left:In this example, rings of degree seven vertices (grey) are separated by growing bands of degree sixvertices (white), causing the hyperbolic radii of circles to decay. The bands of degree six verticescan grow surprisingly quickly without the triangulation becoming recurrent [211]. Right: In thisexample, half-spaces of the 8-regular (grey) and 6-regular (white) triangulations have been gluedtogether along their boundaries; the circles corresponding to the 6-regular half-space are containedinside a horodisc and have decaying hyperbolic radii.Let G be CP hyperbolic and let (P, P \u2020) be a double circle packing of G in the hyperbolicplane. We write rH(v) for the hyperbolic radius of the circle P (v). For each subset A \u2282 V (G),we define diamH(A) to be the hyperbolic diameter of the set of hyperbolic centres of the circlesin P corresponding to vertices in A, and define areaH(A) to be the hyperbolic area of the unionof the circles in P corresponding to vertices in A. Since (P, P \u2020) is unique up to isometries of thehyperbolic plane, rH(v), diamH(A), and areaH(A) do not depend on the choice of (P, P\u2020).We say that a network G has bounded local geometry if there exists a constant M such thatdeg(v) \u2264 M for every vertex v of G and M\u22121 \u2264 c(e) \u2264 M for every edge e of G. Given a planenetwork G, we let F be the set of faces of G, and defineM = MG = max{supv\u2208Vdeg(v), supf\u2208Fdeg(f), supe\u2208Ec(e), supe\u2208Ec(e)\u22121}.Given a spanning tree F and two vertices x and y in G, we write \u0393F(x, y) for the unique path in Fconnecting x and y.Theorem 10.1.3 (Free diameter exponent). Let G be a transient, simple, 3-connected, properplane network with MG < \u221e, let F be the free uniform spanning forest of G, and let e = (x, y)be an edge of G. Then there exist positive constants k1 = k1(M, rH(x)), increasing in rH(x), andk2 = k2(M) such thatk1R\u22121 \u2264 P (diamH(\u0393F(x, y)) \u2265 R) \u2264 k2R\u2212126910.1. Introductionfor every R \u2265 1.Given a spanning forest F of G in which every component is a one-ended tree and an edge e ofG, the past of e in F, denoted pastF(e), is defined to be the unique finite connected componentof F \\ {e} if e \u2208 F and to be the empty set otherwise. The following theorem is equivalent toTheorem 10.1.3 by duality (see Sections 10.2.4 and 10.5.6).Theorem 10.1.4 (Wired diameter exponent). Let G be a transient, simple, 3-connected, properplane network with MG < \u221e, let F be the wired uniform spanning forest of G, and let e = (x, y)be an edge of G. Then there exist positive constants k1 = k1(M, rH(x)), increasing in rH(x), andk2 = k2(M) such thatk1R\u22121 \u2264 P (diamH(pastF(e)) \u2265 R) \u2264 k2R\u22121for every R \u2265 1.By similar methods, we are also able to obtain a universal exponent of 1\/2 for the tail of thearea of pastF(e), where F is the WUSF of G.Theorem 10.1.5 (Wired area exponent). Let G be a transient, simple, 3-connected, proper planenetwork with MG < \u221e, let F be the wired uniform spanning forest of G, and let e = (x, y) bean edge of G. Then there exist positive constants k1 = k1(M, rH(x)), increasing in rH(x), andk2 = k2(M) such thatk1R\u22121\/2 \u2264 P (areaH(pastF(e)) \u2265 R) \u2264 k2R\u22121\/2for every R \u2265 1.The exponents 1 and 1\/2 occurring in Theorems 10.1.4 and 10.1.5 should, respectively, be com-pared with the analogous exponents for the survival time and total progeny of a critical branchingprocess whose offspring distribution has finite variance (see e.g. [173, \u00a7 5,12]).For uniformly transient proper plane graphs (that is, proper plane graphs in which the escapeprobabilities of the random walk are bounded uniformly away from zero), the hyperbolic radii ofcircles in (P, P \u2020) are bounded away from zero uniformly (Proposition 10.5.16). This implies thatthe hyperbolic metric and the graph metric are rough-isometric (Corollary 10.5.17). Consequently,Theorems 10.1.3\u201310.1.5 hold with the graph distance and counting measure appropriately (Corol-lary 10.5.18). This yields the following appealing corollary, which applies, for example, to planarCayley graphs of co-compact Fuchsian groups.Corollary 10.1.6 (Free length exponent). Let G be a uniformly transient, simple, 3-connected,proper plane network with MG <\u221e and let F be the free uniform spanning forest of G. Let p > 0be a uniform lower bound on the escape probabilities of G. Then there exist positive constantsk1 = k1(M,p) and k2 = k2(M,p) such that27010.2. Background and definitionsk1R\u22121\/2 \u2264 P (|\u0393F(x, y)| \u2265 R) \u2264 k2R\u22121\/2for every edge e = (x, y) of G and every R \u2265 1.See [31, 52, 150, 170, 180, 210] and the survey [29] for related results on Euclidean lattices.OrganisationSection 10.2 contains definitions and background on those notions that will be used throughoutthe paper. Experienced readers are advised that this section also includes proofs of a few sim-ple folklore-type lemmas and propositions. Section 10.3 contains the proofs of Theorems 10.1.1and 10.1.2, as well as the upper bound of Theorem 10.1.4 in the case that G is a triangulation.Section 10.5 completes the proofs of Theorems 10.1.3\u201310.1.5 and Corollary 10.1.6; the most sub-stantial component of this section is the proof of the lower bound of Theorem 10.1.4. We concludewith some remarks and open problems in Section 10.6.10.2 Background and definitions10.2.1 NotationWe write E\u2192 for the set of oriented edges of a network G = (V,E). An oriented edge e \u2208 E\u2192 isoriented from its tail e\u2212 to its head e+, and has reversal \u2212e. For each r\u2032, r > 0, and z \u2208 C, wedefine the ballBz(r) = {z\u2032 \u2208 C : |z \u2212 z\u2032| < r}and the annulusAz(r, r\u2032) = {z\u2032 \u2208 C : r \u2264 |z\u2032 \u2212 z| \u2264 r\u2032}.We write dC for the Euclidean metric on C and write dH for the hyperbolic metric on D.10.2.2 Uniform Spanning ForestsWe begin with a succinct review of some basic facts about USFs, referring the reader to [44] andChapters 4 and 10 of [173] for a comprehensive overview. For each finite, connected graph G, wedefine USTG to be the uniform measure on the set of spanning trees of G (i.e., connected subgraphsof G containing every vertex and no cycles). More generally, for a finite network G = (G,w), wedefine USTG to be the probability measure on spanning trees of G for which the measure of eachtree is proportional to the product of the conductances of the edges in the tree.An exhaustion of an infinite network G is a sequence \u3008Vn\u3009n\u22650 of finite, connected subsets ofV such that Vn \u2286 Vn+1 for all n \u2265 0 and \u222anVn = V . Given such an exhaustion, let the networkGn be the subgraph of G induced by Vn together with the conductances inherited from G. Thefree uniform spanning forest measure FUSFG is defined to be the weak limit of the sequence27110.2. Background and definitions\u3008USTGn\u3009n\u22651, so thatFUSFG(S \u2282 F) = limn\u2192\u221eUSTGn(S \u2282 T ).for each finite set S \u2282 E, where F is a sample of FUSFG and T is a sample of USTGn . For each n, wealso construct a network G\u2217n from G by gluing (wiring) every vertex of G \\Gn into a single vertex,denoted \u2202n, and deleting all the self-loops that are created. We identify the set of edges of G\u2217n withthe set of edges of G that have at least one endpoint in Vn. The wired uniform spanning forestmeasure WUSFG is defined to be the weak limit of the sequence \u3008USTG\u2217n\u3009n\u22651, so thatWUSFG(S \u2282 F) = limn\u2192\u221eUSTG\u2217n(S \u2282 T )for each finite set S \u2282 E, where F is a sample of WUSFG and T is a sample of USTG\u2217n . Theseweak limits were both implicitly proven to exist by Pemantle [190], although the WUSF was notconsidered explicitly until the work of Ha\u00a8ggstro\u00a8m [109]. Both measures are easily seen to beconcentrated on the set of essential spanning forests of G, that is, cycle-free subgraphs ofG including every vertex such that every connected component is infinite. The measure FUSFGstochastically dominates WUSFG for every infinite network G.The Spatial Markov PropertyLet G = (V,E) be a finite or infinite network and let A and B be subsets of E. We write (G\u2212B)\/Afor the (possibly disconnected) network formed from G by deleting every edge in B and contracting(i.e., identifying the two endpoints of) every edge in A. We identify the edges of (G \u2212 B)\/A withE \\B. Suppose that G is finite, and thatUSTG(A \u2286 T,B \u2229 T = \u2205) > 0.Then, given the event that T contains every edge in A and none of the edges in B, the conditionaldistribution of T is equal to the union of A and the UST of (G \u2212 B)\/A. That is, for every eventA \u2286 {0, 1}E ,USTG(T \u2208 A | A \u2282 T, B \u2229 T = \u2205) = UST(G\u2212B)\/A(T \u222aA \u2208 A ).This is the spatial Markov property of the UST. Taking limits over exhaustions, we obtain acorresponding spatial Markov property for the USFs: If G = (V,E) is an infinite network and Aand B are subsets of E such thatWUSFG(A \u2286 F, B \u2229 F = \u2205) > 0and the network (G\u2212B)\/A is locally finite, thenWUSFG(F \u2208 A | A \u2282 F, B \u2229 F = \u2205) = WUSF(G\u2212B)\/A(F \u222aA \u2208 A ). (10.2.1)27210.2. Background and definitions(If (G\u2212B)\/A is not connected, we define WUSF(G\u2212B)\/A to be the product of the WUSF measureson the connected components of (G \u2212 B)\/A.) A similar spatial Markov property holds for theFUSF.The UST and USFs also enjoy a strong form of the spatial Markov property. Let G be afinite or infinite network, and suppose that H is a random element of {0, 1}E , which we think ofas a random subgraph of G. We say that a random element K of {0, 1}E , defined on the sameprobability space as H, is a local set for H if for every set W \u2286 E, the event {K \u2286W} is, up to anull set, measurable with respect to the \u03c3-algebra generated by the collection of random variables{H(e) : e \u2208W}.For example, suppose that F is a random spanning forest of a network G, and that u and vare vertices of G. Let K = K(F) be the random set of edges that is equal to the path connectingu and v in F if such a path exists, and otherwise equal to all of E. Then K is a local set for F,since, for every proper subset W of E, the event {K \u2286W} is equal (up to the null event in whichF is not a forest) to the event that u and v are in the same component of F and that the uniquepath connecting u and v in F is contained in W , which is clearly measurable with respect to therestriction of F to W .An example of a random set K that is not a local set for F is, for instance, the edges of Ftouching a given vertex v. If v has degree at least 2 and e is some edge touching v, then the event{K \u2282 {e}} cannot be determined just by knowing F(e). Note also that Proposition 10.2.1 does nothold for this K \u2014 because knowing K implies that F(e) = 0 for all edges e that touch v but e 6\u2208 K.Given a random subgraph H of a network G and a local set K for H, we write FK to denotethe \u03c3-algebra generated by K and the random collection of random variables {H(e) : e \u2208 K}. Wealso write Ko = {e \u2208 E : e \u2208 K, H(e) = 1} and Kc = K \\Ko for the sets of edges in K that areincluded (open) in H and not included (closed) in H respectively.Proposition 10.2.1 (Strong Spatial Markov Property for the WUSF). Let G = (V,E) be aninfinite network, let F be a sample of the wired uniform spanning forest of G, and let K be a localset for F that is either finite or equal to all of E almost surely. Conditional on FK and the eventthat K is finite, let F\u02c6 be a sample of the wired uniform spanning forest of (G \u2212 Kc)\/Ko. LetF\u2032 = Ko \u222a F\u02c6 if Ko is finite and F\u2032 = F if Ko = E. Then F and F\u2032 have the same distribution.Proof. Expand the conditional probabilityWUSFG(F \u2208 A | FK)1(K 6= E)=\u2211W1,W2\u2282E finiteWUSFG(F \u2208 A | Ko = W1,Kc = W2)1 (Ko = W1,Kc = W2) .Since K is a local set for F, the right-hand side is equal to\u2211W1,W2\u2282E finiteWUSFG(F \u2208 A |W1 \u2286 F,W2 \u2229 F = \u2205)1 (Ko = W1,Kc = W2) .27310.2. Background and definitionsApplying the spatial Markov property givesWUSFG(F \u2208 A | FK)1(K 6= E)=\u2211W1,W2\u2282E