Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Martínez Dibene, Guillermo Elías
Doctor of Philosophy - PhD
University of British Columbia
The purpose of this thesis is to investigate the Uniform Spanning Forest (USF) in the nearestneighbour integer lattice Z^{d+1} = ZxZ^d with the family of conductances c((n, x), (n', x')) = e^{λ max(n,n')}, for all (n, x) ~ (n', x'), where ~ stands here for adjacency of vertices and λ > 0 is a fixed parameter. The random walk corresponding to this assignment of conductances resembles a discrete version of Brownian motion with drift. These conductances are not bounded, neither from below nor from above. This entails that many results known in the literature are not applicable. Our results include: 1. Estimate, up to multiplicative constants, the Green's function of this network. 2. Both the Wired and Free uniform spanning forest measures coincide for these assignment of conductances via a coupling argument. Call USF the resulting measure. 3. For d = 1; 2 we will show that USF consists of a tree; for d ≥ 3; it consists of infinitely many infinite trees. 4. For every d; every component of USF is one-ended. 5. For d ≥ 3; there exists a metric η on Z^{d+1} such that the probability that z, z' belong to the same USF-component is bounded above and below by multiplicative constants of η(z - z')^{-(d-2)}: The upper bound is then generalised to the probability that all the vertices in a finite set should belong to the same component. 6. Collapse every tree in USF to a point and denote by D(z, z') the number of edges that separate the USF-components at z and z': Almost surely, D(z, z') = U+2308 (d-2)/4 U+2309
Uniform spanning forest on the integer latticewith drift in one coordinate.byGuillermo El´ıas Mart´ınez-Dibene.M.Sc., Mathematics, Centro de Investigacio´n en Matema´ticas, 2013.B.Sc., Mathematics, Universidad de Guanajuato, 2008.A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies.(Mathematics.)THE UNIVERSITY OF BRITISH COLUMBIA.(Vancouver.)July 2020c©Guillermo El´ıas Mart´ınez-Dibene, 2020The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:Uniform spanning forest on the integer lattice with drift in one coordinate.submitted by Guillermo Mart´ınez-Dibene in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Mathematics.Examining Committee:Martin Thomas Barlow, MathematicsSupervisorGordon Slade, MathematicsSupervisory Committee MemberHarry Joe, StatisticsUniversity ExaminerAndrew Rechnitzer, MathematicsUniversity ExaminerAdditional Supervisory Committee Members:Ed Perkins, MathematicsSupervisory Committee MemberiiAbstractThe purpose of this thesis is to investigate the Uniform Spanning Forest (USF) in the nearest-neighbour integer lattice Zd+1 = Z× Zd with the family of conductancesc((n, x), (n′, x′)) = eλmax(n,n′), for all (n, x) ∼ (n′, x′),where ∼ stands here for “adjacency” of vertices and λ > 0 is a fixed parameter. Therandom walk corresponding to this assignment of conductances resembles a discrete versionof Brownian motion with drift. These conductances are not bounded, neither from belownor from above. This entails that many results known in the literature are not applicable.Our results include:• Estimate, up to multiplicative constants, the Green’s function of this network.• Both the Wired and Free uniform spanning forest measures coincide for these assign-ment of conductances via a coupling argument. Call USF the resulting measure.• For d = 1, 2 we will show that USF consists of a tree; for d ≥ 3, it consists of infinitelymany infinite trees.• For every d, every component of USF is one-ended.• For d ≥ 3, there exists a metric η on Zd+1 such that the probability that z, z′ belongto the same USF-component is bounded above and below by multiplicative constantsof η(z − z′)−(d−2). The upper bound is then generalised to the probability that all thevertices in a finite set should belong to the same component.• Collapse every tree in USF to a point and denote by D(z, z′) the number of edges thatseparate the USF-components at z and z′. Almost surely, D(z, z′) =¢d− 24•.iiiLay SummaryIn this thesis we will study a mathematical model known as the Uniform Spanning Forest.This model can be understood using electrical networks and a type of probabilistic processes,known as random walks, that govern the behaviour of the current flowing along the wires.The electrical network we will be studying is one in which the underlying random walk hasa drift towards the right. This modification of the unbiased case creates drastic differencesin the electrical network: edge conductances are exponential. Thus, the conductances ofthe network are neither bounded below nor above, as such, some general results in theliterature for uniform spanning forests fail to hold. This creates a rich mathematical structurewhose properties are ultimately different from that of the uniform case and the methods weemployed to study them were created anew (in a sense) or with major modifications of theexisting ones.ivPrefaceThis is an original work by the author, Guillermo El´ıas Mart´ınez Dibene. Chapters 2 through7 are in preparation to be submitted for publication.vContentsAbstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Network random walk and electrical laws. . . . . . . . . . . . . . . . . . . . 21.2 The uniform spanning trees: finite case. . . . . . . . . . . . . . . . . . . . . . 61.3 The uniform spanning trees: infinite case. . . . . . . . . . . . . . . . . . . . . 61.4 Loop-erased random walk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Wilson’s algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Wilson’s algorithm with stacks and the construction of WSF. . . . . . . . . . 91.7 Wilson’s algorithm rooted at infinity. . . . . . . . . . . . . . . . . . . . . . . 111.8 Research in uniform spanning forests. . . . . . . . . . . . . . . . . . . . . . . 121.9 Definition of Γd(λ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.9.1 Biased random walk. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.10 Summary of results of the thesis. . . . . . . . . . . . . . . . . . . . . . . . . 151.11 On the exposition of this work. . . . . . . . . . . . . . . . . . . . . . . . . . 181.12 Strong Markov property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Green’s function bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26viContents2.1 Basic statistics of Γd(λ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Continuous time network random walk. . . . . . . . . . . . . . . . . . . . . . 282.3 Some estimates of sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4 Some properties of (network) random walk. . . . . . . . . . . . . . . . . . . . 352.5 A hitting time for the continuous time network random walk. . . . . . . . . . 392.6 Statement and proof of the Theorem. . . . . . . . . . . . . . . . . . . . . . . 443 The number of trees in USF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.1 The transition operator and harmonic functions. . . . . . . . . . . . . . . . . 523.2 Coupling of random walks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3 The Liouville property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4 Liouville property and the uniform spanning forest measure. . . . . . . . . . 623.5 The number of components. . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 Crossings of NRW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1 Dimensions d = 1, 3, 4, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Two elementary results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3 Setup for Γ2(λ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.1 Some preliminary sums of the Green’s function of Γ2(λ). . . . . . . . 724.3.2 Estimates on the Green’s function on some regions of Γ2(λ). . . . . . 754.4 Dimension d = 2; “logarithmic scale of crossings” of Γ2(λ). . . . . . . . . . . 815 One endedness for Γd(λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.1 One endedness for Γ1(λ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 One endedness for Γ2(λ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.2.1 Invariance of the chronology of Wilson’s algorithm rooted at infinity. 865.2.2 Preliminary estimates in the proof of one-endedness for Γ2(λ). . . . . 895.2.3 The proof of the main theorem. . . . . . . . . . . . . . . . . . . . . . 995.3 One endedness for Γd(λ), with d ≥ 3. . . . . . . . . . . . . . . . . . . . . . . 1075.4 On the rays of Γd(λ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136 Probability of belonging in the same component . . . . . . . . . . . . . . . 1156.1 The probability that two vertices are connected in the USF. . . . . . . . . . . 1156.2 An upper bound on the probability that finitely many points are USF-connected.1297 The separation between components: transitions in dimensions d =2, 6, 10, ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.1 Random relations and stochastic dimension. . . . . . . . . . . . . . . . . . . 137viiContents7.2 Tail triviality of random relations. . . . . . . . . . . . . . . . . . . . . . . . . 1427.3 The WSF connectedness relation. . . . . . . . . . . . . . . . . . . . . . . . . 1437.4 The main theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1448 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153A Graphs: definitions and assumptions . . . . . . . . . . . . . . . . . . . . . . 154B Random spanning subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.1 Random spanning subgraphs of a network. . . . . . . . . . . . . . . . . . . . 163B.2 Weak convergence on SG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165C Electrical networks and the space of flows . . . . . . . . . . . . . . . . . . . 167C.1 Flows and energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167C.2 The divergence and gradient on a network. . . . . . . . . . . . . . . . . . . . 171C.3 Potentials and currents: Ohm’s law, Kirchhoff’s cycle law and some conse-quences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174C.4 The orthogonal decomposition of `2−(E; r). . . . . . . . . . . . . . . . . . . . 175C.5 Voltages and their currents: effective conductance and resistance. . . . . . . 