finiteWUSF(G\u2212W2)\/W1(F\u02c6 \u222aW1 \u2208 A)1 (Ko = W1,Kc = W2)= WUSF(G\u2212Kc)\/Ko(F\u02c6 \u222aKo \u2208 A)1(K 6= E),from which the claim follows immediately.Similar strong spatial Markov properties hold for the FUSF and UST, and admit very similarproofs.10.2.3 Random walk, effective resistancesGiven a network G and a vertex u of G, we write PGu for the law of the simple random walk onG started at u, and will often write simply Pu if the choice of network is unambiguous. For eachset of vertices A, we let \u03c4A be the first time the random walk visits A, letting \u03c4A =\u221e if the walknever visits A. Similarly, \u03c4+A is defined to be the first positive time that the random walk visitsA. The conductance c(u) of a vertex u is defined to be the sum of the conductances of the edgesemanating from u.Let A and B be sets of vertices in a finite network G. The effective conductance between Aand B in G is defined to beCeff(A\u2194 B;G) =\u2211v\u2208Ac(v)Pv(\u03c4B < \u03c4+A ),while the effective resistance Reff(A \u2194 B;G) is defined to be the reciprocal of the effective con-ductance, Reff(A \u2194 B;G) = Ceff(A \u2194 B;G)\u22121. Now suppose that G is an infinite network withexhaustion \u3008Vn\u3009n\u22650 and let A and B be finite subsets of V . Let \u3008Gn\u3009n\u22650 and \u3008G\u2217n\u3009n\u22650 be definedas in Section 10.2.2. The free effective resistance between A and B is defined to be the limitRFeff(A\u2194 B; G) = limn\u2192\u221eReff(A\u2194 B; Gn),while the wired effective resistance between A and B is defined to beRWeff (A\u2194 B; G) = limn\u2192\u221eReff(A\u2194 B; G\u2217n).Free and wired effective conductances are defined by taking reciprocals. The free effective resistancebetween two, possibly infinite, sets A and B is defined to be the limit of the free effective resistancesbetween A \u2229 Vn and B \u2229 Vn, which are decreasing in n. We also defineReff(A\u2194 B \u222a {\u221e}) = limn\u2192\u221eReff(A\u2194 B \u222a {\u2202n};G\u2217n).27410.2. Background and definitionsSee e.g. [173] for further background on electrical networks.We will make frequent use of Rayleigh\u2019 monotonicity principle [173, Chapter 2.4], whichstates that the effective conductance between any two sets in a network is an increasing functionof the edge conductances. In particular, it follows that the effective conductance between two setsdecreases when edges are deleted from the network (which corresponds to taking the conductanceof those edges to zero), and increases when edges are contracted (which corresponds to taking theconductance of those edges to infinity).The proof of Theorem 10.1.2 will require the following simple lemma.Lemma 10.2.2. Let A and B be sets of vertices in an infinite network G. ThenRWeff (A\u2194 B; G) \u2264 3 max{Reff(A\u2194 B \u222a {\u221e}; G) , Reff (B \u2194 A \u222a {\u221e}; G)} .Proof. Let M = max{Reff(A\u2194 B \u222a {\u221e}; G), Reff(B \u2194 A \u222a {\u221e}; G)}. Recall that for any threesets A, B and C in G\u2217n (or any other finite network) [173, Exercise 2.33],Reff(A\u2194 B \u222a C;G\u2217n)\u22121 \u2264 Reff(A\u2194 B;G\u2217n)\u22121 +Reff(A\u2194 C;G\u2217n)\u22121.Letting C = {\u2202n} and taking the limit as n\u2192\u221e, we obtain thatM\u22121 \u2264 Reff(A\u2194 B \u222a {\u221e};G)\u22121 \u2264 RWeff (A\u2194 B;G)\u22121 +Reff(A\u2194\u221e;G)\u22121.Rearranging, we have thatReff(A\u2194\u221e;G) \u2264 MRWeff (A\u2194 B;G)RWeff (A\u2194 B;G)\u2212M.By symmetry, the inequality continues to hold when we exchange the roles of A and B. Combiningboth of these inequalities with the triangle inequality for effective resistances [173, Exercise 9.29]yields thatRWeff (A\u2194 B;G) \u2264 Reff(A\u2194\u221e;G) +Reff(B \u2194\u221e;G) \u2264 2MRWeff (A\u2194 B;G)RWeff (A\u2194 B;G)\u2212M,which rearranges to give the claimed inequality.Kirchhoff\u2019s Effective Resistance Formula.The connection between effective resistances and spanning trees was first discovered by Kirchhoff[154] (see also [63]).Theorem 10.2.3 (Kirchhoff\u2019s Effective Resistance Formula). Let G be a finite network. Then forevery e \u2208 E\u2192USTG(e \u2208 T ) = c(e)Reff(e\u2212 \u2194 e+;G).27510.2. Background and definitionsTaking limits over exhaustions, we also have the following extension of Kirchhoff\u2019s formula:WUSFG(e \u2208 F) = c(e)RWeff (e\u2212 \u2194 e+;G). (10.2.2)A similar equality holds for the FUSF.The method of random paths.We say that a path \u0393 in a network G is simple if it does not visit any vertex more than once.Given a probability measure \u03bd on simple paths in a network G, we define the energy of \u03bd to beE(\u03bd) = 12\u2211e\u2208E\u21921c(e)(\u03bd(e \u2208 \u0393)\u2212 \u03bd(\u2212e \u2208 \u0393))2.Effective resistances can be estimated using random paths as follows (see [173, \u00a73] and [? ]). Inparticular, if G is an infinite network and A and B are two finite sets of vertices in G, thenReff(A\u2194 B \u222a {\u221e};G) = min\uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3E(\u03bd) :\u03bd a probability measure on simplepaths in G starting in A that areeither infinite or finite and end in B\uf8fc\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8fe .Obtaining resistance bounds by defining flows using random paths in this manner is referred toas the method of random paths. It will be convenient to use the following weakening of themethod of random paths. Given the law \u00b5 of a random subset W \u2282 V (G), defineE(\u00b5) =\u2211v\u2208V\u00b5(v \u2208W )2.Lemma 10.2.4 (Method of random sets). Let A and B be two finite sets of vertices in an infinitenetwork G, and let \u00b5 be a measure on subsets W \u2282 V (G) such that the subgraph of G induced byV almost surely contains a path starting at A that is either infinite or finite and ends at B. ThenReff(A\u2194 B \u222a {\u221e};G) \u2264 supe\u2208Ec(e)\u22121E(\u00b5). (10.2.3)Proof. Given W , let \u0393 be a simple path connecting A to B that is contained in W . Then, letting\u03bd be the law of \u0393,E(\u03bd) \u2264 supe1c(e)\u2211e\u2208E\u2192\u03bd(e \u2208 \u0393)2.Letting \u0393\u2032 be an independent random path with the same law as \u0393, the sum above is exactly theexpected number of oriented edges that are used by both \u0393 and \u0393\u2032. Since these paths are simple,they each contain at most one oriented edge emanating from v for each vertex v \u2208 V . It followsthat the number of oriented edges included in both paths is at most the number of vertices included27610.2. Background and definitionsin both paths. This yields thatE(\u03bd) \u2264 supe\u2208E1c(e)\u2211v\u2208V\u03bd(v \u2208 \u0393)2 \u2264 supe\u2208E1c(e)\u2211v\u2208V\u00b5(v \u2208W )2 = supe\u2208E1c(e)E(\u00b5).10.2.4 Plane Graphs and their USFsGiven a graph G = (V,E), let G be the metric space defined as follows. For each edge e of G,choose an orientation of e arbitrarily and let {I(e) : e \u2208 E} be a set of disjoint isometric copies ofthe interval [0, 1]. The metric space G is defined as a quotient of the union\u22c3e I(e) \u222a V , where weidentify the endpoints of I(e) with the vertices e\u2212 and e+ respectively, and is equipped with thepath metric.Let S be an orientable surface without boundary, which in this paper will always be a domainD \u2286 C\u222a{\u221e}. A proper embedding of a graph G into S is a continuous, injective map z : G\u2192 Ssatisfying the following conditions:1. (Every face is a topological disc.) Every connected component of the complement S\\z(G),called a face of (G, z), is homeomorphic to the disc. Moreover, for each connected componentU of S \\z(G), the set of oriented edges of G that have their left-hand side incident to U formseither a cycle or a bi-infinite path in G.2. (z is locally finite.) Every compact subset of S intersects at most finitely many edges ofz(G). Equivalently, the preimage z\u22121(K) of every compact set K \u2286 S is compact in G.A locally finite, connected graph is planar if and only if it admits a proper embedding into somedomain D \u2286 C \u222a {\u221e}. A plane graph G = (G, z) is a planar graph G together with a specifiedembedding z : G \u2192 D \u2286 C \u222a {\u221e}. A plane network G = (G, z, c) is a planar graph togetherwith a specified embedding and an assignment of positive conductances c : E \u2192 (0,\u221e).Given a pair G = (G, z) of a graph together with a proper embedding z of G into a domain D,the dual G\u2020 of G is the graph that has the faces of G as vertices, and has an edge drawn betweentwo faces of G for each edge incident to both of the faces in G. By drawing each vertex of G\u2020 inthe interior of the corresponding face of G and each edge of G\u2020 so that it crosses the correspondingedge of G but no others, we obtain an embedding z\u2020 of G\u2020 in D. The edge sets of G and G\u2020 arein natural correspondence, and we write e\u2020 for the edge of G\u2020 corresponding to e. If e is oriented,we let e\u2020 be oriented so that it crosses e from right to left as viewed from the orientation of e. IfG = (G, z, c) is a plane network, we assign the conductances c\u2020(e\u2020) = c(e)\u22121 to the edges of G\u2020.USF Duality.Let G be a plane network with dual G\u2020. For each set W \u2286 E, let W \u2020 := {e\u2020 : e \/\u2208 W}. If G isfinite and t is a spanning tree of G, then t\u2020 is a spanning tree of G\u2020: the subgraph t\u2020 is connected27710.2. Background and definitionsbecause t has no cycles, and has no cycles because t is connected. Moreover, the ratio\u220fe\u2208t c(e)\u220fe\u2020\u2208t\u2020 c\u2020(e\u2020)=\u220fe\u2208Ec(e)does not depend on t. It follows that if T is a random spanning tree of G with law USTG, thenT \u2020 is a random spanning tree of G\u2020 with law USTG\u2020 . This duality was extended to infinite properplane networks by BLPS.Theorem 10.2.5 ([44, Theorem 12.2 and Proposition 12.5]). Suppose that G is an infinite properplane network with locally finite dual G\u2020. Then if F is a sample of FUSFG, the subgraph F\u2020 is anessential spanning forest of G\u2020 with law WUSFG\u2020. In particular, F is connected almost surely if andonly if every component of F\u2020 is one-ended almost surely.10.2.5 Circle packingWe now give some background on circle packing. The carrier of a circle packing P , denotedcarr(P ), is the union of all the discs of P and of all the faces of G(P ), so that the embedding z ofG(P ) defined by drawing straight lines between the centres of tangent circles is a proper embeddingof G(P ) into carr(P ). Similarly, the carrier of a double circle packing (P, P \u2020) is defined to be theunion of all the discs in P \u222aP \u2020. Given a domain D \u2282 C\u222a{\u221e}, a circle packing P (or double circlepacking (P, P \u2020)) is said to be in D if its carrier is D. In particular, a (double) circle packing P issaid to be in the plane if its carrier is the plane C and in the disc if its carrier is the open unitdisc D. The following theorems, which we stated in the introduction, are the cornerstones of thetheory for infinite proper plane graphs.Theorem 10.2.6. [He-Schramm Existence and Uniqueness Theorem [120\u2013122, 206]] Let G be aninfinite, simple, 3-connected, proper plane graph with locally finite dual. Then G admits a doublecircle packing either in the plane or the disc, but not both, and this packing is unique up to Mo\u00a8biustransformations and reflections of the plane or the disc as appropriate.Theorem 10.2.7. [He-Schramm Recurrence Theorem [120, 121]] Let G be an infinite, simple, 3-connected, proper plane graph with bounded degrees and codegrees. Then G is CP parabolic if andonly if it is recurrent for simple random walk.Let us now explain how these theorems follow from the more general statements given in [120].In that paper, He considers disc patterns of simple proper plane triangulations, which are likecircle packings except that circles corresponding to adjacent vertices intersect at a specified angle,between 0 and pi, rather than being tangent. Given a proper plane network G = (V,E) with withlocally finite dual and face set F , we can form a proper plane triangulation T with vertex setV \u222a F by adding a vertex inside each face of G and connecting this vertex to each vertex in theboundary of the face. It is easily verified that T is simple if and only if G is simple and 3-connected.Moreover, a double circle packing of T , either in the plane or the disc, can now be obtained using27810.3. Connectivity of the FUSFTheorem 1.3 of [120] by requiring that the angles between circles corresponding to the originaledges of G are 0 and pi\/2 between circles corresponding to edges of T that are in V \u00d7 F . It isstraightforward to check that conditions (C1) and (C2) of Theorem 1.3 of [120] hold, and that thepacking obtained this way is indeed the double circle packing of G. Thus, the existence statementof Theorem 10.2.6 follows. The uniqueness of this packing, as formulated in Theorem 10.2.6, isthe content of Theorems 1.1 and 1.2 of [120]. Lastly, Theorem 10.2.7 is a direct consequence ofTheorem 1.3 of [120] together with Theorems 2.6 and 8.1 of [121].Besides their results that we have already mentioned, He and Schramm [122] also proved thatevery simple triangulation of a countably-connected domain D can be circle packed in a circledomain, that is, a domain D\u2032 such that every connected component of C\u222a {\u221e} \\D\u2032 is a disc or apoint.Given a double circle packing (P, P \u2020) of a plane graph G in a domain D \u2286 C, we write diamC(A)for the Euclidean diameter of the set {z(v) : v \u2208 A}, and write dC(A,B) for the Euclidean distancebetween the sets {z(v) : v \u2208 A} and {z(v) : v \u2208 B}. We write r(v) and r(f) for the Euclidean radiiof the circles P (v) and P \u2020(f). If (P, P \u2020) has carrier D, we write \u03c3(v) for 1 \u2212 |z(v)|, which is thedistance between z(v) and the boundary of D, and write rH(v) for the hyperbolic radius of P (v).(Recall that Euclidean circles in D are also hyperbolic circles with different centres and radii; seee.g. [13] for further background on hyperbolic geometry.)We will also use the Ring Lemma of Rodin and Sullivan [200]; see [116] and [2] for quantitativeversions. In Section 10.5.1 we formulate and prove a version of the Ring Lemma for double circlepackings, Theorem 10.5.1.Theorem 10.2.8 (The Ring Lemma). There exists a sequence of positive constants \u3008km : m \u2265 3\u3009such that for every circle packing P of every simple triangulation T , and every pair of adjacentvertices u and v of T , the ratio of radii r(v)\/r(u) is at most kdeg(v).An immediate corollary of the Ring Lemma is that, whenever P is a circle packing in D of aCP hyperbolic proper plane triangulation T and v is a vertex of T , the hyperbolic radius of Pv isbounded above by a constant C = C(deg(v)).10.3 Connectivity of the FUSFIn this section we prove Theorem 10.1.2 and show it easily implies Theorem 10.1.1. We will write\u0016, \u0017 and \u0010 for inequalities that hold up to a positive multiplicative constant depending only onsupf\u2208F deg(f) and supe\u2208E c(e)\u22121.10.3.1 PreliminariesWe begin with some preliminary estimates that will be used in the proof. We first reduce thestatement of Theorem 10.1.2 to the case where the graph is simple and has no peninsulas.27910.3. Connectivity of the FUSFLet G be a bounded codegree proper plane network with locally finite dual G\u2020. Recall that apeninsula of G is a finite connected component of G \\ {v} for some vertex v \u2208 V . In order toapply the theory of circle packing, we first reduce to the case in which G is simple and does notcontain a peninsula; this will ensure that the triangulation T = T (G) formed by drawing a starinside each face of G is simple. We will then use the circle packing of T to analyse the WUSF of G.Lemma 10.3.1. Let G be a network such that every vertex v of G is contained in a peninsula ofG, and let F be a sample of WUSFG. Then F is connected and one-ended a.s.Proof. Let v0 be an arbitrary vertex of G, and for each i \u2265 1 let vi be a vertex of G such that vi\u22121is contained in a finite connected component of G \\ {vi}. Let u be another vertex of G and let \u03b3be a path from v0 to u in G. If u is contained in an infinite connected component of G \\ {vi}, then\u03b3 must pass through the vertex vi. Since \u03b3 is finite, we deduce that there exists an integer I suchthat u is contained in a finite connected component of G \\ {vi} for all i \u2265 I. Thus, any infinitesimple path from u in G must visit vi for all i \u2265 I. We deduce that every essential spanning forestof G is connected and one-ended.Thus, to prove Theorem 10.1.2, it suffices to consider the case that there is some vertex of Gthat is not contained in a peninsula. In this case, let G\u2032 be the simple plane network formed fromG by first splitting every edge e of G into a path of length 2 with both edges given the weight c(e),and then deleting every peninsula of the resulting network. The assumption that G has boundedcodegrees and edge resistances bounded above ensures that G\u2032 does also (indeed, the maximumcodegree of G\u2032 is at most twice that of G).Lemma 10.3.2. Let G be a plane network and let G\u2032 be as above. Then every component of theWUSF of G is one-ended a.s. if and only if every component of the WUSF of G\u2032 is one-ended a.s.Proof. Let F be a sample of WUSFG, and let F\u2032 be the essential spanning forest of G\u2032 defined asfollows. For each edge e \u2208 F such that neither endpoint of e is contained in a peninsula of G, letboth of the edges of the path of length two corresponding to e in G\u2032 be included in F\u2032. For eachedge e \/\u2208 F such that neither endpoint of e is contained in a peninsula of G, choose uniformly andindependently exactly one of the two edges of the path of length two corresponding to e in G\u2032 tobe included in F\u2032. Then every component of F\u2032 is one-ended if and only if every component of F isone-ended, and it is easily verified that F\u2032 is distributed according to WUSFG\u2032 .Thus it suffices to consider the case when G is simple and has no peninsulas. This assumptionallows us to circle pack G as follows. Let T be the triangulation obtained by adding a vertex insideeach face of G and drawing an edge between this vertex and each vertex of the face it correspondsto. The assumption that G is simple and does not contain a peninsula ensures that T is a simpletriangulation. We identify the vertices of T with V (G) \u222a F (G), where F (G) is the set of faces ofG. Let P = {P (v) : v \u2208 V (G)} \u222a {P (f) : f \u2208 F (G)} be a circle packing of T in either the plane28010.3. Connectivity of the FUSFor the unit disc. (Note that this is not the double circle packing of G, which is less useful for us atthis stage since we are not assuming that G has bounded degrees.)We proceed with two geometric lemmas. For each r\u2032 > r > 0 and z in the carrier of P letVz(r, r\u2032) be the set of vertices v of G such that either the intersection of the circle P (v) with theannulus Az(r, r\u2032) is non-empty or the circle P (v) is contained in the ball Bz(r) and there is a facef incident to v such that the intersection P (f) \u2229Az(r, r\u2032) is non-empty.Lemma 10.3.3 (Existence of a uniformly large number of disjoint annuli). Suppose that T is CPhyperbolic so that the carrier of P is D. Then the following hold:1. There exists a decreasing sequence \u3008rn\u3009n\u22650 with rn \u2208 (0, 1\/4) such that for every z \u2208 D with|z| \u2265 1\u2212 rn, the setsVz(ri, 2ri) 1 \u2264 i \u2264 nare disjoint.2. If G has bounded degrees, then there exists a constant C = C(MG) such that we may takern = C\u2212n in (1).Proof. We first prove item (1). We construct the sequence recusively, letting r0 = 1\/8. Supposethat \u3008ri\u3009ni=0 satisfying the conclusion of the lemma have already been chosen, and consider the setof verticesKn ={v \u2208 V (G) : r(v) \u2265 rn4or r(f) \u2265 rn4for some face f incident to v}.Since the carrier of P has finite area Kn is a finite set. We define rn+1 to bern+1 = sup{r \u2264 rn\/4 : P (v) \u2286 B0(1\u2212 3r) for all v \u2208 Kn}which is positive since Kn is finite. We claim that \u3008ri\u3009n+1i=0 continues to satisfy the conclusion ofthe lemma. That is, we claim that Vz(rn+1, 2rn+1) \u2229 Vz(ri, 2ri) = \u2205 for every 1 \u2264 i \u2264 n and everyz \u2208 D with |z| \u2265 1\u2212 rn+1.Indeed, let z be such that |z| \u2265 1 \u2212 rn+1 and let v \u2208 Vz(rn+1, 2rn+1). By definition ofVz(rn+1, 2rn+1), the intersection P (v) \u2229 Bz(2rn+1) is non-empty, and we deduce that P (v) is notcontained in the closure of B0(1\u2212 3rn+1). By definition of rn+1, it follows that v \/\u2208 Kn and hencethat r(v) \u2264 rn\/4 and r(f) \u2264 rn\/4 for every face f incident to v. Thus, since rn+1 \u2264 rn\/4,P (v) \u2286 Bz(2rn+1 + 2rn\/4) \u2286 Bz (rn) and P (f) \u2286 Bz (2rn+1 + 2rn\/4) \u2286 Bz (rn)for every face f incident to v. It follows that Vz(rn+1, 2rn+1) \u2229 Vz(ri, 2ri) = \u2205 for every 1 \u2264 i \u2264 nas claimed.To prove (2), observe that, by the Ring Lemma (Theorem 10.2.8), there exists a constantk = k(M) such that for every C > 1, every z with |z| \u2265 1 \u2212 C\u2212m, and every 0 \u2264 n \u2264 m,28110.3. Connectivity of the FUSFevery circle in P that either has centre in Az(C\u2212n, 2C\u2212n) or is tangent to some circle with centrein Az(C\u2212n, 2C\u2212n) has radius at most kC\u2212n. Thus, this set of circles is contained in the ballBz((2 + 4k)C\u2212n). It follows that taking C = 1\/(4 + 8k) suffices.Lastly, we estimate the energy of a random set of vertices that we will frequently use.Lemma 10.3.4. Let z be a point in the carrier of P (which may be either C or D) and U be auniform random variable on the interval [1, 2] and, for each r > 0, let \u00b5r be the law of the randomset of vertices Wz(Ur) := Vz(Ur, Ur). ThenE(\u00b5r) \u0016 1uniformly in r > 0.Proof. For a vertex v of G to be included in Wz(Ur), the circle {z\u2032 \u2208 C : |z\u2032\u2212 z| = Ur} must eitherintersect the circle P (v) or intersect P (f) for some face f incident to v. We note that the unionof P (v) and all the P (f) incident to P (v) is contained in the ball of radius r(v) + 2 maxf\u223cv r(f)around z(v). Since the codegrees of G are bounded, the Ring Lemma implies that r(f) \u0016 r(v) forall incident v \u2208 V and f \u2208 F , and so\u00b5r(v \u2208Wz(Ur)) \u2264 1rmin(2r(v) + 4 maxf3vr(f), r)\u0016 1rmin{r(v), r}. (10.3.1)We claim that \u2211v\u2208Vz(r,2r)min{r(v), r}2 \u2264 16r2. (10.3.2)To see this, replace each circle of a vertex in Vz(r, 2r) that has radius larger than r with acircle of radius r that is contained in the original circle and intersects Bz(2r): The circles inthis new set still have disjoint interiors, are contained in the ball Bz(4r), and have total areapi\u2211v\u2208Vz(r,2r) min(r(v), r)2, yielding (10.3.2). The claim follows from (10.3.1) and (10.3.2) by defi-nition of the energy E(\u00b5r).We are now ready to prove Theorem 10.1.2.10.3.2 Proof of Theorem 10.1.2Let G be a simple proper plane network with bounded codegrees and bounded edge resistances. ByLemma 10.3.2 we can assume that G does not contain a peninsula. Let F be a sample of WUSFGand given an edge e = (x, y) let A e be the event that x and y are in distinct infinite connectedcomponents of F \\ {e}. It is clear that every component of F is one-ended a.s. if and only ifWUSFG(e \u2208 F ,A e) = 0 (10.3.3)for every edge e of G.28210.3. Connectivity of the FUSFConsider the triangulation T obtained from G and its circle packing P = {P (v) : v \u2208 V (G)} \u222a{P (f) : f \u2208 F (G)}, as described in the previous section. By applying a Mo\u00a8bius transformation,we normalise P by setting the centres z(x) and z(y) to be on the negative and positive real axesrespectively, setting the circles P (x) and P (y) to have the origin as their tangency point and, inthe parabolic case, fixing the scale by setting z(y)\u2212 z(x) = 1.Let \u03b5 > 0 be arbitrarily small. If T is CP hyperbolic, let V\u03b5 be the set of vertices of G with|z(v)| \u2264 1\u2212 \u03b5. Otherwise, T is CP parabolic and we define V\u03b5 to be the set of of vertices of G with|z(v)| \u2264 \u03b5\u22121. We also denote by E\u03b5 the set of edges that both endpoints are in V\u03b5.Let Be\u03b5 be the event that every component of F \\ {e} intersects V \\ V\u03b5. On the event Be\u03b5 , wedefine \u03b7x to be the rightmost