176C.5.1 Voltages from a point to a set. . . . . . . . . . . . . . . . . . . . . . . 177C.5.2 Voltages between two sets. . . . . . . . . . . . . . . . . . . . . . . . . 181C.5.3 Flux and Kirchhoff’s node law. . . . . . . . . . . . . . . . . . . . . . 182C.6 Dirichlet’s and Thompson’s principles. . . . . . . . . . . . . . . . . . . . . . 184C.7 Rayleigh’s monotonicty law. . . . . . . . . . . . . . . . . . . . . . . . . . . . 186C.8 Common network reductions. . . . . . . . . . . . . . . . . . . . . . . . . . . 186C.9 Effective resistance and conductance to infinity: Doyle and Snell’s criterion. . 193C.10 Flows to infinity: Terry Lyons’ criterion. . . . . . . . . . . . . . . . . . . . . 194C.11 Variational definitions of R (o,∞) and C (o,∞) . . . . . . . . . . . . . . . . 197C.12 Free and wired effective resistance and conductance between points. . . . . . 198D The spaces D and HD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204D.1 The topological spaces D and HD. . . . . . . . . . . . . . . . . . . . . . . . . 204D.2 Convergence in D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207D.3 The topological space D0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208viiiContentsD.4 The Hilbert space D/R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212D.5 Riesz’ and Royden’s decompositions. . . . . . . . . . . . . . . . . . . . . . . 214D.6 The space HD and the Dirichlet Liouville property. . . . . . . . . . . . . . . 217ixGlossaryIn mathematical mode, we reserve the straight alphabetA B C D E F G H I J K L M N O P Q R S T U V W X Y Zto denote subsets of vertices or of edges. Lower casesa b c d e f g h i j k l m n o p q r s t u v w x y zwill be employed to denote vertices, indices, numbers or other similar quantities.We reserve slant alphabetA B C D E F G H I J K L M N O P Q R S T U V W X Y Zto denote random objects, functions or numbers. Random graphs will often be denoted withthe following bold alphabetA B C D E F G H I J K L M N O P Q R S T U V W X Y Z.CC Classical Coupling, sect. 3.2.DLP Dirichlet-Liouville Property, sect. 3.3.FSF Free Spanning Forest, sect. 1.3.EHI Eliptic Harnack Inequality, sect. 3.3.LE Loop Erasure, see (1.4.1).LP Liouville Property, sect. 3.3.NRW Network Random Walk, see (1.1.2).SLP Strong Liouville Property, sect. 3.3.SRW Standard Random Walk, see 1.8.USF Uniform Spanning Forest, sect. 1.3.UST Uniform Spanning Tree, sect. 1.2.WSF Wired Spanning Forest, sect. 1.3.xAcknowledgementsFirst, I thank my supervisor Martin T. Barlow for his patience and guidance during thisalmost five years. His comprehensive knowledge in the subject was always a guiding lightthrough the many unknowns that presented to me while researching this topic. Secondly, myfellow graduate students who provided a rich and fun environment for research. I would like tomention a few names (anyone not mentioned, please accept my apologies, having a thoroughlist would be endless) Malcolm Bowels, Sara´ı Herna´ndez Torres, Thomas James Hughes,Chen-Chih Lai and Jens Malmsquit, with whom I constantly have enriching mathematicaland philosophical conversations.I am also deeply grateful to my friends outside the world of mathematics. They speciallyhelped me to continue forward and keep my enthusiasm and temper during these years.Again, mentioning everyone would be a too long list, I do thank in particular a few people,Gordon Sien with whom I had very many study sessions, Stephanie Kirk (and her cat)with whom I spent several evenings playing board games or cooking, Mikael Tran for theexperiences shared together and I specially want to thank Gary Ma for accompanying thelast three years as my partner.Finally, I thank my family for their moral support, I know they are very proud.xiDedicationTo my mother, accompanying me from heaven.xiiChapter 1IntroductionThe purpose of this thesis is to investigate different properties of the uniform (random) span-ning trees arising from specific conductances on the nearest-neighbour integer lattice. Recallthe definition of the nearest-neighbour integer lattice (A.21). We will set the conductancesof Zd+1 = Z× Zd to be (depending on a parameter λ > 0)µ((n, x), (n′, x′)) = eλmax(n,n′), (n, x) ∼ (n′, x′).We will denote Γd(λ) = (Zd+1, µ). The network random walk of Γd(λ) has drift in thepositive direction of the first axis while being uniform on the other axes. It is clear thatthese conductances have an exponential decay and growth, and nearly all papers looking atthe Uniform Spanning Tree (UST) on explicit graphs consider the case of unit (or uniformlybounded) conductances. In the upcoming section we provide a skimmed bird’s-eye view ofcurrent trends in Uniform Spanning Forest (USF) research.Before commencing to read the next section, the reader is recommended to check AnnexA for the mathematical definitions related to graphs and the implicit assumptions we willbe making for the remainder of the thesis; in particular, spanning trees are defined in thisAnnex. In Annex B we describe how to construct the space of random spanning subgraphson a given graph; this is the construction it will always remain tacitly assumed wheneverdealing with uniform spanning trees or forests in finite or infinite graphs. In Annex C wehave compiled enough background about electrical networks as to make this document self-contained, most of this material is not used explicitly until Chapter 5. Finally, Annex Dprovides the study of a space of function and it is based on Annex C and its results will bekey in Chapter 3.1Chapter 1. Introduction§ 1.1. Network random walk and electrical laws.Intuitively, a network is a bunch of nodes with wires connecting them, the wires have con-ductances. More formally, we have the following.Definition ( 1.1.1 ) Electrical networks. A weighted graph or electrical network Γ =(G, µ) consists of a graph G and conductances (or weights) defined by means of function µ,often also denoted by c, from E into the set of positive real numbers or zero; in this casevertices are also named nodes and edges wires. The number c(e) or µ(e) is called conductanceor weight of e and r(e) =1c(e)the resistance. Since conductances and resistances are non-negative we may define c(e) = ∞ whenever r(e) = 0 and vice versa, r(e) = ∞ wheneverc(e) = 0. In the case of a directed graph, the only conductances that will be considered arethose for which c(e) = c(−e). Any graph without explicit conductances given can be madeinto a network in a canonical way by choosing the “canonical conductances” c = 1 (thefunction one on E); when a graph is given and no conductances are given, we assume thecanonical conductances are being considered. Thus, every graph is a network. In the graphinduced on a set of vertices A by the network (G, µ) we impose the conductances to be therestriction of µ to E|A and write µ|A if necessary from context, else we simply write µ; thisnetwork is said to be induced on A by the network (G, µ); an analogous definition applieson the graph induced by a set of edges. Given a network Γ = (G, µ), we will say that the(directed or undirected) graph G is underlying the structure of the network; by definition, avertex or an edge of a network is a vertex or an edge of the underlying graph.The conductances of a network essentially allow what amount of current flows throughthe wire. It is well-known experimentally that voltage (which is the potential differencebetween two adjacent nodes) obeys several so-called laws. We are interested in the followingthree:(a) at every node, except those grounded or connected to the battery, the sum of thecurrents meeting at this node is zero1 (“Kirchhoff’s node law”);(b) the sum of the voltages (potential differences) at every node of a circuit is zero (“Kirch-hoff’s voltage law”); and(c) the current passing between two nodes equals the voltage divided by the resistance(“Ohm’s law”).1By Ohm’s law, this is trivially equivalent to the following: the voltages at a node is the weighted average,proportionally to the conductances, of the voltages in the neighbouring nodes.2Chapter 1. IntroductionThen, the network random walk is a Markov chain with state space the nodes of the network,such that the steps (or jumps) of the walk happen with probability proportional to theconductances. Explicitly, we have the following definition (we use Z for the set of integernumbers and Z+ for the non-negative integers):Definition ( 1.1.2 ) Network random walk. Consider any directed connected electricalnetwork (G, µ) (with at least two vertices). One can construct the network random walk(NRW) (in the literature, often referred to as “standard random walk”) in (G, µ) as follows.Consider the number(1.1.3) µ(v) =∑e−=vµ(e), v ∈ Vthat is, “conductance at the vertex v.” Intuitively, if µ(v) = 0, then all wires adjacent tov have zero conductance and, from a network perspective, neither v nor its adjacent wiresbelong to the network. Thus, it only makes sense to assume µ(v) > 0 for every vertex v fromnow onwards. From a physical standpoint, it also only makes sense to assume µ(v) <∞.From now onwards, we will assume that all conductances are chosen to satisfy that forevery vertex v, the conductance at v is positive and finite. This will always be the case if Gis locally finite with positive conductances, for then the sum consist of finitely many positiveterms and is therefore finite.For any edge e, the fractionµ(e)µ(e−)is a number in [0, 1] and clearly∑e−=vµ(e)µ(v)= 1.Therefore, for any vertex v, the function e 7→ µ(e)µ(v)defines a probability measure on theedges adjacent to v. The network random walk is any Markov chain with state space V andtransition probability given by(1.1.4) p(u, v) =1µ(u)∑e−=u,e+=vµ(e).The previous definition implies the following reversibility equation or balance equation (sinceµ(e) = µ(−e))(1.1.5) µ(u)p(u, v) = µ(v)p(v, u) for all vertices u, v.Consider a network random walk (Sn)n∈Z+ of (G, µ). We introduce the following quantitiesused several times in the rest of the thesis,(1.1.6) τH = inf{n ∈ Z+ | Sn ∈ H}, τ+H = inf{n ∈ N | Sn ∈ H}3Chapter 1. Introductionwhere H is any set of vertices. For sake of simplicity, we write τa = τ{a}, τ+a = τ+{a}. We willcall τH the “hitting time of H” (by S) and τ+H the “hitting time of H after the first step”;when H = {a} we replace the word “hitting” by “visit.” When H = V \ K, the expressions“exit time of K” and “exit time of K after the first step” mean the hitting times of H. Ifthere are several random walks S, S ′, etc., under consideration, we will use the notationτH(S), τH(S′), and so on, to differentiate the hitting or visit times of the different randomwalks. We use this notation without further notice unless otherwise explicitly stated.Remark ( 1.1.7 ) (a) We just defined the NRW for directed graphs. This is so since thesteps of the network random walk are always directed. In other words, NRW canonicallytake place in directed networks. If we need to construct an undirected network out ofthe paths of the NRW (such as in Wilson’s algorithm, see below), we can apply thecanonical loss of direction (A.4).