path in F \\ {e} from x to V \\ V\u03b5 when looking at x from y, and \u03b7yto be the leftmost path in F \\ {e} from y to V \\ V\u03b5 when looking at y from x. Note that the paths\u03b7x and \u03b7y are not necessarily disjoint. Nonetheless, concatenating the reversal of \u03b7x with e and \u03b7yseparates V\u03b5 into two sets of vertices, L and R, which are to the left and right of e (when viewedfrom x to y) respectively. See Figure 10.3 for an illustration of the case when \u03b7x and \u03b7y are disjoint(when they are not, L is a \u201cbubble\u201d separated from V \\ V\u03b5).Let K be the set of edges that touch a vertex in L or edges that belong to \u03b7x \u222a \u03b7y. Note thatedges of K do not touch the vertices R. The condition that \u03b7x and \u03b7y are the rightmost andleftmost paths to V \\ V\u03b5 from x and y is equivalent to the condition that K does not contain anyopen path from x to V \\ V\u03b5 other than \u03b7x, and does not contain any open path from y to V \\ V\u03b5other than \u03b7y. It follows that, if we define K = E on the complement of the event Be\u03b5 , then K isa local set for F.10.Let A e\u03b5 denote the event that Be\u03b5 occurs and that K does not contain an open path from x toy, or equivalently, that \u03b7x and \u03b7y are disjoint. Note that A e\u03b5 is measurable with respect to FK (asdefined in Section 10.2.2), and that A e = \u2229\u03b5>0A e\u03b5 . Thus,WUSFG(e \u2208 F ,A e) \u2264WUSFG(e \u2208 F | A e\u03b5 ) = E[WUSFG(e \u2208 F | FK , A e\u03b5 )].Let C denote the set of closed edges of F in K and let O denote the set of open edges of F in K. Bythe strong spatial Markov property (Proposition 10.2.1), conditioned on FK and the event A e\u03b5 , thelaw of F is equal to the union of O with a sample of the WUSF of the network (G\u2212C)\/O obtainedfrom G by deleting all the revealed closed edges and contracting all the revealed open edges. Inparticular, by Kirchhoff\u2019s Effective Resistance Formula (Theorem 10.2.3),WUSFG(e \u2208 F | FK , A e\u03b5 ) \u2264 c(e)RWeff (\u03b7x \u2194 \u03b7y; G\u2212 C) (10.3.4)Since the edge e was arbitrary to prove (10.3.3) (and hence Theorem 10.1.2) it suffices to provethat there is a upper bound on the effective resistance appearing in (10.3.4) that tends to zero as\u03b5\u2192 0 uniformly in FK . We perform this analysis now according to whether T is CP hyperbolic or10Indeed, K can be explored algorithmically, without querying the status of any edge in E \\K, by performing aright-directed depth-first search of x\u2019s component in F and a left-directed depth-first search of y\u2019s component in F,stopping each search when it first leaves V\u03b5.28310.3. Connectivity of the FUSFx yLRx yFigure 10.3: Illustration of the proof of Theorem 10.1.2 in the case that T is CP hyperbolic. Left: On the eventBe\u03b5 , the paths \u03b7x and \u03b7y split V\u03b5 into two pieces, L and R. Right: We define a random set containing a path (solidblue) from \u03b7x to \u03b7y \u222a {\u221e} in G \\ C using a random circle (dashed blue). Here we see two examples, one in which thepath ends at \u03b7y, and the other in which the path ends at the boundary (i.e., at infinity).parabolic.Proof of Theorem 10.1.2, hyperbolic case. Suppose that T is CP hyperbolic and let vx be the end-point of the path \u03b7x and put z0 = z(vx). Let \u3008rn\u3009n\u22650 be as in Lemma 10.3.3 and let n(\u03b5) be themaximum n such that \u03b5 < rn. On the event A e\u03b5 , for each 1\u2212 |z0| \u2264 r \u2264 1\/4, we claim that the setWz0(r), as defined in Lemma 10.3.4, contains a path in G from \u03b7x to \u03b7y \u222a{\u221e} that is contained inR\u222a \u03b7x \u222a \u03b7y, and is therefore a path in G \\ C.Indeed, consider the arc A\u2032(r) = {z \u2208 D : |z\u2212z0| = r}, parameterised in the clockwise direction.Let A(r) be the subarc of A\u2032(r) beginning at the last time that A\u2032(r) intersects a circle correspondingto a vertex in the trace of \u03b7x, and ending at the first time after this time that A\u2032(r) intersects either\u2202D or a circle corresponding to a vertex in the trace of \u03b7y (see Figure 10.3). Thus, on the event A e\u03b5 ,the set of vertices of T whose corresponding circles in P are intersected by A(r) contains a path inT from \u03b7x to \u03b7y \u222a {\u221e}, for every 1\u2212 |z0| \u2264 r \u2264 1\/4. (Indeed, for Lebesgue a.e. 1\u2212 |z0| \u2264 r \u2264 1\/4,the arc A(r) is not tangent to any circle in P , and in this case the set is precisely the trace of asimple path in T .) To obtain a path in G rather than T , we divert the path clockwise around eachface of G. That is, whenever the path passes from a vertex u of G to a face f of G and then to avertex v of G, we replace this section of the path with the list of vertices of G incident to f thatare between u and v in the counterclockwise order. This construction shows that the subgraph ofG \\ C induced by the set Wz0(r) contains a path from \u03b7x to \u03b7y \u222a {\u221e}, as claimed.By Lemma 10.3.3 the measures \u00b5ri are supported on sets contained in the disjoint sets Vz(ri, 2ri).28410.3. Connectivity of the FUSFFurthermore, by Lemma 10.2.4 and Lemma 10.3.4 we haveRWeff (\u03b7x \u2192 \u03b7y \u222a {\u221e}; G \\ C) \u0016 E\uf8eb\uf8ed 1n(\u03b5)n(\u03b5)\u2211i=1\u00b5ri\uf8f6\uf8f8 = 1n(\u03b5)2n(\u03b5)\u2211i=1E(\u00b5ri) \u00161n(\u03b5)and hence, by symmetry,RWeff (\u03b7y \u2192 \u03b7x \u222a {\u221e}; G \\ C) \u0016 1n(\u03b5).Applying Lemma 10.2.2 and (10.3.4), we haveWUSF(e \u2208 F | FK , Be\u03b5) \u0016c(e)n(\u03b5)(10.3.5)which by Lemma 10.3.3 converges to zero as \u03b5\u2192 0, completing the proof of Theorem 10.1.2 in thecase that T is CP hyperbolic. If G has bounded degrees, then combining (10.3.5) with Lemma 10.3.3implies that there exists a positive constant C = C(M) such thatWUSF(e \u2208 F | FK , Be\u03b5) \u2264 Cc(e)log(1\/\u03b5)(10.3.6)for all \u03b5 \u2264 1\/2.Proof of Theorem 10.1.2, parabolic case. Suppose that T is CP parabolic. Let r1 = 1 and define\u3008rn\u3009n\u22651 recursively byrn = 1 + inf{r \u2265 rn\u22121 : V0(r, 2r) \u2229 V0(rn\u22121, 2rn\u22121) = \u2205}.Since V0(rn\u22121, 2rn\u22121) is finite, this infimum is finite. By definition, the sets V0(ri, 2ri) are disjoint.A similar analysis to the hyperbolic case shows that, on the event A e\u03b5 , for each r \u2265 1, the set W0(r)contains a path in G from \u03b7x to \u03b7y that is contained in R \u222a \u03b7x \u222a \u03b7y, and is therefore a path inG \\ C. For each \u03b5 > 0, let n(\u03b5) be the maximal n such that rn \u2264 \u03b5\u22121. Then on the event A e\u03b5 , byLemma 10.2.4 and Lemma 10.3.4,RFeff(\u03b7x \u2194 \u03b7y; G \\ C) \u0016 E\uf8eb\uf8ed 1n(\u03b5)n(\u03b5)\u2211i=1\u00b5ri\uf8f6\uf8f8 \u2264 1n(\u03b5)2n(\u03b5)\u2211i=1E(\u00b5ri) \u00161n(\u03b5).Thus, by (10.3.4),WUSFG(e \u2208 F | FK , Be\u03b5) \u0016c(e)n(\u03b5). (10.3.7)The right hand side converges to zero as \u03b5\u2192 0, completing the proof of Theorem 10.1.2.Remark 10.3.5. Since the random sets used in the CP parabolic case above are always finite, theycan also be used to bound free effective resistances. Therefore, by repeating the proof above withthe FUSF in place of the WUSF, we deduce that every component of the FUSF of G is one-ended28510.4. Critical Exponentsalmost surely if T is CP parabolic. Since the FUSF stochastically dominates the WUSF, and anessential spanning forest with one-ended components does not contain any strict subgraphs thatare also essential spanning forests, we deduce that if T is CP parabolic, then the FUSF and WUSFof G coincide. In particular, using [44, Theorem 7.3], we obtain the following.Theorem 10.3.6. Let T be a CP parabolic proper plane triangulation. Then T does not admitnon-constant harmonic functions of finite Dirichlet energy.10.3.3 The FUSF is connected.Proof of Theorem 10.1.1. First suppose that G\u2020 is locally finite. Since G has bounded degrees andbounded conductances, G\u2020 has bounded codegrees and bounded resistances. Thus, Theorem 10.1.2implies that every component of the WUSF on G\u2020 is one-ended a.s. and consequently, by Theo-rem 10.2.5, that the FUSF of G is connected a.s.Now suppose that G does not have locally finite dual. In this case, we form a plane network(G\u2032, c\u2032) from G by adding edges to triangulate the infinite faces of G while keeping the degreesbounded. We enumerate these additional edges \u3008ei\u3009i\u22651 and define conductancesc\u2032(ei) = 2\u2212i\u22121RFeff(e\u2212i \u2194 e+i ;G)\u22121.Since G\u2032 has bounded degrees, bounded conductances and locally finite dual, its FUSF is connecteda.s. By Kirchhoff\u2019s Effective Resistance Formula (Theorem 10.2.3), Rayleigh monotonicity and theunion bound, the probability that the FUSF of G\u2032 contains any of the additional edges ei is at most\u2211i\u22650c\u2032(ei)RFeff(e\u2212i \u2194 e+i ;G\u2032) \u2264\u2211i\u22650c\u2032(ei)RFeff(e\u2212i \u2194 e+i ;G) \u2264 1\/2.In particular, there is a positive probability that none of the additional edges are contained in theFUSF of G\u2032. The conditional distribution of the FUSF of G\u2032 on this event is FUSFG by the spatialMarkov property, and it follows that the FUSF of G is connected a.s.10.4 Critical Exponents10.4.1 The Ring Lemma for double circle packingsIn this section we extend the Ring Lemma to double circle packings. While unsurprising, we wereunable to find such an extension in the literature.Theorem 10.4.1 (Ring Lemma). There exists a family of positive constants \u3008kn,m : n \u2265 3,m \u2265 3\u3009such that if (P, P \u2020) is a double circle packing in C \u222a {\u221e} of a simple, 3-connected plane graph Gand v is a vertex of G, then for every f \u2208 F incident to v such that P (v) does not contain \u221e, thenr(v)\/r(f) \u2264 kdeg(v),maxg\u22a5v deg(g)28610.4. Critical ExponentsFigure 10.4: Proof of the Double Ring Lemma. If two dual circles are close but do not touch, there must be manyprimal circles contained in the crevasse between them. This forces the two dual circles to each have large degree.The right-hand figure is a magnification of the left-hand figure.where g \u22a5 v means that the face g is incident fo v.As for triangulations, the Ring Lemma immediately implies that whenenever a simple, 3-connected, CP hyperbolic proper plane network G with bounded degrees and codegrees is cir-cle packed in D, the hyperbolic radii of the discs in P \u222a P \u2020 are bounded above by a constantC = C(MG,M\u2020G).Proof of Theorem 10.5.1. Let n be the degree of v and let m be the maximum degree of thefaces incident to v in G. We may assume that r(v) = 1. Note that for each two distinct discsP \u2020(f), P \u2020(f \u2032) \u2208 P \u2020 that are not tangent, there is at most one disc P (u) \u2208 P that intersects bothP \u2020(f) and P \u2020(f \u2032), while if P \u2020(f) and P \u2020(f \u2032) are tangent, there exist exactly two discs in P thatintersect both P \u2020(f) and P \u2020(f \u2032). For each two faces f and f \u2032 incident to v, the complement\u2202P (v) \\(P \u2020(f1) \u222a P \u2020(f2))is either a single arc (if P \u2020(f) and P \u2020(f \u2032) are tangent), or is equal tothe union of two arcs (if P \u2020(f) and P \u2020(f \u2032) are not tangent).We claim that there exists an function \u03c8m(\u00b7, \u00b7) : (0,\u221e)2 \u2192 (0, 2pi], increasing in both coordi-nates, such that if f and f \u2032 are two distinct faces of G incident to v, then each of the (one ortwo) arcs forming the complement \u2202P (v) \\ (P \u2020(f1) \u222a P \u2020(f2)) have length at least \u03c8m(r(f), r(f \u2032)).Indeed, let r(f) and r(f \u2032) be fixed, and suppose that one of the arcs forming the complement\u2202P (v) \\ P \u2020(f) \u222a P \u2020(f \u2032) is extremely small, with length \u03b5. Let the primal circles incident to f beenumerated v1, . . . , vdeg(f), where v1 = v, P (v2) is the primal circle that is tangent to P (v) andintersects P \u2020(f) on the same side as the small arc, P (v3) is the next primal circle that is tangentto P (v2) and intersects P\u2020(f), and so on. Since \u03b5 is small, P (v2) must also be very small, as itdoes not intersect P \u2020(f \u2032). Similarly, if \u03b5 is sufficiently small, P (v3) must also be small, since italso does not intersect P \u2020(f \u2032). See Figure 10.5. Applying this argument recursively, we see that,if \u03b5 is sufficiently small, then the circles P (v2), . . . , P (vdeg(f)) are collectively too small to contain\u2202P \u2020(f) \\ P (v) in their union, a contradiction. We write \u03c8m(r(f), r(f \u2032)) for the minimal \u03b5 that isnot ruled out as impossible by this argument.Let the faces incident to the vertex v be indexed in clockwise order f1, . . . , fn, where n = deg(v)28710.4. Critical Exponentsand f1 has maximal radius among the faces incident to v. For each face f incident to v, the arc\u2202P (v)\u2229P \u2020(f) has length 2 tan\u22121(r(f)). Since \u2202P (v) = \u222ani=1(\u2202P (v)\u2229P \u2020(fi)), we deduce that r(f1)is bounded below by tan(pi\/n). By definition of \u03c8m, we have that r(fk) satisfies\u03c8m(tan(pi\/n), r(fk)) \u2264 \u03c8m (r(f1), r(fk)) \u2264 k\u22121\u2211i=22 tan\u22121(r(fi))(10.4.1)for all 3 \u2264 k \u2264 n. For each such k, (10.5.1) yields an implicit upper bound on r(fk) which convergesto zero as r(f2) converges to zero. This in turn yields a uniform lower bound on r(f2): if r(f2) weresufficiently small, the bound (10.5.1) would imply that\u2211nk=2 2 tan\u22121(r(fk)) would be less than pi,a contradiction (since we have trivially that 2 tan\u22121(r(f1)) \u2264 pi). We obtain uniform lower boundson r(fk) for each 3 \u2264 k \u2264 n by repeating the above argument inductively.10.4.2 Good embeddings of planar graphsIf G is a plane graph and (P, P \u2020) is a double circle packing of G in a domain D \u2286 C, then drawingstraight lines between the centres of the circles in P yields a proper embedding of G in D in whichevery edge is a straight line. We call such an embedding a proper embedding of G with straightlines in D. Following [17], a proper embedding of a graph G with straight lines in a domain D \u2286 Cis said to be \u03b7-good if the following conditions are satisfed:1. (No near-flat, flat, or reflexive angles.) All internal angles of every face in the drawingare at most pi \u2212 \u03b7. In particular, every face is convex.2. (Adjacent edges have comparable lengths.) For every pair of edges e, e\u2032 sharing acommon endpoint, the ratio of the lengths of the straight lines corresponding to e and e\u2032 inthe drawing is at most \u03b7\u22121.The Ring Lemma (Theorem 10.5.1) has the following immediate corollary.Corollary 10.4.2. The straight line embedding given by any double circle packing of a plane graphG with bounded degrees and bounded codegrees in a domain D \u2286 C is \u03b7-good for some positive\u03b7 = \u03b7(MG).We remark that, in contrast, the embedding of G obtained by circle packing the triangulation T (G)formed by drawing a star inside each face of G (and then erasing these added vertices) does notnecessarily yield a good embedding of G.For the remainder of this section and in Sections 10.5.3, 10.5.5 and 10.5.6, G will be a fixedtransient, simple, 3-connected proper plane network with bounded codegrees and bounded localgeometry, and (P, P \u2020) will be a double circle packing of G in D. We will write \u0016, \u0017 and \u0010 todenote inequalities or equalities that hold up to positive multiplicative constants depending onlyupon M. We will also fix an edge e = (x, y) of G and, by applying a Mo\u00a8bius transformation ifnecessary, normalise (P, P \u2020) by setting the centres z(x) and z(y) to be on the negative real axis28810.5. Critical exponentsand positive real axis respectively and setting the circles P (x) and P (y) to have the origin as theirtangency point.In [17] and [69], several estimates are established that allow one to compare the random walkon a good embedding of a planar graph with Brownian motion. The following estimate, provenfor general good embeddings of proper plane graphs in [21, Theorems 1.4 and 1.5], is of centralimportance to the proofs of the lower bounds in Theorem 10.1.4 and Theorem 10.1.5. Recall that\u03c3(v) is defined to be 1\u2212 |z(v)|.Theorem 10.4.3 (Diffusive Time Estimate). There exists a constant C1 = C1(M) \u2265 1 such thatthe following holds. For each vertex v of G, let \u3008Xn\u3009n\u22650 be a random walk on G started at v, letr(v) \u2264 r \u2264 C\u221211 \u03c3(v) and let Tr be the first time n that |z(Xn)\u2212 z(v)| \u2265 r. ThenEvTr\u2211n=0r(Xn)2 \u0010 r2.It will be shown in a forthcoming paper of Murugan [personal communication] that the constantC1 above can in fact be taken to be 1.Theorem 10.4.4 (Cone Estimate). Let \u03b7 = \u03b7(M) > 0 be the constant from Corollary 10.5.2.There exists a positive constant q1 = q1(M) such that the following holds. For each vertex v of G,let \u3008Xn\u3009n\u22650 be a random walk on G started at v, let 0 \u2264 r \u2264 \u03c3(v) and let Tr be the first time n that|z(Xn)\u2212 z(v)| \u2265 r. Then for any interval I \u2282 R\/(2piZ) of length at least pi \u2212 \u03b7 we havePv(arg(z(XTr)\u2212 z(v)) \u2208 I) \u2265 q1.We also apply the following result of Benjamini and Schramm [47, Lemma 5.3]; their proof wasgiven for simple triangulations but extends immediately to our setting.Theorem 10.4.5 (Convergence Estimate [47]). For every vertex v of G, we have thatPv(\u2223\u2223z(Xn)\u2212 z(v)\u2223\u2223 \u2265 t\u03c3(v) for some n \u2265 0) \u0016 1log t.We remark that the logarithmic decay in Theorem 10.5.5 is not sharp. Sharp polynomialestimates of the same quantity have been obtained by Chelkak [69, Corollary 7.9].10.5 Critical exponents10.5.1 The ring lemma for double circle packingsIn this section we extend the Ring Lemma to double circle packings. While unsurprising, we wereunable to find such an extension in the literature.28910.5. Critical exponentsTheorem 10.5.1 (Ring Lemma). There exists a family of positive constants \u3008kn,m : n \u2265 3,m \u2265 3\u3009such that if (P, P \u2020) is a double circle packing of a simple, 3-connected plane graph G and v is avertex of G such that P (v) does not contain \u221e, thenr(v)\/r(f) \u2264 kdeg(v),maxg\u22a5v deg(g)for all f \u2208 F incident to v.As for triangulations, the Ring Lemma immediately implies that whenenever a simple, 3-connected, CP hyperbolic proper plane network G with bounded degrees and codegrees is cir-cle packed in D, the hyperbolic radii of the discs in P \u222a P \u2020 are bounded above by a constantC = C(MG,M\u2020G).Proof of Theorem 10.5.1. Let n be the degree of v and let m be the maximum degree of thefaces incident to v in G. We may assume that r(v) = 1. Note that for each two distinct discsP \u2020(f), P \u2020(f \u2032) \u2208 P \u2020 that are not tangent, there is at most one disc P (u) \u2208 P that intersects bothP \u2020(f) and P \u2020(f \u2032), while if P \u2020(f) and P \u2020(f \u2032) are tangent, there exist exactly two discs in P thatintersect both P \u2020(f) and P \u2020(f \u2032). For each two faces f and f \u2032 incident to v, the complement\u2202P (v) \\(P \u2020(f1) \u222a P \u2020(f2))is either a single arc (if P \u2020(f) and P \u2020(f \u2032) are tangent), or is equal tothe union of two arcs (if P \u2020(f) and P \u2020(f \u2032) are not tangent).We claim that there exists an function \u03c8m(\u00b7, \u00b7) : (0,\u221e)2 \u2192 (0, 2pi], increasing in both coordi-nates, such that if f and f \u2032 are two distinct faces of G incident to v, then each of the (one ortwo) arcs forming the complement \u2202P (v) \\ (P \u2020(f1) \u222a P \u2020(f2)) have length at least \u03c8m(r(f), r(f \u2032)).Indeed, let r(f) and r(f \u2032) be fixed, and suppose that one of the arcs forming the complement\u2202P (v) \\ P \u2020(f) \u222a P \u2020(f \u2032) is extremely small, with length \u03b5. Let the primal circles incident to f beenumerated v1, . . . , vdeg(f), where v1 = v, P (v2) is the primal circle that is tangent to P (v) andintersects P \u2020(f) on the same side as the small arc, P (v3) is the next primal circle that is tangentto P (v2) and intersects P\u2020(f), and so on. Since \u03b5 is small, P (v2) must also be very small, as itdoes not intersect P \u2020(f \u2032). Similarly, if \u03b5 is sufficiently small, P (v3) must also be small, since italso does not intersect P \u2020(f \u2032). See Figure 10.5. Applying this argument recursively, we see that,if \u03b5 is sufficiently small, then the circles P (v2), . . . , P (vdeg(f)) are collectively too small to contain\u2202P \u2020(f) \\ P (v) in their union, a contradiction. We write \u03c8m(r(f), r(f \u2032)) for the minimal \u03b5 that isnot ruled out as impossible by this argument.Let the faces incident to the vertex v be indexed in clockwise order f1, . . . , fn, where n = deg(v)and f1 has maximal radius among the faces incident to v. For each face f incident to v, the arc\u2202P (v)\u2229P \u2020(f) has length 2 tan\u22121(r(f)). Since \u2202P (v) = \u222ani=1(\u2202P (v)\u2229P \u2020(fi)), we deduce that r(f1)is bounded below by tan(pi\/n). By definition of \u03c8m, we have that r(fk) satisfies\u03c8m(tan(pi\/n), r(fk)) \u2264 \u03c8m (r(f1), r(fk)) \u2264 k\u22121\u2211i=22 tan\u22121(r(fi))(10.5.1)29010.5. Critical exponentsFigure 10.5: Proof of the Double Ring Lemma. If two dual circles are close but do not touch, theremust be many primal circles contained in the crevasse between them. This forces the two dualcircles to each have large degree. The right-hand figure is a magnification of the left-hand figure.for all 3 \u2264 k \u2264 n. For each such k, (10.5.1) yields an implicit upper bound on r(fk) which convergesto zero as r(f2) converges to zero. This in turn yields a uniform lower bound on r(f2): if r(f2) weresufficiently small, the bound (10.5.1) would imply that\u2211nk=2 2 tan\u22121(r(fk)) would be less than pi,a contradiction (since we have trivially that 2 tan\u22121(r(f1)) \u2264 pi). We obtain uniform lower boundson r(fk) for each 3 \u2264 k \u2264 n by repeating the above argument inductively.10.5.2 Good embeddings of planar graphsIf G is a plane graph and (P, P \u2020) is a double circle packing of G in a domain D \u2286 C, then drawingstraight lines between the centres of the circles in P yields a proper embedding of G in D in whichevery edge is a straight line. We call such an embedding a proper embedding of G with straightlines in D. Following [17], a proper embedding of a graph G with straight lines in a domain D \u2286 Cis said to be \u03b7-good if the following conditions are satisfed:1. (No near-flat, flat, or reflexive angles.) All internal angles of every face in the drawingare at most pi \u2212 \u03b7. In particular, every face is convex.2. (Adjacent edges have comparable lengths.) For every pair of edges e, e\u2032 sharing acommon endpoint, the ratio of the lengths of the straight lines corresponding to e and e\u2032 inthe drawing is at most \u03b7\u22121.The Ring Lemma (Theorem 10.5.1) has the following immediate corollary.Corollary 10.5.2. The straight line given by any double circle packing of a plane graph G withbounded degrees and bounded codegrees in a domain D \u2286 C is \u03b7-good for some positive \u03b7 = \u03b7(MG).We remark that, in contrast, the embedding of G obtained by circle packing the triangulation T (G)formed by drawing a star inside each face of G (and then erasing these added vertices) does notnecessarily yield a good embedding of G.29110.5. Critical exponentsFor the remainder of this section and in Sections 10.5.3, 10.5.5 and 10.5.6, G will be a fixedtransient, simple, 3-connected proper plane network with bounded codegrees and bounded localgeometry, and (P, P \u2020) will be a double circle packing of G in D. We will write \u0016, \u0017 and \u0010 todenote inequalities or equalities that hold up to positive multiplicative constants depending onlyupon M. We will also fix an edge e = (x, y) of G and, by applying a Mo\u00a8bius transformation ifnecessary, normalise (P, P \u2020) by setting the centres z(x) and z(y) to be on the negative real axisand positive real axis respectively and setting the circles P (x) and P (y) to have the origin as theirtangency point.In [17] and [69], several estimates are established that allow one to compare the random walkon a good embedding of a planar graph with Brownian motion. The following estimate, proven forgeneral