(b) Although G is assumed connected, it can still happen that it may have several zero-conductance wires, potentially making the network random walk to be split into dif-ferent classes or regions in the graph. To prevent this situation, we assume the con-ductances are chosen so that the network random walk is irreducible. This will alwaysbe the case if G is connected and the conductances never take value zero.(c) In the standard lattice of dimension d (A.21), the standard random walk can be de-scribed as a process (Sn)n∈Z+ such that S0 is some “initial point” and for positiveintegers n, Sn = Sn−1 + Yn, where for any unit vector u in Zd, P (Yn = u) =12d, thefamily (Yn)n∈N being independent. Indeed, this clearly defines a Markov chain withstate space Zd and its transition density is p(x, y) =12dif ‖x− y‖ = 1 (Euclideannorm) and zero otherwise.Definition ( 1.1.8 ) Harmonic functions. A function h : V→ R is said to be harmonicat x ifh(x) =∑y∈Vp(x, y)h(y) =∑e−=xµ(e)µ(x)h(e+).We will often say that h is harmonic on a subset of vertices A if h is harmonic at everyx ∈ A. A function h : V→ R is said to be superharmonic if we replace the first = by ≥ .Observe that Kirchhoff’s node law is modelled mathematically by harmonic functions.Definition ( 1.1.9 ) Green’s function. Let (G, µ) be a network. Suppose S = (Sn)n∈Z+is the network random walk on G. We will often employ the notationPx (Sn = y) = P (Sn = y | S0 = x) ,4Chapter 1. Introductionand similarly for the expectationEx (f(Sn)) = E (f(Sn) | S0 = x) .In particular,Px (S1 = y) = ExÄ1{S1=y}ä.The Green’s function “of the network G” is, by definition, the function G : V2 → [0,∞]given by(1.1.10) G(x, y) =∑n∈Z+Px (Sn = y) = Ex( ∞∑n=01{Sn=y}), (x, y) ∈ V2,where the second equality is just a consequence of Lebesgue-Tonelli’s theorem. By definition,the Green’s function is the expected number of visits to y for a network random walk startedat x. More generally, if V0 is a subset of V, we set τ to be the exit time of V0 by the networkrandom walk S. We define the Green’s function restricted to leaving V0 by(1.1.11) GV0(x, y) =∞∑m=0Px (Sm = y,m < τ) .Because of known reasons of periodicity, we introduce the following two definitions, whichwill be required later on.Definition ( 1.1.12 ) Lazy Markov chains. From the general theory of Markov chains,given a chain X having transition probability p, we can define the λ-lazy chain Xλ, for anyfixed 0 < λ < 1, as a chain with transition density(1.1.13) pλ(x, y) =λ+ (1− λ)p(x, x) if x = y,(1− λ)p(x, y) if x 6= y.Given a transition kernel p, we will use the adjective “lazy” to refer to functions and randomelements associated with pλ, in particular pλ is called “lazy transition kernel” relative to pand Xλ the “lazy (version of the) random walk X.” We remark once and for all that lazyversions are useful when p(x, x) = 0 since then pλ(x, x) = λ > 0 and every lazy version isaperiodic.Definition ( 1.1.14 ) Continuous time network random walk. Let S be the networkrandom walk on some network Γ, satisfying the usual assumptions (A.22). Suppose thatS0 = o is some fixed vertex of the network. Suppose that N = (Nt)t≥0 is a Poisson processin Z+ with unitary constant intensity which is independent of the process S. We define thecontinuous time network random walk to be the process S˜ =ÄS˜tät≥0 given by S˜t = SNt .Define the continuous time network random walk transition probability by(1.1.15) qSt (o, v) = PoÄS˜t = vä, (t, z) ∈ [0,∞)× V.5Chapter 1. Introduction§ 1.2. The uniform spanning trees: finite case.Recall the definition of spanning tree of (A.12). Given a finite connected graph G, thenumber of subgraphs is also finite and thus, there are only finitely many spanning trees inthat graph. The uniform spanning tree is, by definition, the probability measure on the setof spanning trees of G satisfyingP(t) =1# of spanning trees.More generally, if t is a tree of G and there is an assignment of conductances on G, we maydefine weight(t) to be the product of all conductances of the edges of t. Then, we may alsodefine the following probability measure on trees of GP(t) =weight(t)k,where k is the sum of all weight(t) as t runs on the set of spanning trees of G so that P isensured to be a probability measure. The first of these definitions is what is naturally called“uniform spanning tree measure” on G and we will denoted it as UST from now onwards.By lack of a better name, we will employ the same nomenclature and notation in the caseof the network, although, it is admitted that different trees will have different probabilitiesof being chosen. Any tree-valued random object UST-distributed will be denoted by T.To the naked eye, it may seem that spanning trees and electrical networks have littleto do with each other. In fact, there exists a very deep and rich structure amalgamatingboth concepts. The first instance of this deep connection is present in Kirchhoff’s effectiveresistance formula (1847) which states that the probability of the event {e ∈ T} equals thecurrent passing through e (when the nodes adjacent to e are connected to a battery andgrounded and unit amount of current flows in the entire network).§ 1.3. The uniform spanning trees: infinite case.Suppose now that (G, µ) is an infinite network satisfying the usual assumptions (A.22). TheUST on G can no longer be defined as a quotient and thus, some sort of limit is needed. Thefirst paper to introduce UST on infinite graphs was Robin Pemantle’s [Pem91]. Pemantlework with the graph Zd, although his idea could have been applied to many networks. Heconsidered Zd-balls corresponding to the norm ‖x‖∞ = max1≤i≤d |xi| of centre at the origin andradii n ∈ N (these are symmetric parallelotopes with edges parallel to the axes of Zd). Using6Chapter 1. Introductionequivalences of random walk and electrical networks, he showed that the uniform spanningtree measure µn on the trees of Bn can first be extended canonically to a measure on thespanning forests of Zd and then that the sequence of extensions converges weakly to somemeasure µ, which, naturally, is the uniform spanning forest measure on spanning forests ofZd; see Annex B for more on random spanning subgraphs on a graph. As before, we willemploy the notation USF to denote uniform spanning forest measure and F will denote anyrandom object USF-distributed.Later, Itai Benjamini, Russel Lyons, Yuval Peres and Oded Schramm, in [BLPS01] andusing tools unavailable to Pemantle, extended his result from Zd to general networks. As theauthors stated, the tools they used that were unavailable to Pemantle are Wilson’s algorithm,the “mass-transport principle” and the main result of [LPS03] regarding the intersection ofa random walk and a loop-erased random walk.Based on motivations from Statistical mechanics, Benjamini, Lyon, Peres and Schrammconsidered two ways of extending the uniform spanning forest measure to general networks.They started by considering an exhaustion of the graph created by considering increasingfamilies of finite sets of vertices and then considering the networks induced on these sets. If(Gn) is any such exhaustion, they showed that the uniform spanning forests measure of Gn,extended in the obvious way to G, defines a sequence of measures that converge weakly tosome measure µF , the F stands for “free” since there are no constraints in this approach.They named µF as “free uniform spanning forest measure.” The second way to construct theuniform spanning forest measure is by considering constraints in the boundary of Gn. Indeed,the considered a sequence (Vn) of families of finite sets of vertices and then constructed thenetwork GWn , which the network induced on Vn with the boundary of Vn wired into avertex (A.11). Again, they showed that (the extension to the spanning subgraphs of G) ofthe uniform spanning forest measures of GWn converge to a measure µW and they called thismeasure the “wired uniform spanning forest measure.” Several properties of µF and µW werestudied, in particular, necessary and sufficient conditions were given as to when µF = µW .We will employ FSF and WSF instead of µF and µW , respectively.§ 1.4. Loop-erased random walk.Recall the definition of a path in a graph (A.7).Definition ( 1.4.1 ) Loop-erasure of paths. Let γ = (vj)0≤j≤m be a path of vertices.The loop-erasure of γ, denoted as LE(γ), is the path obtained from γ by deleting cycles inchronological order. More specifically, it is defined as follows. Set γ0 = γ.7Chapter 1. Introduction(LE) Suppose we are at stage k and we receive a path γk−1 = (u1, . . . , umk) . If no vertex ofthis path is repeated, then return LE(γ) = γk−1. Else, consider τk to be the first indexj such that the vertex uj appears twice in γk−1. Then, let σk to be the last index jsuch that uj = uτk , consider now the path γk = (u1, . . . , uτk , uσk+1, . . . , umk) . Repeat.Remark ( 1.4.2 ) (a) At first sight it may seem that this algorithm does not erase cyclesin chronological order and in fact, it does not. Yet, it produces the same path. The gistof this equivalence lies in the fact that the expressed algorithm deletes at once the cycles(vτk , . . . , vσk) while any algorithm deleting cycles chronologically will eventually deleteall cycles contained between vτk and vσk and then, ultimately, delete all remainingvertices between vτk and vσk .