good embeddings of proper plane graphs in [21], is of central importance to the proofs of thelower bounds in Theorem 10.1.4 and Theorem 10.1.5. Recall that \u03c3(v) is defined to be 1\u2212 |z(v)|.Theorem 10.5.3 (Diffusive Time Estimate). There exists a constant C1 = C1(M) \u2265 1 such thatthe following holds. For each vertex v of G, let \u3008Xn\u3009n\u22650 be a random walk on G started at v, letr(v) \u2264 r \u2264 C\u221211 \u03c3(v) and let Tr be the first time n that |z(Xn)\u2212 z(v)| \u2265 r. ThenEvTr\u2211n=0r(Xn)2 \u0010 r2.It will be shown in a forthcoming paper of Murugan [personal communication] that the constantC1 above can in fact be taken to be 1.Theorem 10.5.4 (Cone Estimate). Let \u03b7 = \u03b7(M) > 0 be the constant from Corollary 10.5.2.There exists a positive constant q1 = q1(M) such that the following holds. For each vertex v of G,let \u3008Xn\u3009n\u22650 be a random walk on G started at v, let 0 \u2264 r \u2264 \u03c3(v) and let Tr be the first time n that|z(Xn)\u2212 z(v)| \u2265 r. Then for any interval I \u2282 R\/(2piZ) of length at least pi \u2212 \u03b7 we havePv(arg(z(XTr)\u2212 z(v)) \u2208 I) \u2265 q1.We also apply the following result of Benjamini and Schramm [47, Lemma 5.3]; their proof wasgiven for simple triangulations but extends immediately to our setting.Theorem 10.5.5 (Convergence Estimate [47]). For every vertex v of G, we have thatPv(|z(Xn)\u2212 z(v)| \u2265 t\u03c3(v) for some n \u2265 0) \u0016 1log t.We remark that the logarithmic decay in Theorem 10.5.5 is not sharp. Sharp polynomialestimates of the same quantity have been attained by Chelkak [69, Corollary 7.9].29210.5. Critical exponents10.5.3 Preliminary estimatesFor each vertex v of G, let aH(v) denote the hyperbolic area of P (v). Note that, since the hyperbolicradii of the discs in P are bounded from above by the Ring Lemma (Theorem 10.5.1),aH(v) \u0010 rH(v)2 \u0010 \u03c3(v)\u22122r(v)2 (10.5.2)for every vertex v of G. For the same reason, there exists a constant s = s(M) \u2208 (0, 1\/2] such thatfor every \u03b5 > 0, the set {v \u2208 V : z(v) \u2208 A0(1\u2212 \u03b5, 1\u2212 s\u03b5)}disconnects e from \u221e in G. For each \u03b5 > 0, letW\u03b5 :={v \u2208 V : z(v) \u2208 A0(1\u2212 \u03b5, 1\u2212 s3\u03b5)}.In the following section, we will wish to estimate sums of the form\u2211u\u2208AaH(u)Pu(\u03c4B <\u221e) (10.5.3)where A and B are subsets of V . In this section, we prove that, when A is a subset of W\u03b5 for some\u03b5 > 0, we can estimate the sum (10.5.3) in terms of geometric quantities.Lemma 10.5.6 (Cone and half-plane estimate). Let r \u2208 (0, 1], and let \u03b7 = \u03b7(M) > 0 be theconstant from Corollary 10.5.2. There exist positive constants \u03b41 = \u03b41(M, r) and q2 = q2(M, r),both increasing in r, such that the following holds. For each vertex v of G and r \u2208 (0, 1] let H1 bethe coneH1 = H1(v) ={z \u2208 C : | arg z \u2212 arg z(v)| \u2264 pi\/2\u2212 \u03b7\/4},and let H2 = H2(v, r) be the half-plane containing z(v) whose boundary is the unique straight linewith distance r\u03c3(v) to z(v) that is orthogonal to the line connecting z(v) to the origin. Then, letting\u03c4\u03b5\u03c3(v) = \u03c4W\u03b5\u03c3(v) be the first time the walk visits W\u03b5\u03c3(v), we have thatPv(z(Xn) \u2208 H2 for all n and z(Xn) \u2208 H1 for all n \u2265 \u03c4\u03b5\u03c3(v))\u2265 q2. (10.5.4)for all \u03b5 \u2264 \u03b41.Proof. Let H\u20321 and H\u2032\u20321 be the slightly thinner cones {z \u2208 C : | arg(z) \u2212 arg(z(v))| \u2264 pi\/2 \u2212 \u03b7\/3}and {z \u2208 C : | arg(z) \u2212 arg(z(v))| \u2264 pi\/2 \u2212 \u03b7\/2}. By Theorem 10.5.5 there exists some small\u03b4\u2032 = \u03b4\u2032(M) > 0 such that for each vertex u \u2208 H\u2032\u20321 with \u03c3(u) \u2264 \u03b4\u2032\u03c3(v), the random walk started at uhas probability at least 1\/2 never to leave H\u20321.Define numbers \u3008\u03c1i\u3009i\u22650 and stopping times \u3008Ti\u3009i\u22650 by letting T0 = 0 and recursively setting \u03c1ito be the distance between z(XTi) and the boundary of H2 \u2229 D, and Ti to beTi = min{n \u2265 0 : |z(Xn)\u2212 z(XTi\u22121)| \u2265 \u03c1i\u22121}.29310.5. Critical exponentsLet Ai be the event that \u03c3(XTi+1) \u2264 \u03c3(XTi) \u2212 \u03c1i sin(\u03b7\/2), dC(XTi+1 , \u2202H2) \u2265 dC(XTi , \u2202H2) +\u03c1i sin(\u03b7\/2) and z(XTi+1) \u2208 H\u2032\u20321. Applying Theorem 10.5.4 yields thatPv(Ai | z(XTi) \u2208 H\u2032\u20321) \u2265 q1 and hence Pv( n\u22c2i=0Ai) \u2265 qn1 (10.5.5)for every n \u2265 0. Let r\u2032 = min{r, \u03b4\u2032, sin(\u03b7\/2)} and let n0 = d1\/r\u20322e. If XTi has distance at leastr\u2032\u03c3(v) from the boundary of D \u2229 H2, then \u03c1i sin(\u03b7\/2) \u2265 r\u20322\u03c3(v). It follows thatPv(z(Xn) \u2208 H2 for all n, limn\u2192\u221e z(Xn) \u2208 H\u20321) \u2265 12Pv( n0\u22c2i=0Ai)\u2265 12qn01 .Next, applying Theorem 10.5.5, let \u03b41 = \u03b41(M, r) > 0 to be sufficiently small that for eachvertex u with z(u) \/\u2208 H1 and \u03c3(u) \u2264 \u03b41\u03c3(v), the random walk started at u has probability at mostqn01 \/4 ever to visit H\u20321. We conclude by observing that the probability appearing in (10.5.4) is atleastPv(z(Xn) \u2208 H2 for all n, limn\u2192\u221e z(Xn) \u2208 H\u20321) \u2212 Pv(z(X\u03c4W\u03b5\u03c3(v) ) \/\u2208 H1, limn\u2192\u221e z(Xn) \u2208 H1),which is at least qn01 \/4 for all \u03b5 \u2264 \u03b41. We conclude by setting q2 = qn01 \/4.Remark. Although it suffices for our purposes, this argument is rather wasteful. With further effortone can take both q2 and \u03b41 to be polynomials in 1\/r.Let p, r \u2208 (0, 1]. We say that a set A \u2282 V is (p, r)-escapable if for every vertex v \u2208 A, therandom walk started at v has probability at least p of not returning to A after first leaving the setof vertices whose corresponding discs in P have centres contained in the Euclidean ball of radiusr\u03c3(v) about z(v). To avoid trivialities, we also declare the empty set to be (p, r)-escapable for allp and r.Corollary 10.5.7. Let r \u2208 (0, 1]. There exist positive constants \u03b42 = \u03b42(M, r) and p1 = p1(M, r),both increasing in r, such that the set{v \u2208 V : (1\u2212 \u03b42)\u03b5 \u2264 \u03c3(v) \u2264 \u03b5}is (p1, r)-escapable for every \u03b5 > 0.Proof. Let \u03b7 = \u03b7(M) be the constant appearing in Corollary 10.5.2, and let\u03b42 = min{r2sin\u03b72, e\u22122C , 1\/4},where C = C(M) is the implicit constant in Theorem 10.5.5. Let \u03b5 > 0 be arbitrary, let v \u2208 {v \u220829410.5. Critical exponentsV : (1\u2212 \u03b42)\u03b5 \u2264 \u03c3(v) \u2264 \u03b5}, and let T be the stopping timeT = min{n \u2265 0 : |z(Xn)\u2212 z(v)| \u2265 r\u03c3(v)}.Letting q1 = q1(M) be the constant from Theorem 10.5.4, we have thatPv(\u03c3(XT ) \u2264 \u03c3(v)\u2212 r sin(\u03b7\/2)\u03c3(v)) \u2265 q1. (10.5.6)On the event appearing in the left-hand side of (10.5.6), the half-plane H2(XT , \u03b42), defined inLemma 10.5.6, is disjoint from the ball {z \u2208 C : 1\u2212 |z| \u2265 (1\u2212 \u03b42)\u03b5}. Thus, the claim follows from(10.5.6) and Lemma 10.5.6 by taking p = q1(M)q2(M, \u03b42), where q2 is as in Lemma 10.5.6. (Herewe are using Lemma 10.5.6 only to lower bound the probability of the first clause defining the eventin the left-hand side of (10.5.4).)Lemma 10.5.8. Let A be (p, r)-escapable for some p \u2208 (0, 1) and r \u2208 (0, C\u221211 ), where C1 = C1(M)is the constant appearing in Theorem 10.5.3. Then for every vertex u \u2208 V ,Eu[\u2211n\u22650aH(Xn)1(Xn \u2208 A)]\u0016 r2p(1\u2212 r)2 . (10.5.7)Proof. Define the sequences of stopping times \u3008T\u2212i \u3009i\u22650 and \u3008T+i \u3009i\u22650 by letting T\u22120 = \u03c4A and recur-sively lettingT+i = min{n \u2265 T\u2212i : |z(Xn)\u2212 z(XT\u2212i )| \u2265 r\u03c3(XT\u2212i )}and T\u2212i = min{n \u2265 T+i\u22121 : Xn \u2208 A}. ThenEu[\u2211n\u22650aH(Xn)1(Xn \u2208 A)]\u2264\u2211i\u22650Eu[1(T\u2212i <\u221e)T+i \u22121\u2211n=T\u2212iaH(Xn)].Since \u03c3(Xn) \u2265 (1\u2212 r)\u03c3(XT\u2212i ) for all T\u2212i \u2264 n < T+i , the Diffusive Time Estimate (Theorem 10.5.3),the strong Markov property, and (10.5.2) imply thatEu[ T+i \u22121\u2211n=T\u2212iaH(Xn)\u2223\u2223\u2223\u2223\u2223T\u2212i <\u221e, XT\u2212i]\u0016 (1\u2212 r)\u22122\u03c3(XT\u2212i )\u22122Eu[ T+i\u2211n=T\u2212ir(Xn)2\u2223\u2223\u2223\u2223\u2223T\u2212i <\u221e, XT\u2212i]\u0016 r2(1\u2212 r)2 .Meanwhile, since A is (p, r)-escapable, we have that Pu(T\u2212i < \u221e) \u2264 (1 \u2212 p)i. Combining theseestimates yields the desired inequality (10.5.7).29510.5. Critical exponentsWe say that a set A \u2282 V is C-short-lived ifEu[\u2211n\u22650aH(Xn)1(Xn \u2208 A)]\u2264 Cfor every u \u2208 V . Note that if A is a C-short-lived set of vertices, then every subset of A is alsoC-short-lived. While Lemma 10.5.8 states that escapable sets are short-lived, we also have thatunions of boundedly many short-lived sets are short-lived with a larger constant. In particular,letting s = s(M) be as above and letting \u03b4 = \u03b42(M, C\u221211 (M)), we have thatW\u03b5 \u2286d3 log1\u2212\u03b4(s)e\u22c3m=0{v \u2208 V : (1\u2212 \u03b4)m+1\u03b5 \u2264 \u03c3(v) \u2264 (1\u2212 \u03b4)m\u03b5}for every \u03b5 > 0. Thus, combining Corollary 10.5.7 and Lemma 10.5.8 immediately yields thefollowing.Corollary 10.5.9. There exist a constant C2 = C2(M) > 0 such that the set W\u03b5 is C2-short-livedfor every \u03b5 > 0.Lemma 10.5.10. Let A and B be two sets of vertices in G, and suppose that A is finite andC-short-lived for some C > 0. Then\u2211v\u2208AaH(v)Pv(\u03c4B <\u221e) \u0016 CC Feff(A\u2194 B).Proof. Let the stopping times \u03c4i be defined recursively by setting \u03c40 = \u03c4A and \u03c4i+1 = min{t > \u03c4i :Xt \u2208 A}, so that \u03c4i is the ith time the random walk \u3008Xn\u3009n\u22650 visits A. Then\u2211v\u2208AaH(v)Pv(\u03c4B <\u221e) \u2264\u2211v\u2208A\u2211i\u22650aH(v)Pv(B hit between time \u03c4i and \u03c4i+1)=\u2211v\u2208A\u2211u\u2208A\u2211i\u22650aH(v)Pv(\u03c4i <\u221e, X\u03c4i = u)Pu(\u03c4B < \u03c4+A ).Reversing time then yields that\u2211v\u2208AaH(v)Pv(\u03c4B <\u221e) \u0016\u2211v\u2208A\u2211u\u2208A\u2211i\u22650aH(v)Pu(\u03c4i <\u221e, X\u03c4i = v)Pu(\u03c4B < \u03c4+A ).By exchanging the order of summation, we obtain that\u2211v\u2208AaH(v)Pv(\u03c4B <\u221e) \u0016\u2211u\u2208AEu[\u2211n\u22650aH(Xn)1(Xn \u2208 A)]Pu(\u03c4B < \u03c4+A )\u0016 C\u2211u\u2208Ac(u)Pu(\u03c4B < \u03c4+A ).29610.5. Critical exponentsTo conclude, let \u3008Vj\u3009j\u22651 be an exhaustion of V and let Gj be the subgraph of G induced by Vj ,with conductances inherited from G, and observe that\u2211u\u2208Ac(u)PGu (\u03c4B < \u03c4+A ) = limj\u2192\u221e\u2211u\u2208Ac(u)PGu (\u03c4B < min{\u03c4+A , \u03c4V \\Vj})\u2264 limj\u2192\u221e\u2211u\u2208Ac(u)PGju (\u03c4B < \u03c4+A ) = CFeff(A\u2194 B).Recall that diamC(A) and dC(A,B) denote the Euclidean diameter of {z(v) : v \u2208 A} and theEuclidean distance between {z(v) : v \u2208 A} and {z(v) : v \u2208 B} respectively.Lemma 10.5.11. Let A and B be disjoint sets of vertices in G. ThenC Feff(A\u2194 B) \u0016diamC(A)2min{diamC(A), dC(A,B)}2 ,with the convention that the right-hand side is 1 if diamC(A) = 0.Proof. Let D = diamC(A). If D = 0 then A is a single vertex v and CFeff(A \u2194 B) \u2264 c(v) \u0016 1, soassume not. Recall the extremal length characterisation of the free effective conductance [173,Exercise 9.42]: For each function ` : E \u2192 [0,\u221e) assigning a non-negative length to every edge e ofG, let d` be the shortest path pseudometric on G induced by `. ThenC Feff(A\u2194 B) = inf{\u2211e\u2208E c(e)`(e)2d`(A,B)2: ` : E \u2192 [0,\u221e), dl(A,B) > 0}.Let W be the set of vertices v of G whose corresponding circles intersect the D-neighbourhood ofz(A) in C. Define lengths by setting `(e) to be`(e) =\uf8f1\uf8f2\uf8f3 min{|z(e\u2212)\u2212 z(e+)|, D} if e has an endpoint in W0 otherwise.Thend`(A,B) \u2265 min{D, dC(A,B))}> 0 (10.5.8)while, since |z(e\u2212)\u2212 z(e+)| = r(e\u2212) + r(e+),\u2211ec(e)`(e)2 \u0016\u2211v\u2208W\u2211v\u2032\u223cvmin{r(v) + r(v\u2032), D}2 \u0016 \u2211v\u2208Wmin{r(v), D}2, (10.5.9)where the Ring Lemma (Theorem 10.5.1) is used in the second inequality. As in the