(b) The same procedure allows one to define LE(γ) for unending paths γ = (vj)j∈Z+ suchthat no vertex is repeated infinitely many times for then σk can be defined in theprevious iteration of steps. Obviously, for infinite paths, the previous constructionwill never end and does not constitute an algorithm, but the mathematical object stillexists. Indeed, if γk−1 = (u1, u2, . . .), we may define τk and σk as in (LE), then it isclear that γk = (u1, . . . , uτk)∨ γ˜k, for some infinite path γ˜k. Also, from the constructionthe portion of path (u1, . . . , uτk) does not contain cycles, then it will be present in allγk+m, m ≥ 1. It is now obvious how to define LE(γ) in this case.Suppose S denotes the network random walk of a network Γ. By definition, for everyω, S(ω) = (Sn(ω))n∈Z+ is an infinite path. If the network random walk is transient, thenfor almost every ω, no vertex in S(ω) is repeated infinitely often, then LE(S(ω)) is well-defined. The random object ω 7→ LE(S(ω)) is naturally known as loop-erased random walk,we will denote it as LE(S). It was first introduced by Gregory Lawler in [Law80]. In thispaper, Lawler proved that LE(S), with S the network random walk of the standard Euclideanlattice Zd, converges in distribution to the Wiener measure for d ≥ 5. In other words, heproved a central limit theorem for LE(S). To be more specific, if LE(S(ω)) = (vj)j∈Z+ , then wedefine the continuous-time continuous-path Yt(ω) by setting Yj(ω) = vj and if j ≤ t ≤ j + 1then t 7→ Yt(ω) will be the line segment joining vj and vj+1 in Rd. What Lawler showed wasthat Yt converges weakly to the Wiener measure (Brownian motion) after an appropriaterescaling.§ 1.5. Wilson’s algorithm.Wilson’s algorithm for generating uniform spanning trees in a finite connected network tiesUST with: network random walk and loop-erased random walk. David Wilson published his8Chapter 1. Introductionalgorithm in [Wil96]. The algorithm is actually fairly easy to describe:Step 0. Order the vertices as v0, . . . , vq, and set t0 = {v0}.Step 1. Suppose tk−1 is constructed. Call the vertices that are end-points of edges in tk−1“searched.” Let vαk the first vertex among v0, . . . , vq not yet searched. Clearly, αk ≥ k.Run an independent network random walk started at vαk until its hitting time of tk−1.Erase the cycles in chronological order of this section of random walk. Set tk to be theunion of the resulting loop-erased walk with tk−1.Step 2. If there are no more vertices to go (if all vertices have been searched), the finaltree so constructed, say T = tk, is distributed as uniform spanning tree of the givennetwork.A straightforward modification of this algorithm produces another algorithm which generatesuniform spanning trees on general recurrent infinite networks. It was later generalised totransient infinite networks by Benjamini, Lyons, Peres, Schramm [BLPS01]. (They calledthis version “Wilson’s algorithm rooted at infinity”, see below.)§ 1.6. Wilson’s algorithm with stacks and the construc-tion of WSF.Here we explain a way to use Wilson’s algorithm in a finite network, embedded in a morecomplicated structure, and such that the resulting tree is essentially independent of theordering of the vertices of the network. We assume that the underlying graph of this networkis simple (multiple edges and self-loops banned). The main reason for this is that thenetwork random walk was defined using transitions between neighbouring vertices, as such,if two vertices were connected by many edges, we added up their conductances to define thetransition probability. We could have called such a random walk a vertex -NRW, and define,in a similar fashion, something that we could have called the edge-NRW, where we followedwhat edges were used. This, however, would be awkward to do since then the transitionkernel of the underlying Markov chain would then be defined on the edges of the network,as opposed to pairs of states, which is the only way (known to the author) that kernels ofMarkov chains are ever defined. Also, it is obvious why we ban self-loops, to construct a treewe must ignore them.Suppose first that some vertex is fixed, call it r for “root.” And for the time being, assumethat some ordering is given to the vertices in V\{r}, that is, assume these vertices are writtenas a finite sequence (v1, . . . , vq). For every vertex j = 1, . . . , q, consider an independent family9Chapter 1. Introduction(Y(j)i)i∈N of random elements, the families are assumed mutually independent as well, suchthat Y(j)i follows the law induced on the neighbours of vj by µ, that is to say, if (p(v, u))(v,u)∈V2is the probability kernel of the network random walk, then P(Y(j)i = u)= p(vj, u). We willsay that(Y(j)i)i∈N is the “stack of neighbours under vertex vj.” Under the root we assumewe have an empty stack.Starting at v1, we take the first random variable out of the stack, namely, u = Y(1)1 , andthen the random walk moves from v1 to u. We then think that we eliminate Y(1)1 from thetop of the stack (we “pop it”) so that the new top element of the stack under v1 is Y(1)2 . Ifu = vj, we now take the first element on top of the stack under vj, namely Y(j)1 and we moveto this vertex and then pop Y(j)1 . Iterating this procedure, we can produce a network randomwalk started at v1. This random walk must eventually reach the root r, and since there is anempty stack under the root, this iteration stops here. Observe that if at whatever step ofpopping top elements of the stacks, we are about to start a cycle u = Y(j1)i1 ∼ . . . ∼ Y (jk)ik = u,then all these elements Y jhih are on top of the stacks and will be popped one by one untilwe return to Y(jk)ik= u, which is the initial position of the cycle. Thus, we may think thatdeleting a cycle is a one-step procedure in this popping iteration. We call this “popping acycle.”At any given time, the top of the stacks determine a graph, which will be called the“visible graph.” If the visible graph has no cycles to pop, then it is a tree. Otherwise, westart with vertex v1, the random walk moves, it will be popping cycles until it reaches theroot. This produces a path L1, which is clearly the loop-erasure of the section of networkrandom walk started at v1 until its visit time to the root. We continue with the next vertex,v2. Observe that if the popping procedure started at v2 produces a path Y(2)1 ∼ . . . ∼ Y (j)isuch that Y(j)i is the first vertex in this path that belongs to L1, then we stop the poppingprocedure started at v2, this clearly produces the LE of the network random walk started atv2 until its hitting time with L1. We proceed in the same way with the vertices v3, . . . , vq, inthat order. Observe that the ordering of the vertices produces a finite sequence of cycles thatwill be popped. The following result will be called here “Wilson’s algorithm with stacks.”Proposition ( 1.6.1 ) Fix a root r ∈ V. With probability one, there are finitely many cyclesthat can ever be popped. Any two ordering of the searches of the vertices in V\{r} will pop allthese cycles and the popping procedure will leave the same visible graph for the two orderings.This visible graph sitting on top of the stacks is a spanning tree and the distribution of thisrandom tree is that of the UST of the given network.A detailed proof is given in [Bar16, Proposition 1.7]. Observe that Wilson’s algorithmrun with stacks on a finite graph produces always the same tree provided the root is fixedin advance. We will make use of this algorithm later on in the thesis, see (5.2.1).10Chapter 1. Introduction§ 1.7. Wilson’s algorithm rooted at infinity.We mention now the usual way to construct WSF on an infinite network (G, µ). Consideran exhaustion Vq of the vertex set and denote by (Gq, µ) the network induced on Vq withits boundary ∂Vq wired into a vertex. Run Wilson’s algorithm with stacks taking ∂Vq asroot. This produces a random tree Tq. Then, Tqweakly−−−→q→∞ F, where F has distribution WSF,see [LP16], sect. 10.1. This method does produce the law of the WSF in the infinite network(G, µ) but it does not produce a random object (in the sense of an a.s. construction). Ofcourse one can still sample the random object having the law. However, there exists analternative more dynamic method to sample the WSF-distributed random object, which wasfirst introduced in [BLPS01], and which has many interesting theoretical applications. Weexplain further in the next paragraph.Suppose Γ is a network satisfying the usual assumptions (A.22) such that the underlyinggraph is simple (A.8). Denote by V the set of vertices of G and by E that of edges. Supposeξ : N → V is a bijection (an “ordering” of the vertex set). Let (Ω,F ,P) be a probabilityspace in which is possible to define an independent family (Sv)v∈V of network random walksof this network, such that Sv is a random walk started at v. Define inductively the followingrandom spanning subgraphs of G (identified with their edge set):Start. Lξ1 = Fξ1 = LEÄSξ(1)mäm∈Z+ .Inductive step. Having defined Lξk and Fξk, defineLξk+1 = LEÄSξ(k+1)mäm=0,...,τk,Fk+1 = Fk ∪ Lξk+1,where τk = τFξkÄSξ(k+1)ä, that is, τk is the hitting time of the forest currently presentFξk by the random walk Sξ(k+1).Consider finally the random spanning subgraph of G :Fξ =⋃k∈NFξk.What Wilson’s algorithm rooted at infinity [BLPS01, Theorem 5.1] states is that Fξ ∼WSF(G,µ), in other words, the distribution of Fξ is that of the wired spanning forest of thenetwork (G, µ), and this happens independently of the choice of ordering ξ.11Chapter 1. Introduction§ 1.8. Research in uniform spanning forests.The uniform spanning tree is an interesting and rich structure that that has been demon-strated to have many connections to other well studied and interesting topics. For example,using Wilson’s algorithm, it is clear that UST is related to loop-erased random walk. It isimportant to mention that, as stated by Gady Kozma [Koz07]“Of all the non Gaussian models in statistical mechanics, LERW is probable themost tractable.”Because of the close connection between loop-erased random walk and uniform spanningtree, the UST is perhaps the next most tractable.Amongst the known properties of uniform spanning forests in Zd are the following.