proof ofLemma 10.3.4, consider replacing each circle corresponding to a vertex in W that has radius largerthan D with a circle of radius D that is contained in the original circle and intersects the D-neighbourhood of z(A). This yields a set of circles contained in the 3D-neighbourhood of z(A).Comparing the total area of this set of circles with that of the 3D-neighbourhood of z(A) yields29710.5. Critical exponentsthat \u2211v\u2208Wmin{r(v), D}2 \u0016 D2. (10.5.10)We conclude by combining (10.5.8), (10.5.9) and (10.5.10).For each \u03b5 > 0 and m \u2208 Z \\ {0}, we defineUm\u03b5 (0) ={z \u2208 D : 1\u2212 \u03b5 \u2264 |z| \u2264 1\u2212 s3\u03b5 and sgn(m)4|m|5|m|pi \u2264 arg z \u2264 sgn(m)4|m|\u221215|m|\u22121pi}and define Um\u03b5 (\u03b8) to be the rotated set ei\u03b8Um\u03b5 .Lemma 10.5.12. There exist universal constants \u03b43, \u03b44 > 0 and k <\u221e such thatdC(Um\u03b5 (\u03b8), {rei\u03b8 : r \u2265 0})\u2265 (2 + \u03b43)diamC(Um\u03b5 (\u03b8))for all \u03b8 \u2208 [\u2212pi, pi], \u03b5 \u2264 \u03b44 and |m| \u2264 log5\/4(1\/\u03b5)\u2212 k.The constants here are more important than usual since we will later need to estimate thedifference 12dC(Um\u03b5 (\u03b8), {rei\u03b8 : r \u2265 0})\u2212 diamC(Um\u03b5 (\u03b8)).Proof. We begin by calculating, using elementary trigonometry, thatdiamC(Um\u03b5 (\u03b8)) \u22644|m|\u221215|m|pi + \u03b5anddC(Um\u03b5 (\u03b8), {rei\u03b8 : r \u2265 0}) =\uf8f1\uf8f2\uf8f3 1\u2212 \u03b5 |m| \u2264 3(1\u2212 \u03b5) sin(4|m|5|m|pi)|m| \u2265 4.By concavity of the function sin(t) on [0, pi], we have that sin(t) \u2265 2t\/pi for all t \u2208 [0, pi\/2], and inparticulardiamC(Um\u03b5 (\u03b8))dC(Um\u03b5 (\u03b8), {rei\u03b8 : r \u2265 0})\u2264 18(1\u2212 \u03b5)pi +5|m|\u03b52 \u00b7 4|m|(1\u2212 \u03b5) \u226418(1\u2212 \u03b42)pi +4k\u03b442 \u00b7 5k(1\u2212 \u03b44)for all \u03b5 \u2264 \u03b44 and |m| \u2264 log5\/4(1\/\u03b5) \u2212 k. This upper bound is less than pi\/7 < 1\/2 when \u03b44 issufficiently small and k is sufficiently large.10.5.4 Wilson\u2019s algorithmWilson\u2019s algorithm rooted at infinity [44, 228] is a powerful method of sampling the WUSF ofan infinite, transient graph by joining together loop-erased random walks. We now give a very briefdescription of the algorithm (see [173] for a detailed exposition). Let G be a transient network. Let\u03b3 be a path in G that visits each vertex of G at most finitely many times. The loop-erasure isformed by erasing cycles from \u03b3 chronologically as they are created. (The loop-erasure of a random29810.5. Critical exponentswalk path is referred to as loop-erased random walk and was first studied by Lawler [161].) Let{vj : j \u2208 N} be an enumeration of the vertices of G. Let F0 = \u2205 and define a sequence of forests inG as follows:1. Given Fi, start an independent random walk from vi+1. Stop this random walk if it hits theset of vertices already included in Fi, running it forever otherwise.2. Form the loop-erasure of this random walk path and let Fi+1 be the union of Fi with thisloop-erased path.Then the forest F =\u22c3i\u22650 Fi is a sample of the WUSF of G [44, Theorem 5.1].10.5.5 Proof of Theorems 10.1.3, 10.1.4 and 10.1.5Recall that e = (x, y) is a fixed edge. We write \u0017e to denote a lower bound that holds up to apositive multiplicative constant that depends only on M and rH(x), and that is increasing in rH(x).Proof of Theorem 10.1.4. Let F be the WUSF of G. Let Re\u03b5 be the event that pastF(e) contains avertex whose center is within Euclidean distance \u03b5 of the unit circle, and let DeR be the event thatdiamH(pastF(e)) \u2265 R. Then, letting \u03b5(R) = 1 \u2212 tanh(R\/2), we have that Re\u03b5(R)\/2 \u2286 DeR \u2286 Re\u03b5(R).Thus, to prove Theorem 10.1.4, it suffices to prove thatlog(1\/\u03b5)\u22121 \u0016e P(Re\u03b5) \u0016 log(1\/\u03b5)\u22121for all \u03b5 \u2264 1\/2. The proof of Theorem 10.1.2, and in particular of the estimate (10.3.6), adaptsimmediately to yield the desired upper bound (i.e., if the analysis there is carried out using thedouble circle packing of G rather than the circle packing of T ); the remainder of this proof isdevoted to proving the lower bound. This will be done by applying a second moment argument tothe random variableZ\u03b5 =\u2211v\u2208W\u03b5aH(v)1(v \u2208 pastF(e)),which, by definition of W\u03b5, is positive if and only if Re\u03b5 occurs.Lemma 10.5.13. There exists a positive constant \u03b45 = \u03b45(M, rH(x)), increasing in rH(x), suchthat E[Z\u03b5] \u0017e 1 for all \u03b5 \u2264 \u03b45.Proof of Lemma 10.5.13. If we generate F using Wilson\u2019s algorithm, starting with the vertex v,then v is in pastF(e) if and only if the loop-erased random walk from v passes through e = (x, y).In particular, we obtain the lower boundP(v \u2208 pastF(e)) \u2265 Pv(\u03c4x <\u221e, X\u03c4x+1 = y, \u3008Xn\u3009n\u2265\u03c4x+1 disjoint from \u3008Xn\u3009\u03c4xn=0)29910.5. Critical exponentsand hence, decomposing according to the value of \u03c4x,E[Z\u03b5] \u2265\u2211v\u2208W\u03b5\u2211m\u22651aH(v)Pv(\u03c4x = m, Xm+1 = y, \u3008Xn\u3009n\u2265m disjoint from \u3008Xn\u3009m\u22121n=0).Letting \u3008Yn\u3009n\u22650 be a random walk started at x independent of \u3008Xn\u3009n\u22650 and reversing time yieldsthatE[Z\u03b5] \u0017\u2211v\u2208W\u03b5\u2211m\u22651aH(v)Px (Xm = v, Cm) = Ex[\u2211m\u22650aH(Xm)1 (Xm \u2208W\u03b5, Cm)],where Cm is the eventCm ={Y1 = y, and \u3008Xn\u3009mn=1 disjoint from \u3008Yn\u3009n\u22650}.(Note that \u3008Xn\u3009mn=0 does not return to x after time 0 on the event Cm.) Let \u03c41 be the first timethat the random walk \u3008Xm\u3009m\u22650 visits {v \u2208 V : s2\u03b5 \u2264 \u03c3(v) \u2264 s\u03b5}, which is finite a.s. by definition ofs, and let \u03c42 be the first time m after \u03c41 that |z(Xm)\u2212 z(X\u03c41)| \u2265 C\u221211 s2\u03b5, where C1 = C1(M) \u2265 1is the constant from Theorem 10.5.3. Then Xm \u2208W\u03b5 for all \u03c41 \u2264 m \u2264 \u03c42, and soE[Z\u03b5] \u0017 Ex[\u03c42\u2211m=\u03c41aH(Xm)1 (Cm)]\u2265 Ex[\u03c42\u2211m=\u03c41aH(Xm)1(dC(X\u03c41 , {Yn : n \u2265 0}) > \u03b5\/s, C\u03c42) ].The events C\u03c41 \u2229 {dC(X\u03c41 , {Yn : n \u2265 0}) > \u03b5\/s} and C\u03c42 \u2229 {dC(X\u03c41 , {Yn : n \u2265 0}) > \u03b5\/s} are equal,and soE[Z\u03b5] \u0017 Ex\uf8ee\uf8f0 \u03c42\u2211m=\u03c41aH(Xm)1(dC(X\u03c41 , {Yn : n \u2265 0}) > \u03b5\/s,C\u03c41)\uf8f9\uf8fb= Ex\uf8ee\uf8ef\uf8f0Ex\uf8ee\uf8f0 \u03c42\u2211m=\u03c41aH(Xm)\u2223\u2223\u2223\u2223\u2223\u2223 X\u03c41\uf8f9\uf8fb1(dC(X\u03c41 , {Yn : n \u2265 0}) > \u03b5\/s,C\u03c41)\uf8f9\uf8fa\uf8fb .Applying (10.5.2) and the Diffusive Time Estimate (Theorem 10.5.3), we obtain thatEx\uf8ee\uf8f0 \u03c42\u2211m=\u03c41aH(Xm)\u2223\u2223\u2223\u2223\u2223\u2223 X\u03c41\uf8f9\uf8fb \u0010 \u03b5\u22122Ex\uf8ee\uf8f0 \u03c42\u2211m=\u03c41r(Xm)2\u2223\u2223\u2223\u2223\u2223\u2223 X\u03c41\uf8f9\uf8fb \u0010 1,and hence thatE[Z\u03b5] \u0017 Px(dC(X\u03c41 , {Yn : n \u2265 0}) > \u03b5\/s,C\u03c41). (10.5.11)30010.5. Critical exponentsBy the Ring Lemma (Theorem 10.5.1), there exists a positive constant k = k(M) such thatmin{r(x), r(y)} \u2265 rH(x)\/k. Let \u03b41(M, rH(x)\/k) be the constant appearing in Lemma 10.5.6, andlet \u03b45 = \u03b45(M, rH(x)) \u2264 \u03b41(M, rH(x)\/k) be sufficiently small that every vertex u that has \u03c3(u) \u2264 \u03b45and is contained in the cone H1(x) has distance at least \u03b45\/s from the half-plane H2(y, rH(x)\/k) (asdefined in Lemma 10.5.6). Applying Lemma 10.5.6 to both random walks X and Y conditioned onY1 = y yields thatPx(dC(X\u03c41 , {Yn : n \u2265 0}) > \u03b5\/s,C\u03c41) \u0017 q2(M, rH(x)\/k)2 \u0017e 1for all \u03b5 \u2264 \u03b45, concluding the proof.Lemma 10.5.14. E[Z2\u03b5 ] \u0016 E[Z\u03b5] log(1\/\u03b5) for all 0 < \u03b5 \u2264 \u03b44, where \u03b44 is the constant fromLemma 10.5.12.Proof. For each two vertices u and v in G, letH(u, v) :={w \u2208 V : |z(w)\u2212 z(u)| \u2264 |z(w)\u2212 z(v)|}be the set of vertices closer to u than to v with respect to the Euclidean metric on the circle packing.More generally, for a vertex u and a set A \u2282 V , letH(u,A) :={w \u2208 V : |z(w)\u2212 z(u)| \u2264 |z(w)\u2212 z(v)| for some u \u2208 A} = \u22c3v\u2208AH(u, v).Note that for every vertex u and set A,dC(A,H(u,A)) \u2265 infv1,v2\u2208AdC(v2, H(u, v2))\u2212 dC(v1, v2) \u2265 12dC(u,A)\u2212 diamC(A). (10.5.12)Expand E[Z2\u03b5 ] as the sumE[Z2\u03b5 ] =\u2211u,v\u2208W\u03b5aH(u)aH(v)P(u, v \u2208 pastF(e)).If u and v are both in the past of e in F, let w(u, v) be the first vertex at which the unique simplepaths in F from u to e and from v to e meet. ThenE[Z2\u03b5 ] \u2264 2\u2211u,v\u2208W\u03b5aH(u)aH(v)P(u, v \u2208 pastF(e) and w(u, v) \u2208 H(u, v)).Consider generating F using Wilson\u2019s algorithm rooted at infinity, starting first with u and then v.In order for u and v both to be in the past of e and for w(u, v) to be in H(u, v), we must have that30110.5. Critical exponentsthe simple random walk from v hits H(u, v), so thatE[Z2\u03b5 ] \u2264 2\u2211u\u2208W\u03b5aH(u)P(u \u2208 pastF(e))\u2211v\u2208W\u03b5aH(v)Pv(\u03c4H(u,v) <\u221e). (10.5.13)Thus, it suffices to prove that \u2211v\u2208W\u03b5aH(v)Pv(\u03c4H(u,v) <\u221e) \u0016 log(1\/\u03b5) (10.5.14)for all \u03b5 \u2264 \u03b44 and u \u2208W\u03b5.Recall the definition of Um\u03b5 from Lemma 10.5.12. Let k be the universal constant from Lemma 10.5.12,let `(\u03b5) = dlog5\/4(\u03b5) \u2212 ke, and let \u03b8 = arg(u). For each m \u2208 Z with 1 \u2264 |m| \u2264 `(\u03b5), let Sm\u03b5 (\u03b8) bethe set of vertices whose centres are contained in Um\u03b5 (\u03b8), and letS0\u03b5 = S0\u03b5 (u) :={v \u2208W\u03b5 :\u2223\u2223\u2223\u2223arg z(v)z(u)\u2223\u2223\u2223\u2223 \u2264 5kpi\u03b54k}.Then\u2211v\u2208W\u03b5aH(v)Pv(\u03c4H(u,v) <\u221e) =`(\u03b5)\u2211m=\u2212`(\u03b5)\u2211v\u2208Sm\u03b5aH(v)Pv(\u03c4H(u,v) <\u221e)\u2264`(\u03b5)\u2211m=\u2212`(\u03b5)\u2211v\u2208Sm\u03b5aH(v)Pv(\u03c4H(u,Sm\u03b5 ) <\u221e). (10.5.15)Since Sm\u03b5 is contained in W\u03b5, it is C-short-lived for some C = C(M) by Corollary 10.5.9. Thus,applying Lemmas 10.5.10\u201310.5.12 together with (10.5.13) yields that\u2211v\u2208Sm\u03b5aH(v)Pv(\u03c4H(u,Sm\u03b5 ) <\u221e) \u0016diamC(Sm\u03b5 )2(12dC (Sm\u03b5 , u)\u2212 diamC (Sm\u03b5 ))2\u0016 diamC(Um\u03b5 )2(12dC(Um\u03b5 (\u03b8), {rei\u03b8 : r \u2265 0})\u2212 diamC (Um\u03b5 (\u03b8)))2 \u0016 1 (10.5.16)for \u03b5 \u2264 \u03b44 and 1 \u2264 |m| \u2264 `(\u03b5). This in turn yields (10.5.14) when combined with (10.5.15) and thefact that\u2211v\u2208S0\u03b5 aH(v) \u0016 1.Let \u03b44 be the constant from Lemma 10.5.12 and let \u03b45 = \u03b45(M, rH(x)) be the constant fromLemma 10.5.13. The lower bound of Theorem 10.1.4 now follows from Lemmas 10.5.13 and 10.5.14together with the Cauchy-Schwartz inequality, which imply thatP(Z\u03b5 > 0) \u2265 E[Z\u03b5]2E[Z2\u03b5 ]\u0017e 1log(1\/\u03b5)30210.5. Critical exponentsfor all \u03b5 \u2264 min{\u03b44, \u03b45}. Since P(Z\u03b5 > 0) is an increasing function of \u03b5 and min{\u03b44, \u03b45} is anincreasing function of rH(x), it follows thatP(Z\u03b5 > 0) \u2265 P(Zmin{\u03b44,\u03b45}\u03b5 > 0) \u0017e1log(1\/\u03b5)for all \u03b5 \u2264 1\/2.Lemma 10.5.15. E[Z\u03b5] \u0016 1 for all \u03b5 > 0.Proof. By Wilson\u2019s algorithm, P(v \u2208 pastF(e)) \u2264 Pv(\u03c4x <\u221e). Thus, the claim follows immediatelyfrom Corollary 10.5.9 and Lemma 10.5.10, taking A = W\u03b5 \\ {x} and B = {x}, since C Feff(x \u2194V \\ {x}) \u2264 c(x) \u0016 1. The claim can also be proven more directly by a time reversal argumentsimilar to that carried out in the proof of Lemma 10.5.13.Proof of Theorem 10.1.5. We continue to use the notation from the proof of Theorem 10.1.4. Wefirst prove the upper bound. Let R \u2265 1. Applying Markov\u2019s inequality and Lemma 10.5.15 weobtain thatP(R1\/2\u2211k=0Zs\u22123k \u2265 R)\u2264 1RE[R1\/2\u2211k=0Zs\u22123k]\u0016 R\u22121\/2, (10.5.17)and hence thatP(areaH(pastF(e)) \u2265 R)\u2264 P(R1\/2\u2211k=0Zs\u22123k \u2265 R)+ P( \u221e\u2211k=R1\/2Zs\u22123k > 0)\u0016 R\u22121\/2, (10.5.18)where the second inequality follows from Theorem 10.1.4 and (10.5.17).We now obtain a matching