(a) WSF = FSF, see [LP16, Corollary 10.9]. Denote them by USF.(b) If d = 1, 2, 3, 4 then USF-a.s. every realisation (sample) is a tree. By virtue of thisreason, we denote UST in lieu of USF in this dimensions. When d ≥ 5, then USF-a.s.every realisation contains infinitely many infinite components. See [LP16, Theorem10.30].(c) Denote by SRW the network random walk of the graph Zd (this is “standard randomwalk”). Then, in dimension d = 1, 2, 3, 4 two independent SRW-paths in the standardlattice will cross infinitely often (Sn = S′n′ for infinitely many pairs (n, n′)), while ind ≥ 5 they will not cross at all with positive probability.(d) In every dimension, a.s. all components of USF have one end (from every vertex, thereis one and only one ray), see [LP16, Theorem 10.41].(e) In d ≥ 5, collapse every tree of USF of Zd into a single vertex. For two vertices x, y ofZd, denote by D(x, y) the distance between the component Tx containing x and thecomponent Ty containing y in the collapsed graph. (The relation Tx 6= Ty is equivalentto D(x, y) ≥ 1.) Then [BKPS04],maxx,y∈ZdD(x, y) =úd− 14üa.s.(f) In d = 2, the scaling limit of UST is related to SLE processes (see below).12Chapter 1. Introduction(g) In [Ha¨g95], Olle Ha¨ggstro¨m showed that the UST is a limiting member of the FK ran-dom cluster models, where FK stands for the Fortuin-Kastelyn random cluster model.If FK(p, q) denotes the FK random cluster model with parameters p and q, then USTis the limit as p→ 0 and qp→ 0 of FK(p, q).The FK random cluster model includes as special cases percolation and the Ising model,two other very interesting processes. This connects the UST with very important class ofmodels in statistical physics. The study of scaling limits of the UST and LE(S) led OdedSchramm to the discovery of the SLE models [Sch00]. SLE(κ) consists of conformally invariantfamily of curves in the complex plane or the complex unit disc. It is conjectured that thescaling limits of critical FK(p∗, q) in dimension d = 2 will be SLE with an appropriateparameter. This has been proved in a few cases, for example percolation on hexagonallattices.The USF is vigorous in research now; both [BKPS04] and [BLPS01] proposed severalconjectures yet to be resolved, many of which involve UST and related models with itslimits. Resent research includes:(a) Martin Barlow and Robert Masson, in [BM11], looked at the geometry of UST indimension two. They also studied SRW, on UST clusters.(b) M. Barlow, David Croydon and Takashi Kumagai in [BCK17] refined Schramm’s resulton scaling limit of UST.(c) M. Barlow and Antal Jarai in [BJ19] studied the size of the balls of USF for dimensionsd ≥ 5 and give estimates for the probabilities of these sizes.(d) Tom Hutchcroft and Asaf Nachmias [HN19] showed that FSF in a planar graph is a.s.a tree.(e) T. Hutchcroft and Yuval Peres in [HP19] showed that qualitatively, the USF in Zq isvery different from that in Zp for q > p ≥ 4.(f) T. Hutchroft in [Hut18] showed if a property for k-tuples of components in the USF ofZd is stable under finite modifications of the forest, then a.s. every k-tuple will satisfythe property or a.s. no component will.(g) T. Hutchroft in [Hut20] showed that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of atleast quintic volume growth and every bounded degree nonamenable graph.13Chapter 1. Introduction(h) In d = 3, there are current works on the scaling limit of UST. Recently (March 2020) in[ACHS20], Omer Angel, David Croydon, Sara´ı Herandez Torres, and Daisuke Shiraishi,showed that the law of UST is tight under rescaling of the space.There are still several open questions regarding USF in Zd, for example, in d ≥ 5, it is con-jectured that the scaling limit of the components of the USF will be the Brownian continuumrandom tree.§ 1.9. Definition of Γd(λ).Recall at the beginning of this chapter the definition of Γd(λ). For future reference, we restatethe definition of its conductances.(1.9.1) µ((n, x), (n′, x′)) = eλmax(n,n′), for all (n, x) ∼ (n′, x′).As mentioned earlier, the conductances at the vertices of Γd(λ) (1.1.3) either decay or growexponentially, this makes Γd(λ) substantially different from networks studied in the literatureso far. Most the research has focused on the study of USF of Zd with conductances c(e) = 1everywhere. This was then generalised to different cases, among others, 0 < infec(e) ≤supec(e) < ∞ (“bounded conductances”) or assuming that there existed an index α > 0such that for every ball of vertices µ(B (x; r)) ≤ Crα (“polynomial growth”) the constant Cdepending solely on the network but not on the vertices or radii (r ≥ 1). Neither of thesehypotheses hold in Γd(λ).z 1.9.1 Biased random walk.We make special mention of an article appearing in the arXiv.org in 2018. In [SSS+18] ZhanShi, Vladas Sidoravicius, He Song, Longmin Wang, Kainan Xiang investigated properties ofUSF for a random walk they called “biased random walk.” They studied the network Zd withassignment of conductances cλ(e) = λ−|e|, where |e| is the graph-distance from the origin tothe edge e and λ ∈ (0, 1) is a fixed parameter. The network considered in this paper andΓd(λ) both have conductances which are far from uniform. However, the two assignments ofconduces give rise to(a) very different random walks, and(b) very different properties of the uniform spanning forest.14Chapter 1. IntroductionIn [SSS+18] the network random walk has a drift away from the origin, while in our case ithas a uniform drift towards the right. Our assignments of conductances make the networktransitive, while theirs is not. They exploited this fact by noticing that the axes play aspecial roˆle. This also leads to quite different properties of the uniform spanning forest. Forexample, the USF in [SSS+18] has 2d trees if d = 2, 3, while in our case this number is one.Their methods and ours are also different: they used “spectral radius” and “speed” of therandom walk, while we will not mention these. Another display of the difference in natureof the results in their paper when compared to this thesis is when they count the number ofintersections of random walk paths. They estimated these intersections by using a local limittheorem (a variant of central limit theorem). Typically, a local limit theorem approximatesthe probability P (Sn = z) using the density of a normal distribution plus an adjustmentor error term. This error terms are typically of a polynomial decay in ‖z‖ . We will showthat the Green’s function in Γd(λ) has exponential decay in some directions. Thus, a locallimit theorem did not prove useful for us. Finally, we also mentioned another contrast inthe two assignment of conductances. The fact that cλ(e) is bounded from below allowedthem to prove one-endedness using the isoperimetric condition [LP16, Theorem 10.43]. Ourconductances decay exponentially, and so we cannot use the isoperimetric condition for one-endedness; a new technique was derived from scratch to prove it.§ 1.10. Summary of results of the thesis.Each chapter will contain a main theorem, sometimes divided into several cases, for examplearising from dimensionality differences.The first main theorem we will show is regarding the bounds of the Green’s function(1.1.9). If we denote by S = (B,X) the network random walk of Γd(λ), after n steps, theexpected position will be around a point of the form (an, x) ∈ R×Rd with ‖x‖ ≤ cn 12 . Thus,Sn lies inside the “parabola” {(n, x) ∈ Z×Zd; ‖x‖2 ≤ cn}. Notice this parabola has about n d2points, each of them roughly equally likely. This suggests that in this parabola, the expectednumber of times the random walk visits any vertex is about n−d2 . It is also easy to argue thatbacktracking is exponentially costly, so the expected number of visits to points in Z−×Zd isnaturally of the form e−c‖z‖. Finally, there exists a transition between the parabola and thisexponential decay; indeed, if z = (n, x) is a point in which kn12 ≤ ‖x‖ ≤ 2kn 12 (so that therandom walk X needs to deviates from its mean about k times), then the expected number ofvisits to z will be of the form e−ck2 ‖z‖− d2 . Explicitly, we will establish the following theorem.Theorem. (Theorem (2.6.1)) There exist four constants c1, c2, c3, c4 > 0 such that for all15Chapter 1. Introductionz = (n, x) ∈ Z× Zd, z 6= 0, we haveG(z) ≤ c1exp(−c2 ‖z‖) for ‖x‖ > nexp(−c2‖x‖2n)‖z‖− d2 for ‖x‖ ≤ nandG(z) ≥ c3exp(−c4 ‖z‖) for ‖x‖ > n,exp(−c4‖x‖2n)‖z‖− d2 for ‖x‖ ≤ nThe next main result we will establish is the following.Theorem. (Theorem (3.4.2)) On Γd(λ), we have WSF = FSF.This is done using the Liouville property and a coupling argument. We denote USF inlieu of WSF or FSF. Shortly after, we will also show that USF consists of a single tree whend = 1 and d = 2, and it consists of infinitely many infinite trees for d ≥ 3.Theorem. (Theorem (3.5.1)) In d = 1, 2, WSF of Γd(λ) is a.s. one tree; in d ≥ 3, there area.s. infinitely many trees.Using results of [LP16], the proof will essentially reduce to show that∑z∈Zd+1G(z)2 isinfinite in d = 1, 2 and finite in higher dimensions. The Green’s function estimates can beused to prove this theorem, albeit in a rather messy way. Using Fourier inversion theoremtogether with Plancherel’s theorem, we will provide a cleaner, faster proof of this result.After this, we will tackle one of the main results of this thesis, this happens in Chapter4. We will investigate the frequency of crossings between two independent network randomwalks. To be specific, we will show the following.Theorem. (Theorems (4.1.1), (4.1.2), Corollary (4.4.1.2)) Let S and S ′ be two independentnetwork random walks in Γd(λ) started at 0.(a) If d = 1, then a.s. there exists infinitely many n such that Sn = S′n = 0.(b) If d = 2, then a.s. there exist infinitely many pairs (n, n′) such that Sn = S ′n′ .(c) If d ≥ 3, there there exists a positive probability that for no pair (n, n′), Sn = S ′n′ .In Chapter 5 we will tackle another important geometrical property of the WSF, thenumber of ends. An end in a tree can be thought as an infinite path that never backtracks,in other words, as a ray (A.9). Since each component in WSF is known to be infinite, thetrees in the forest always have at least one ray. We will show that a.s. from every vertexstarts one and only one ray.16Chapter 1. IntroductionTheorem. (Theorems (5.1.1) and (5.2.11) and (5.3.1)) In Γd(λ), a.s. every tree in USF hasone end.The proof of this theorem is quite involved, specially for dimension d = 2 in which atechnique was created anew. Let me explain further. For d = 1, we use planar dualityand the rather easy-to-prove fact that Γ1(λ) is its own dual; the idea appears implicitly in[BLPS01]. For d ≥ 3 we will employ a mix of new and already in place techniques. We willshow that if 0 has at least two rays, then in fact, it has infinitely many and we will provethat this is rather difficult in a series of steps. In d = 2, we will employ an entirely newtechnique. The core idea follows. We will show that 0 has only one infinite non-backtrackingpath. To do this, we will show that the opposite happens with arbitrarily small probabilityby showing that the second non-backtracking path starts “far away from the left” of theorigin and the further away the smaller the chances of a second non-backtracking path. Ifz = (−n, x) is a point to the “left” of the origin (that is, n > 0) which is far away, then anindependent network random walk started at z has a very small chance of hitting zero, aboutn−d2 and by Wilson’s algorithm, the event where z is in the second infinite non-backtrackingpath started at zero has probability of order n−d2 . We show that there is a universal smallfraction δ > 0 such that with high probability there are about δnd2 branches started in theregion R = {−n} × B(0;n12)none of which is connected to 0. Then, we will show that anypoint in R has a high chance of hitting these branches before it can even hit zero. The detailsare rather technical and the proof is lengthy, but the geometrical idea was that simple.In Chapter 6 we will show that there exists a function z 7→ η(z) on the vertices ofΓd(λ) satisfying that (z, z′) 7→ η(z − z′) is a metric. In fact, if z = (n, x), then η(z) =max(|n| 12 , ‖x‖)and the probability that z and z′ are in the same component is aboutη(z − z′)−(d−2); the upper bound is then generalised to a finite set of vertices.Theorem. (Theorems (6.1.2) and (6.2.3)) There exists a metric η suchP (z and z′ are in the same USF-component)  η(z − z′)−(d−2).This is done using techniques already established in [BKPS04]. We will show that theprobability on the left is bounded above and below by multiples of∑z′′∈Zd+1G(z, z′′)G(z′, z′′)and we will show, studying different types of sums, that this sum has precisely lower andupper bounds as stated. There are no new techniques but just calculations for this result.In Chapter 7 we will show that almost surely any two spanning trees (USF-components)in Γd(λ) are one edge apart for 3 ≤ d ≤ 6; that every two trees are at most two edges apartfor 7 ≤ d ≤ 10 and there exists two trees two edges apart; and so on, the separation increasesin one for every fourth increase in dimension. Specifically,17Chapter 1. IntroductionTheorem. (Theorem (7.4.1)) Let D(z, z′) be the minimal number of Zd+1-edges outside theUSF-forest that connects the tree at z and that at z′. Almost surely, maxz,z′D(z, z′) =¢d− 24•.The proof of this is done by adaptation of most of the methods of [BKPS04] to thenetwork Γd(λ).§ 1.11. On the exposition of this work.Most of the thesis has been written pursuing the goal of making the intuition behind theproof clear and yet provide mathematically correct statements without many large gaps forthe reader. To achieve this, we have compromised mostly in the extension of this text,admittedly longer than it would usually be if we had followed other styles of presentation.We have decided for this approach of the material since the author, firstly, believes this is thebest way to deliver mathematics, and secondly, hopes this text can be read at the graduatestudent level of mathematics. By personal experience, the author knows that the currentliterature in the topic can be a difficult first read and often the proofs of complicated resultshave hidden all intuition behind them. We also believe that any shortening will come atthe cost of losing at least one of these two principles. Similarly, consistency of the theory ispresented, in particular, we avoided awkward situations such as developing results in somecontexts, say graphs where multiple edges are banned, and later having to admit that thiswas not enough and assume the results were both defined and proved in a more generalcontext. We leave no major gaps to the reader, and it is hoped the annexes at the end willfill any gaps the reader may have and which the author encountered when initially learningabout these topics of random walks and electrical networks.To prevent repetition of the same phrase many times over and again, we often replaceexpressions like “if and only if” with other ones. It will always be clear and obvious fromcontext what we mean. In the example of “if an only if”, we will often replace it with“signify”, “is the same as saying”, “is equivalent to” amongst others. We followed a standardapproach to numbering results. Primarily we followed a global numbering system indicatedby an expression of the form (a.b.c) : a is the chapter number, b the section and c the resultof that section; numberings (a.b.c.d) also will make rarer appearances, the meaning of thefirst three numbers will be the same, and d will mean that this is the dth partial result,usually needed in a longer proof and such that the result may be interesting in and of itself.Local numbering systems (only used to reference steps or equations inside a long proof) arealso used, these will be indicated by the common notation (∗), (∗∗) or with the alphabet:(A), (B), etc.18Chapter 1. IntroductionWe use the terms “σ-algebra” and “σ-field” indistinguishably from each other and we em-ploy common notation from probability. The term “random variable” will mean a real-valuedmeasurable function defined on a probability space; we use “random object” or “random el-ement” to refer to a measurable function with values on some measurable space and definedon a probability space. Given a family of random elements (Zα)α∈J we denote σ(Zα;α ∈ J)to be the minimal σ-field making measurable all the Zα, and if A and B are two σ-algebrasthen A ∨B = σ(A ∩ B; A ∈ A ,B ∈ B). If (Sn)n∈Z+ is any discrete time stochastic pro-cess, we use F Sn = σ(Sk; 0 ≤ k ≤ n). If Z is a random object, defined on a probabilityspace (Ω,F ,P), with values on a measurable space (X,X ), the measure on X given byB 7→ P (Z ∈ B) is known in probability as the “distribution” or “law” of Z; it is simply theimage measure of P by the function Z, in analysis this would be denoted as Z(P), however,it makes more sense, from a probabilistic lens, to denote it as P (Z ∈ dx) .We digress in this paragraph to mention some notation that will be employed throughoutthe rest of the document. Suppose ϕ and σ are two non-negative functions defined on someset Z, the notation ϕ  σ or ϕ(z)  σ(z) for z ∈ Z signifies the existence of constants a, b > 0such that for every z ∈ Z,aϕ(z) ≤ σ(z) ≤ bϕ(z);it is obvious that this defines an equivalence relation on the set of functions defined on Z. Ingeneral, positive constants appearing on the proofs will be denoted with the letter c (lowerand upper case used) and affixes such as sub and superscripts, and apostrophes, will beemployed to distinguish different constants, unless an assertion to the contrary is explicitlyexpressed. All constants may depend on the dimension and on the parameter λ in thedefinition of Γd(λ). Often we will need to handle many constants in a long proof, to preventthe affixes to grow unreasonably, we will often claim multiple times in a proof, the existenceof a constant c > 0 satisfying certain properties, it is tacitly assumed that the reader knowsthat different claims of existence of constants c will mean that the constants themselves maybe different, even when the same letter c is employed multiple times in the proof. We willnot, however, and contrary to what is somewhat common, use the same letter c to denotedifferent constants inside a given sentence, in other words, if in a given equation the constantc appears multiple times, then in that equation, all these c are the same constant.§ 1.12. Strong Markov property.This section arguably contains no new material, but is here for completeness and providea proof of the strong Markov property for Markov processes indexed on Z2+ (as opposed toZ+). We recall the following definition. A set of events P in a probability space is called19Chapter 1. Introductiona pi-system if the relation J,K ∈P implies J ∩ K ∈P. A set L of events in a probabilityspace is called a λ-system if:(a) Ω ∈ L ,(b) the relation A ∈ L implies A{ ∈ L , and(c) the two relations An ↑ A, each An ∈ L , imply A ∈ L .The following is a well-known theorem (“Dynkin’s theorem”), it can be found in [Ash72,4.1.2]: if a λ-system contains a pi-system, then it also contains the σ-algebra generated bythe pi-system. Perhaps, the most used application of Dynkin’s theorem is the following result:if µ and ν are two finite measures on a measure space (X,X ) such that µ(X) = ν(X) andµ = ν on a pi-system X0 that generates X , then µ = ν. (The set of all elements G ∈ Xsuch that µ(G) = ν(G) is a λ-system.)Proposition ( 1.12.1 ) Let (Ω,F ,P) be a probability space and suppose that H0 and K0are two subsets of F such that, for all H ∈H0 and all K ∈ K0,P (H ∩K) = P (H)P (K) .If H0 and K0 are pi-systems, then H = σ(H0) and K = σ(K0) are independent σ-fields.The proof is a simple application of Dynkin’s theorem; we leave the details to the reader.