lower bound. Let \u03b44 be the constant from Lemma 10.5.12 andlet \u03b45 = \u03b45(M, rH(x)) be the constant from Lemma 10.5.13, and let R0 = R0(M, rH(x)) =(4\/9) log21\/s(1\/min{\u03b44, \u03b45}), so that that s\u22123R1\/2\/2 is less than both \u03b44 and \u03b45 for all R \u2265 R0.Let R \u2265 R0 and letW =R1\/2\u22c3k= 12R1\/2Ws\u22123k and let Z =R1\/2\u2211k= 12R1\/2Zs\u22123k .The argument used to derive (10.5.13) in the proof of Lemma 10.5.14 also yields thatE[Z2] \u2264 2\u2211u\u2208WaH(u)P(u \u2208 pastF(e))\u2211v\u2208WaH(v)Pv(\u03c4H(u,v) <\u221e). (10.5.19)Let u \u2208 W and let \u03b8 = arg(z(u)). Let `(s\u22123k) and the sets Sms\u22123k(\u03b8) be defined as in the proof of30310.5. Critical exponentsLemma 10.5.14. Then,\u2211v\u2208WaH(v)Pv(\u03c4H(u,v) <\u221e) \u2264R1\/2\u2211k= 12R1\/2`(s\u22123k)\u2211m=\u2212`(s\u22123k)\uf8ee\uf8f0 \u2211v\u2208Sm\u03b5aH(v)P(\u03c4H(u,Sm\u03b5 ) <\u221e)\uf8f9\uf8fb . (10.5.20)The arguments yielding the estimate (10.5.16) in the proof of Lemma 10.5.14 also yield that thesums appearing in the square brackets on the right-hand side of (10.5.20) are bounded above by aconstant depending only on M. It follows that\u2211v\u2208WaH(v)Pv(\u03c4H(u,v) <\u221e) \u0016 Rfor all u \u2208W , and hence that E[Z2] \u0016 RE[Z].Next, Lemma 10.5.15 implies that E[Z] \u0017e R1\/2, while Theorem 10.1.4 implies that P(Z > 0) \u0016R\u22121\/2. Thus, there exists a positive constant C = C(M) such that E[Z | Z > 0] \u2265 CR for allsufficiently large R. Applying the second moment estimate above, the Paley-Zigmund Inequalityimplies thatP(Z \u2265 C2R\u2223\u2223\u2223\u2223 Z > 0) \u2265 E[Z | Z > 0]24E[Z2 | Z > 0] = E[Z]24E[Z2]P(Z > 0) \u0017e 1. (10.5.21)Combining (10.5.21) with the lower bound of Theorem 10.1.4 yields thatP(areaH(pastF(e)) \u2265C2R)\u2265 P(Z \u2265 C2R)\u0017e R\u22121\/2for all R \u2265 R0. Since the probability on the left hand side is decreasing in R, we conclude thatP(areaH(pastF(e)) \u2265 R) \u0017e R\u22121\/2as claimed.Proof of Theorem 10.1.3. Let F be a spanning forest of G and let e\u2020 be the edge of G\u2020 dual toe = (x, y). The past of e\u2020 in the dual forest F\u2020 is contained in the region of the plane bounded bye and \u0393F(x, y), so thatdiamH(\u0393F(x, y)) \u2265 diamH(pastF\u2020(e\u2020)).Meanwhile, if pastF\u2020(e\u2020) is non-empty, then every edge in the path \u0393F(x, y) is incident to a face ofG that is in pastF\u2020(e\u2020). By the Ring Lemma (Theorem 10.5.1), the hyperbolic radii of circles inP \u222a P \u2020 are bounded above. We deduce that there exists a constant C = C(M) such thatdiamH(\u0393F(x, y)) \u2264 diamH(pastF\u2020(e\u2020)) + C.We deduce Theorem 10.1.3 from Theorem 10.1.4 by applying these estimates when F is the FUSF30410.5. Critical exponentsof G and F\u2020 is the WUSF of G\u2020.10.5.6 The uniformly transient caseRecall that a graph is said to be uniformly transient if p = infv\u2208V Pv(\u03c4+v = \u221e) is positive. Ifthe graph has bounded degrees, this is equivalent to the property that Ceff(v \u2192 \u221e) is boundedaway from zero uniformly in v.Proposition 10.5.16. Then there exists a constant C = C(M) such thatrH(v) \u2265 1Cexp(\u2212CReff(v \u2192\u221e)) .Proof. By applying a Mo\u00a8bius transformation if necessary, we may assume that the circle P (v) iscentered at the origin. By the Ring Lemma (Theorem 10.5.1), rH(v) is bounded above by a constantdepending only on the maximum degree and codegree of G. Applying [107, Corollary 3.3] togetherwith Theorem 10.5.1 yields that Reff(v \u2192 \u221e) \u2265 c log(1\/r(v)) for some constant c = c(M). Sincethe hyperbolic radii are bounded, the Euclidean radius r(v) is comparable to rH(v).We are now ready to prove Corollary 10.1.6. In the rest of this subsection, we will use \u0016,\u0017 and\u0010 to denote inequalities or equalities that hold up to positive multiplicative constants dependingonly on M and p.Proof of Corollary 10.1.6. Let F be a spanning forest of G and let e\u2020 be the edge of G\u2020 dual toe = (x, y). The past of e\u2020 in the dual forest F\u2020 is contained in the region of the plane bounded bye and \u0393F(e). The non-amenability of the hyperbolic plane implies that the perimeter of any set isat least a constant multiple of its area, and so\u2211v\u2208\u0393F(x,y)rH(v) \u0017 areaH(pastF\u2020(e\u2020)). (10.5.22)On the other hand, if pastF\u2020(e\u2020) is non-empty, then every edge in the path \u0393F(x, y) is incident toa face of G that is in pastF\u2020(e\u2020). We deduce that if pastF\u2020(e\u2020) is non-empty thenareaH(pastF\u2020(e\u2020))\u0017\u2211v\u2208\u0393F(x,y)aH(v). (10.5.23)Note that neither estimate (10.5.22) or (10.5.23) required uniform transience. Proposition 10.5.16and the Ring Lemma (Theorem 10.5.1) imply that|\u0393F(x, y)| \u0010\u2211v\u2208\u0393F(x,y)rH(v) \u0010\u2211v\u2208\u0393F(x,y)aH(v).Combining the above estimates in the case that F is the FUSF of G and F\u2020 is the WUSF of G\u2020allows us to deduce Corollary 10.1.6 from Theorem 10.1.5.30510.5. Critical exponentsLet (X1, d1) and (X2, d2) be metric spaces and let \u03b1, \u03b2 be positive. A (not necessarily continu-ous) function \u03c6 : X1 \u2192 X2 is said to be an (\u03b1, \u03b2)-rough isometry if the following hold.1. (\u03c6 roughly preserves distances.) \u03b1\u22121d1(x, y) \u2212 \u03b2 \u2264 d2(\u03c6(x), \u03c6(y)) \u2264 \u03b1d1(x, y) + \u03b2 for allx, y \u2208 X1.2. (\u03c6 is almost surjective.) For every x2 \u2208 X2, there exists x1 \u2208 X1 such that d2(\u03c6(x1), x2) \u2264 \u03b2.See [173, \u00a72.6] for further background on rough isometries. We write dG for the graph distance onV .Corollary 10.5.17. Let G be a uniformly transient, simple, 3-connected, proper plane network withbounded codegrees and bounded local geometry. Let dG denote the graph distance on V , let (P, P\u2020)be a double circle packing of G in D, and let z(v) be the centre of the disc in P corresponding tothe vertex v. Then there exist positive constants \u03b1 = \u03b1(M,p) and \u03b2 = \u03b2(M,p) such that z is an(\u03b1, \u03b2)-rough isometry from (V, dG) to (D, dH).Proof. Proposition 10.5.16 implies that for every vertex v of G, rH(v) is bounded both above andaway from zero by positive constants. Almost surjectivity is immediate. For each two vertices uand v in G, the shortest graph distance path between them induces a curve in D (by going alongthe hyperbolic geodesics between the centres of the circles in the path) whose hyperbolic length is\u0017 dG(u, v).Conversely, let \u03b3 be the hyperbolic geodesic between z(u) and z(v), and consider the set W ofvertices w of G such that either P (w) intersects \u03b3 or P \u2020(f) intersects \u03b3 for some face f incidentto w. Let d be the length of \u03b3. Since all circles in (P, P \u2020) have a uniform upper bound on theirhyperbolic radii, we deduce that all circles in W are contained in a hyperbolic neighbourhood ofconstant thickness about \u03b3, and hence the total area of these circles is \u0016 d. Since the radii of thecircles are also bounded away from zero, we deduce that the cardinaility of W is also \u0016 d. SinceW contains a path in G from u to v, we deduce that dG(u, v) is \u0016 d as required.We summarise the situation for uniformly transient graphs in the following corollary, which fol-lows immediately by combining Corollary 10.5.17 with Theorems 10.1.3\u201310.1.5 and Corollary 10.1.6.We write diamG for the graph distance diameter of a set of vertices in G.Corollary 10.5.18 (Graph distance exponents). Let G be a uniformly transient, simple, 3-connected,proper plane network with bounded codegrees and bounded local geometry, and let F be the free uni-form spanning forest of G. Let p > 0 be a uniform lower bound on the escape probabilities of G.Then there exist positive constants k1 = k1(M,p) and k2 = k2(M,p) such thatk1R\u22121 \u2264 FUSFG(diamG(\u0393F(x, y)) \u2265 R) \u2264 k2R\u22121,k1R\u22121 \u2264 WUSFG(diamG(pastF(e)) \u2265 R) \u2264 k2R\u22121,k1R\u22121\/2 \u2264 WUSFG(|pastF(e)| \u2265 R) \u2264 k2R\u22121\/2, andk1R\u22121\/2 \u2264 FUSFG(|\u0393F(x, y)| \u2265 R) \u2264 k2R\u22121\/230610.6. Closing remarks and open problemsFigure 10.6: The double circle packing of N\u00d7 Z4.for every edge e = (x, y) of G and every R \u2265 1.10.6 Closing remarks and open problems10.6.1 RemarksRemark 10.6.1 (Non-universality in the parabolic case.). Unlike in the CP hyperbolic case, theexponents governing the behaviour of the USTs of CP parabolic, simple, 3-connected proper planegraphs with bounded codegrees and bounded local geometry are not universal, and need not existin general.Indeed, consider the double circle packing of the proper plane quadrangulation with underlyinggraph N\u00d7Z4, pictured in Figure 10.6. Let the packing be normalised to be symmetric under rotationby pi\/2 about the origin and to have r(0, 0) = 1. It is possible to compute that r(i, j) = (3 + 2\u221a2)iand hence that |z(i, j)| is comparable to (3 + 2\u221a2)i for every (i, j) \u2208 N \u00d7 Z4. Suppose that theedges connecting (i, j) to (i \u00b1 1, j) are given weight 1 for every (i, j) \u2208 N \u00d7 Z4, while the edgesconnecting (i, j) to (i, j \u00b1 1) are given weight c for each (i, j) \u2208 N \u00d7 Z4. It can be computed thatthe probability that a walk started at (i, 0) hits (0, 0) without ever changing its second coordinateis a(c)i := (1 + c \u2212 \u221ac2 + 2c)i. Let e = ((0, 0), (0, 1)). By running Wilson\u2019s algorithm rooted at(0, 0) starting from the vertices (i, 0) and (i, 1), we see thatUST(pastT (e) \u2229 {i} \u00d7 Z4 6= \u2205) \u2265 P(i,0)(\u03c4(0,0) < \u03c4N\u00d7{1,2,3})P(i,1)(\u03c4(0,1) < \u03c4N\u00d7{0,2,3})\u00b7P(0,1)(X1 = (0, 0))=c2c+ 1a(c)2i.the right-hand side is exactly the probability that the random walk from (i, 0) hits (0, 0) withoutever changing its second coordinate, and that the random walk from (i, 1) hits (0, 1) without everchanging its second coordinate and then steps to (0, 0).30710.6. Closing remarks and open problemsLet q(c) be the probability that a random walk started at (i, j) visits every vertex of {i} \u00d7 Z4before changing its vertical coordinate, which tends to one as c \u2192 \u221e. Let Y be a loop-erasedrandom walk from (0, 0). It can be computed that the probability that a random walk started from(i, j) visits {0}\u00d7Z4 before hitting the trace of Y is at most b(c)i := (1\u2212 q(c))i. Thus, by Wilson\u2019salgorithm and a union bound,UST(pastT (e) \u2229 {i} \u00d7 Z4 6= \u2205) \u2264 4b(c)i.It follows that there exist positive constants k(c), \u03b1(c) and \u03b2(c) such that \u03b1(c) \u2192 0 as c \u2192 0,\u03b2(c)\u2192\u221e as c\u2192\u221e, andk(c)\u22121R\u2212\u03b1(c) \u2264 UST (diamC(pastT (e)) \u2265 R) \u2264 k(c)R\u2212\u03b2(c).Thus, by varying c, we obtain CP parabolic proper plane networks with bounded codegrees andbounded local geometry with different exponents governing the diameter of the pasts of edges intheir USTs: If c is large the diameter has a light tail, while if c is small the diameter has a heavy tail.Furthermore, by varying the weight of ((i, j), (i, j\u00b11)) as a function of i in the above example (i.e.,making c small at some scales and large at others), it is possible to construct a simple, 3-connected,CP parabolic proper plane network G with bounded codegrees and bounded local geometry suchthatlogUSTG(diamC(pastT (e)) \u2265 R)log(R)does not converge as R\u2192\u221e for some edge e of G. The details are left to the reader.Similar constructions show that the behaviour of WUSFG(diamH(pastF(e)) \u2265 R \u00b7 rH(x)) is notuniversal over simple, 3-connected, CP hyperbolic proper plane network G with bounded codegreesand bounded local geometry in the regime that rH(x) is small.10.6.2 Open problemsIt is natural to ask to what extent the assumption of planarity in Theorem 10.1.1 can be relaxed.Part (1) of the following question was suggested by R. Lyons.Question 10.6.2. Let G be a bounded degree proper plane graph.1. Let H be a finite graph. Is the free uniform spanning forest of the product graph G \u00d7 Hconnected almost surely?2. Let G\u2032 be a bounded degree graph that is rough isometric to G. 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