(For fixed H, the set of K satisfying the relation P (H ∩K) = P (H)P (K) is a λ-system,so the relation holds for any H ∈ H0 and all K ∈ K , the same argument then gives theconclusion).Proposition ( 1.12.2 ) Suppose Z and Z ′ are two independent random objects defined onsome probability space (Ω,F ,P) with values in corresponding measurable spaces (X,X ) and(X′,X ′). Suppose that H ⊂ σ(Z) and H ′ ⊂ σ(Z ′) are two σ-algebras. If ϕ : X → R andϕ′ : X′ → R are two measurable bounded functions (relative to the Borel sets and X or X ′,respectively) and K = H ∨H ′ thenE (ϕ(Z)ϕ′(Z ′) | K ) = E (ϕ(Z) |H )E (ϕ′(Z ′) |H ′) .The proof is also a standard application of Dynkin’s theorem. (By the usual way ofstarting with indicator functions and then a passage to general non-negative measurablefunctions using monotone class arguments suffices to show the result when ϕ and ϕ′ areindicator functions. In this case, we need to showP ({Z ∈ A} ∩ {Z ′ ∈ A′} ∩ E) =∫EE (1A(Z) |H )E (1A′(Z ′) |H ′) dP20Chapter 1. Introductionfor E ∈ K . Again, Dynkin’s theorem shows it suffices to establish it for events E = F ∩ F′with F ∈H and F′ ∈H ′. This case holds by independence.)Recall the following general construction. Suppose Z is a random object, defined on someprobability space (Ω,F ,P), with values in a measurable space (X,X ). If Y is a randomvariable, then there exists a measurable function (relative to the Borel sets of R and X ),say g, such that ∫{Z∈A}Y dP =∫Ag(x)P (Z ∈ dx)(The proof is easy: the left hand side defines a measure on A which is absolutely continuouswith respect to the distribution of Z, the result follows upon invoking Lebesgue-Nikodymtheorem.) We will write E (Y | Z = x) to denote g(x); this function is unique up to nullevents of Z. We may call g a “version of Y given Z.”Consider a discrete time Markov process (Sn,Fn)n∈Z+ , with values on a metrisable sepa-rable space E, endowed with Borels sets E as σ-algebra. Let η be an a.s. finite stopping timefor this Markov process, that is to say, we assume {η ≤ n} ∈ Fn for every n ∈ Z+. We mayconstruct the σ-field Fη of all events G ∈ E such that G ∩ {η ≤ n} ∈ Fn. We may call Fηthe “stopped filtration up to time η relative to the given Markov process.” In a similar way,we may construct Sη : ω 7→ Sη(ω)(ω), defined in this way almost everywhere. If no filtrationis specified, it is assumed the canonical filtration is used F Sn = σ(Sk; 0 ≤ k ≤ n). Considernow the product space EZ+ , which is also a metrisable separable space, and its Borel σ-field,which coincides with⊗n∈Z+E . For any bounded measurable function ϕ : EZ+ → R, thereexists a version of the conditional expectation E(ϕ (Sj)j∈Z+∣∣∣S0 = x) ; denote this functionas Ex(ϕ (Sj)j∈Z+). The strong Markov property states thatE (ϕ ((Sj)j≥η)|Fη) = ESηÄϕÄ(Sj)j∈Z+ää.We will prove an extension of this result to two simultaneously occurring stopping times fortwo independent Markov processes.Proposition ( 1.12.3 ) Suppose S = (Sn)n∈Z+ and S′ = (S ′n)n∈Z+ are two independentMarkov processes, defined on some probability space (Ω,F ,P) with values on some metris-able, separable space E, endowed with Borel σ-field E . Let ψ : EZ+ ×EZ+ → R be a boundedmeasurable function, relative to the Borel sets of both R and EZ+ × EZ+ . Denote by vψ aversion of ψÄSj, S′j′ä(j,j′)∈Z2+given (S0, S′0), that isvψ(x, x′) = EÅψÄSj, S′j′ä(j,j′)∈Z2+∣∣∣∣S0 = x, S ′0 = x′ã ,21Chapter 1. Introductionfor (x, x′) ∈ E2. Suppose η and η′ are two stopping times, η relative to S and η′, to S ′. Denoteby F Sη and FS′η′ the corresponding stopped filtrations up to times η and η′ for the processesS and S ′. With ψ as before,EÄψ(Sj, S′j′)j≥η,j′≥η′∣∣∣F Sη ∨F S′η′ ä = vψ(Sη, S ′η).Proof. As is usual, if the result holds true for an indicator function, say ψ = 1E×E′ , thena usual passage to any measurable bounded function using monotone class arguments willcomplete the proof. In this case, we may apply (1.12.2) with ϕ = 1E, ϕ′ = 1E′ , H = F Sη ,H ′ = F S′η′ . This gives, together with the strong Markov property applied once for S andonce for S ′,EÄ1E(Sj)j≥η1E′(S ′j′)j′≥η′∣∣∣F Sη ∨F S′η′ ä = ESη Ä1E(Sj)j∈Z+äES′η′ Ä1E′(S ′j′)j′∈Z+ä .Thus, it only remains to show thatvψ(x, x′) = ExÄ1E(Sj)j∈Z+äEx′ Ä1E′(S′j′)j′∈Z+ä.Denote by u(x, x′) the expression on the right hand side above. Then, u(x, x′) is a versionof the condition expectation 1E×E′(Sj, S ′j′)(j,j′)∈Z2+ given (S0, S′0) if and only if∫Au(x, x′)P((S0, S ′0) ∈ d(x, x′)) =∫{(S0,S′0)∈A}1E×E′(Sj, S ′j′)(j,j′)∈Z2+dP,for every Borel set A of E × E. Thus, by proceeding in the usual way, starting with anindicator function and employing monotone class arguments, suffices to show the case inwhich A is a rectangle, namely A = G × G′. Also observe that independence guaranteesP((S0, S′0) ∈ d(x, x′)) = P(S0 ∈ dx)P(S ′0 ∈ dx′). In this case,∫G×G′u(x, x′)P((S0, S ′0) ∈ d(x, x′))=∫GExÄ1E(Sj)j∈Z+äP(S0 ∈ dx)∫G′Ex′ Ä1E′(S′j′)j′∈Z+äP(S ′0 ∈ dx′)= PÄS0 ∈ G, (Sj)j∈Z+ ∈ EäPÄS ′0 ∈ G′, (S ′j′)j′∈Z+ ∈ E′ä= PÄS0 ∈ G, (Sj)j∈Z+ ∈ E, S ′0 ∈ G′, (S ′j′)j′∈Z+ ∈ E′ä=∫{(S0,S′0)∈G×G′}1E×E′(Sj, S ′j′)(j,j′)∈Z2+dP,as desired. The proof is complete.22Chapter 1. IntroductionFor the following proposition, we will need two additional definitions. Since we will bedealing with sums of random objects, it is necessary to assume a linear structure on the space.Consider then E = Rd, both the proofs and concepts below apply equally well when Rd isreplaced by a separable (real) Fre´chet space. An E-valued stochastic process (Sn,Fn)n∈Z+is said to have independent increments if for every pair (j, k) ∈ Z+ ×N, we have Sk − Sj isindependent of Fj. Every process with independent increments is a Markov process. (Theproof of this fact can be derived as follows: if f in any bounded measurable function, thenE (f(Sj+k) | Fj) = E (ϕ(Sj+k − Sj, Sj) | Fj) ,where ϕ(x, y) = f(x + y), and we may apply the usual way and reduce to ϕ = 1A×B,which then follows from definition.) An E-valued stochastic process (Yn,Fn)n∈Z+ is said tohave time homogeneity or stationarity if the distribution of the EZ+-valued random object(Yn)n∈Z+ is the same as that of (Yn+k)n∈Z+ for every k ∈ N. A necessary and sufficientcondition for a process to be stationary is that for all times n1 < . . . < nk and every h ∈ N,(Yn1 , . . . , Ynk) ∼ (Yn1+h, . . . , Ynk+h) ,that is to say, they have the same distribution. (The proof of this easy: stationarity signifiesthat for every h and every Borel set A of EZ+ and every h ∈ N,PÄ(Yn)n∈Z+ ∈ Aä= PÄ(Yn+h)n∈Z+ ∈ Aä,we can then apply Dynkin’s theorem to the pi-system of rectangles.) Thus, if an E-valuedstochastic process (Sn,Fn)n∈Z+ possesses stationary independent increments, then we mayset Yk = Sk+1 − Sk, and we see that(a) Yk is independent of Fk−1 while being measurable relative to Fk;(b) S0 is measurable relative to F0; and(c) there is a distribution on E that each Yk follows.There is a partial reciprocal result, namely, if S0 = 0, and (Yk)k∈N is a family of independentE-valued random objects satisfying the three conditions above for a given filtration (Fn)n∈Z+ ,then (Sn,Fn)n∈Z+ is a process with independent and stationary increments. (The proof iseasy, suffices to show stationarity on rectangles by Dynkin’s theorem, and independence ofthe Yk readily gives it.) In particular, a network random walk Γd(λ) started at zero, is aZd-valued process which possesses stationary independent increments.23Chapter 1. IntroductionProposition ( 1.12.4 ) Let E be as above. Suppose (Sn)n∈Z+ and (S′n)n∈Z+ are two E-valued stochastic processes with stationary independent increments. Suppose that η and η′are stopping times relative to S and S ′, respectively. Then, F Sη ∨ F S′η′ is independent ofG Sη ∨ G S′η′ , with G Sη = σ(Sη+j − Sη; j ∈ Z+), and a corresponding definition for G S′η′ with(S ′, η′) replacing (S, η). In particular, if A is an event depending on the paths (Sj)0≤j≤η and(S ′j′)0≤j′≤η′ and B, depending on the paths (Sη+j − Sη)j∈Z+ and (S ′η′+j′ − S ′η′)j′∈Z+ , then Aand B are independent.Proof. Observe that F Sη ∨F S′η′ is generated by all events A ∩ A′, with A ∈ σ(Sn;n ∈ Z+)and A′ ∈ σ(S ′n;n ∈ Z+) such that for any pair (m,m′) ∈ Z2+, we have(∗) A ∩ A′ ∩ {η = m, η′ = m′} ∈ F Sm ∨F S′m′ .Similarly, G Sη ∨ G S′η′ is, by definition, generated by the events(∗∗) ¶(Sη+1 − Sη, . . . , Sη+k − Sη) ∈ G, ÄS ′η′+1 − S ′η′ , . . . , S ′η′+k′ − Sη′ä ∈ G′© ,where k and k′ run on all non negative integers, G is a Borel set of Ek, and G′ is a Borel setof Ek′. Each of the set of events satisfying (∗) or (∗∗) is a pi-system. Thus, (1.12.1) shows itsuffices to prove independence for these type of events. So, let A∩A′ as in (∗) and considertwo non negative integers k, k′, together with two Borel sets, G of Ek and G′ of Ek′. Then,by denoting ∆m,j(S) = Sm+j − Sm, and with an obvious modification for ∆m′,j′(S ′),PÄA ∩ A′ ∩ (Sη+1 − Sη, . . . , Sη+m − Sη) ∈ G,ÄS ′η′+1 − S ′η′ , . . . , S ′η′+m′ − Sη′ä ∈ G′ä=∑(m,m′)PÄA ∩ A′ ∩ {η = m, η′ = m′} ∩ ¶(∆m,j(S))j ∈ G, (∆m′,j′(S ′))j′ ∈ G′©ä ,where the indices run on (m,m′) ∈ Z2+, 1 ≤ j ≤ k and 1 ≤ j′ ≤ k′. Clearly,{∆m,j(S) ∈ G,∆m′,j′(S ′) ∈ G′} ∈ G Sm ∨ G S′m′ ,and stationarity givesP (∆m,j(S) ∈ G,∆m′,j′(S ′) ∈ G′) = P (∆0,j(S) ∈ G,∆0,j′(S ′) ∈ G′) .For the time being, we assume F Sm ∨F S′m′ to be independent of G Sm ∨ G S′m′ , we will prove thisresult after concluding the proposition with it. In this case,PÄA ∩ A′ ∩ (Sη+1 − Sη, . . . , Sη+m − Sη) ∈ G,ÄS ′η′+1 − S ′η′ , . . . , S ′η′+m′ − Sη′ä ∈ G′ä=∑(m,m′)P (A ∩ A′ ∩ {η = m, η′ = m′})P Ä(∆m,j(S))j ∈ G, (∆m′,j′(S ′))j′ ∈ G′ä= PÄ(∆0,j(S))j ∈ G, (∆0,j′(S ′))j′ ∈ G′ä ∑(m,m′)P (A ∩ A′ ∩ {η = m, η′ = m′})= PÄ(∆0,j(S))j ∈ G, (∆0,j′(S ′))j′ ∈ G′äP (A ∩ A′) .24Chapter 1. IntroductionNext, take A = A′ = Ω, this givesPÄ(∆η,j(S))j ∈ G, (∆η′,j′(S ′))j′ ∈ G′ä= PÄ(∆0,j(S))j ∈ G, (∆0,j′(S ′))j′ ∈ G′ä.Heretofore we showed that F Sη ∨F S′η′ is independent of G Sη ∨G S′η′ provided that for every pair(m,m′) ∈ Z2+, F Sm ∨F S′m′ is independent of G Sm ∨ G S′m′ . To show the independence betweenthese two σ-algebras we proceed in a similar way as before: all reduces to showP (A ∩ A′ ∩ B ∩ B′) = P (A ∩ A′)P (B ∩ B′) ,where A ∈ F Sm, A′ ∈ F S′m′ , B ∈ G Sm and B′ ∈ G S′m′ . Independence between the processes Sand S ′ showP (A ∩ A′ ∩ B ∩ B′) = P (A ∩ B)P (A′ ∩ B′) ,P (A ∩ A′)P (B ∩ B′) = P (A)P (A′)P (B)P (B′) ,and independent increments showP (A ∩ B)P (A′ ∩ B′) = P (A)P (B)P (A′)P (B′) .The proof is complete.25Chapter 2Green’s function boundsThe purpose of this chapter is to obtain upper and lower bounds on the Green’s function(1.1.10).§ 2.1. Basic statistics of Γd(λ).We will often write z ∈ Zd+1 = Z × Zd as z = (n, x1, . . . , xd) = (n, x) and we will referto n as the “drifted coordinate” or the “zeroth coordinate.” A unit vector in Zd+1 has onecoordinate equal to 1 or -1 and all the rest equal to zero; for simplicity, u and ui will denoteunit vectors in Zd+1 and ui is such that its ith coordinate is non zero (0 ≤ i ≤ d). Thefunctions pri : Zd+1 → Z, for i = 0, 1, . . . , d, are the projections and are defined in theobvious way by(2.1.1) (n, x1, . . . , xd) 7→{n if i = 0,xi otherwise .We will call Zui the “ith factor” as well as “ith axis” (i = 0, . . . , d). Thus, Z∗ui is theith factor with the origin removed. Let λ > 0, fixed throughout the rest of this document.Recall Γd(λ) is the graph Zd+1 with conductances (1.9.1). From these conductances, onecan get the transition kernel for network random walk. The probability that the next stepshould be u ∈ Zd+1, a unit vector with our conventions, is given by(2.1.2) p(u) =12d+ 1 + eλif u = (−1, 0) or u = ui, i = 1, . . . , d,eλ2d+ 1 + eλif u = (1, 0).26Chapter 2. Green’s function boundsThe12-lazy version of the above transition density is, by definition (1.1.12),(2.1.3) pL(u) =12(2d+ 1 + eλ)if u = (−1, 0) or u = ui, i = 1, . . . , deλ2(2d+ 1 + eλ)if u = (1, 0)12if u = 0.Below, we will use both distributions (2.1.2) and (2.1.3) according to which one gives thebest fit for the problem tackled.Denote by ζ · εt the measure on Zd such that it has total mass equal to ζ at the vertext. Notice that if Y =ÄY (0), . . . , Y (d)äis a random element of Zd+1 with distribution (2.1.2),then (i = 1, . . . , d)Y (0) ∼ 12d+ 1 + eλÅε−1 + 2d · ε0 + eλ · ε1ã,Y (i) ∼ 12d+ 1 + eλÅε−1 + (2d− 1 + eλ) · ε0 + ε1),(2.1.4)which immediately allows one to conclude thatE (Y ) = (a, 0) =(eλ − 12d+ 1 + eλ, 0),Cov (Y ) =[σ20 00 σ2Id]=eλ + 12d+ 1 + eλ0022d+ 1 + eλId ,(2.1.5)where Id is the identity matrix on Rd. From the Central Limit Theorem, it follows immedi-ately that if (Yn) is a family of independent random elements following (2.1.2),1√nn∑j=1(Yj − (a, 0)) weakly−−−→n→∞ Norm(0;[σ20 00 σ2Id]).Consider Y following law (2.1.2). The Fourier transform of Y is(2.1.6) ϕY (h) = EÄei(h|Y )ä=eλeih0 + e−ih02d+ 1 + eλ+22d+ 1 + eλd∑j=1coshj,where h = (h0, . . . , hd) ∈ [−pi, pi]d+1. Observe also ϕY (h) = 1 if and only if 1 = eih0 = coshjfor j = 0, . . . , d, and this is the same as saying that h0 = . . . = hd = 0.27Chapter 2. Green’s function bounds§ 2.2. Continuous time network random walk.Recall the definition of the continuous time network random walk (1.1.14). In Γd(λ), we canwrite(2.2.1) S˜t =ÄB˜t, X˜tä, (t ≥ 0).The following properties come from well-known facts regarding Poisson processes or areotherwise given a reference. Setp = P (Y = u) = (2d+ 1 + eλ)−1.(a) B˜ and X˜ are independent continuous-time Markov processes.(b) X˜t = XMt , where X is a standard random walk in Zd and M = (Mt)t≥0 is a Poissonprocess in Z+ with constant intensity equal to 2dp =2d2d+ 1 + eλ.(c) B˜t = BLt , where B is a biased random walk in Z with jumps to ±1, L = (Lt)t≥0 is aPoisson process in Z+ with constant intensity equal to σ20 = (1 + eλ)p =1 + eλ2d+ 1 + eλ,and the probability of B having a positive jump iseλ1 + eλ.(d) Let qX be the transition probability of X˜, that is, qX(t, y) = qXt (y) = P0ÄX˜t = yä. Then[Bar17, Theorem 5.17] provides upper bounds(2.2.2) qX(t, y) ≤2dexp(− ‖y‖22dpe2t)if ‖y‖ ≤ 2dpet,12dexpÇ−2dpt− ‖y‖ logÇ ‖y‖2dpetååif ‖y‖ ≥ 2dpet;and lower bounds, valid for ‖y‖ ≥ 2dpt > 0,(2.2.3) qX(t, y) ≥ c expÇ−c′ ‖y‖ñ1 + logÇ ‖y‖2dptåôå.(e) By Theorem 6.19 bearing in mind Definition 3.28, Corollary 3.30 and Definition 5.19 of[Bar17] there exists four constants ci > 0 (i = 1, . . . , 4) such that for all (t, y) for which2dpt > ‖y‖ ,28Chapter 2. Green’s function boundsqXt (y) ≤ c1t−d2 e−c2‖y‖2t ,qXt (y) ≥ c3t−d2 e−c4‖y‖2t .(2.2.4)(f) There exists a pair of universal constants c, c′ > 0 such that for all r, T ≥ 1PÇsup0≤s≤T∥∥∥X˜s∥∥∥ ≥ rå ≤ ce−c′ r2T .This follows from Theorem 4.33 of [Bar17] applied to the standard random walk (soβ = 2), you will need to recall Definitions 4.14 and 4.18, and use Lemma 4.20, whichholds with α = d and β = 2 for the standard random walk on Zd.We remark, once and for all, that in Γd(λ) we have the following relation: for whateverthe vertices z and z′ may be, G(z, z′) = G(0, z′ − z) (here G is the Green’s function ofΓd(λ)). The proof of this relation is immediate bearing in mind that in Γd(λ) one hasSn = S0 + Y1 + . . .+ Yn.We conclude this section with the following easy remark.Remark ( 2.2.5 ) Let c > 0.(a) If z = (n, x) satisfies |n| ≤ c ‖x‖ , then ‖x‖ ≤ ‖z‖ ≤ √1 + c2 ‖x‖ .(b) If z = (n, x) satisfies ‖x‖ ≤ c|n|, then |n| ≤ ‖z‖ ≤ √1 + c2|n|.(c) There exists c′ > 0 such that for all z ∈ Zd+1, e−c‖z‖ ≤ c′ ‖z‖− d2 .§ 2.3. Some estimates of sums.We collect together here some estimates of sums which will be used later.All sums over x in this section will be taken on the lattice Zd and sums over jon the integers Z.Proposition ( 2.3.1 ) Let L ∈ N and f : [L,∞)→ R∗+ be a monotone function such for apair a, b > 0, we have af(j + 1) ≤ f(j) ≤ bf(j + 1) for integers j ≥ L. Let r > L+ 1. Then,∑L≤‖x‖≤rf(‖x‖) ∫L≤‖x‖≤rdx f(‖x‖) = σdr∫Ldt f(t)td−1,where σd is the “surface area” of the d-dimensional sphere and any implicit constant maydepend solely on dimension and on the pair (a, b) but not on L or r or otherwise on f.29Chapter 2. Green’s function boundsProof. Since the function x 7→ f(‖x‖) is radial, the equality in the conclusion of the propo-sition is a well-known fact from measure theory. Set dre to be the smallest integer k ≥ r.We have∑L≤‖x‖≤rf(‖x‖) ≤dre∑j=L+1∑j−1≤‖x‖≤jf(‖x‖) dre∑j=L+1f(j)jd−1 r∫Ldt f(t)td−1.And, similarly,∑L≤‖x‖≤rf(‖x‖) ≥ ∑L+1≤j≤r∑j−1≤‖x‖≤jf(‖x‖)  ∑L+1≤j≤rf(j)jd−1 r∫Ldt f(t)td−1,and this completes the proof.In the following proposition, we will consider sums as in the previous one for the func-tion f(t) = tae−γtb, and regard the parameters a and b as constant. We are aiming atapplying the previous Proposition (2.3.1). The exponential term makes the application notstraightforward and thus, some sort of localisation will be employed.Proposition ( 2.3.2 ) Let a, b, γ > 0 and r > 2. Then,∑‖x‖≤r‖x‖a e−γ‖x‖b ra+d if γrb ≤ 1γ−a+db if γrb ≥ 1 = minÄrb, γ−1äa+db .(Any implicit constant may depend on a, b and dimension but not on γ or on r.)Proof. Consider the two cases.Case 1: γrb ≤ 1. In this case, 0 ≤ γ ‖x‖b ≤ 1 on the ball ‖x‖ ≤ r, thus e−1 ≤ e−γ‖x‖b ≤ 1on this ball. Then ∑‖x‖≤r‖x‖a e−γ‖x‖b  ∑‖x‖≤r‖x‖a .We can apply (2.3.1) with f(t) = ta for t ≥ 1. This gives∑‖x‖≤r‖x‖a e−γ‖x‖b r∫1dt ta+d−1 =ra+d − 1a+ d ra+d.Case 2: γrb > 1. There exists a unique integer J ≥ 2 such that 2J−1 < r ≤ 2J . Then,∑‖x‖≤r‖x‖a e−γ‖x‖b ≤J∑j=1∑2j−1≤‖x‖≤2j‖x‖a e−γ‖x‖b .30Chapter 2. Green’s function boundsThe set of points x ∈ Zd satisfying 2j−1 ≤ ‖x‖ ≤ 2j has cardinality bounded above andbelow by universal constant multiples of 2jd (the constants may depend on dimension).Then, for a constant c > 0,∑‖x‖≤r‖x‖a e−γ‖x‖b ≤ cJ∑j=12j(a+d)e−γ2(j−1)b J∫0dt 2t(a+d)e−γ2tb.Apply the change of variables s(t) = 2t, to obtainJ∫0dt 2t(a+d)e−γ2tb 2J∫1ds sa+d−1e−γsb.Consider now the change of variables u(s) = γsb, this gives2J∫1ds sa+d−1e−γsb=1bγ−a+db2Jb∫1du ua+db−1e−u  γ−a+db ,completing the proof of the upper bound.To prove the lower bound we proceed in a similar manner as in the upper bounding.∑‖x‖≤r‖x‖a e−γ‖x‖b ≥J−1∑j=1∑2j−1<‖x‖≤2j‖x‖a e−γ‖x‖b≥ cJ−1∑j=12j(a+d)e−γ2jb J∫1dt 2t(a+d)e−γ2tb.The proof of the lower bound now proceeds as that for the upper bound.With the two cases completed, the proof of the proposition is also complete.The following proposition handles the case a = 0. We need to exclude the origin in thesum for the term corresponding to x = 0 in e−γ‖x‖bis one and, thus, it has no decay orgrowth.Proposition ( 2.3.3 ) Let b, γ > 0 and r > 2. Then∑0<‖x‖≤re−γ‖x‖b  min Ärb, γ−1ä db ;thus, if r−b ≤ γ ≤ 1, ∑‖x‖≤re−γ‖x‖b  γ− db .(Any implicit constant may depend on b and dimension but not on γ or on r.)31Chapter 2. Green’s function boundsProof. The second statement is an immediate consequence of the first one bearing in mindthat inside the stated range of γ, 1 ≤ γ−1 ≤ rb and that∑‖x‖≤re−γ‖x‖b= 1 +∑0<‖x‖≤re−γ‖x‖b.The proof of the first statement is a simple adaptation of that of (2.3.2) by just setting a = 0there. As a matter of fact, no other change is needed.Proposition ( 2.3.4 ) Let a, b > 0. There exists a pair of constants c1, c2 > 0 such that forall t ∈ Zd and all m ∈ N,(a) if ‖t‖ > bm,e−a‖t‖ ≤ ∑‖x‖≤bme−aÄ ‖x‖2m+‖x−t‖ä≤ c1e−c2‖t‖;(b) if ‖t‖ ≤ bm,e−a‖t‖2m ≤ ∑‖x‖≤bme−aÄ ‖x‖2m+‖x−t‖ä≤ c1e−c2‖t‖2m .Proof. The proof of the lower bounds follow by considering the term corresponding to x = 0in the first case and x = t, in the second. Let us handle the two cases for the upper boundsseparately.Case 1: ‖t‖ > bm. Then,∑‖x‖≤bme−aÄ ‖x‖2m+‖x−t‖ä≤ ∑‖x‖≤bm‖x−t‖≤ ‖t‖2e−aÄ ‖x‖2m+‖x−t‖ä+∑‖x−t‖> ‖t‖2e−aÄ ‖x‖2m+‖x−t‖äIn the first of these two sums, notice ‖x‖ ≥ ‖t‖2, whence‖x‖2m>b4‖t‖ , so∑‖x‖≤bm‖x−t‖≤ ‖t‖2e−aÄ ‖x‖2m+‖x−t‖ä≤ e−ab4 ‖t‖ ∑‖x−t‖≤ ‖t‖2e−a‖x−t‖ ≤Ñ∑x∈Zde−a‖x‖ée−ab4‖t‖.The second sum is easier since we bound e−a‖x‖2m ≤ 1 and e−a‖x−t‖ ≤ e−a4 ‖t‖e−a2 ‖x−t‖.Case 2: ‖t‖ ≤ bm. Consider the regions (j ∈ N)Rj = {x ∈ Zd | (j − 1)b2m ≤ ‖x‖2 ≤ jb2m}.32Chapter 2. Green’s function boundsThen, the relation ‖x‖ ≤ bm is the same as x ∈m⋃j=1Rj. Suppose first that t ∈ R1.The conclusion to be reached in this case is that the sum is bounded by a constantindependent of t and m. This is easy:∑‖x‖≤bme−aÄ ‖x‖2m+‖x−t‖ä≤ ∑x∈Zde−a‖x‖.Suppose then that t ∈ RJ for some J > 1. The conclusion is equivalent to show thatthe sum is bounded above by ce−c′J for two constants c, c′ > 0 independent of m andt. Here we have∑‖x‖≤bme−aÄ ‖x‖2m+‖x−t‖ä≤m∑j=1e−a(j−1)b2 ∑x∈Rje−a‖x−t‖.If j ≤ J4, then‖x− t‖ ≥ ‖t‖ − ‖x‖ ≥(√J − 1−»j)b√m ≥( J2− J4)b√J ≥√2− 12bJ,and thus, ∑x∈Rje−a‖x−t‖ ≤ e−Ä(√2−1)ab4äJ ∑x∈Zde−a2‖x‖.Hence, for a pair of constants c, c′ > 0,∑1≤j≤J4∑x∈Rje−a‖x−t‖ ≤ cJ4e−Ä(√2−1)ab4äJ ≤ c′e−Ä(√2−1)ab8äJ.IfJ4< j ≤ m, we can use simpler bounds, in the inner sum ∑x∈Rje−a‖x−t‖ ≤ ∑x∈Zde−a‖x‖,while the exponential in the outer sum can be bounded as e−a(j−1)b2 ≤ eab2e−ab24 J .Substituting these estimates gives the desired conclusion of this item of proposition.Having established both results, the proof is complete.Proposition ( 2.3.5 ) For every α > 1 and all integers m ≥ 2,1α− 1m−(α−1) ≤ ∑k≥mk−α ≤ 2α−1α− 1m−(α−1).33Chapter 2. Green’s function boundsProof. Simply notice that t 7→ t−α is decreasing and thus ∑k≥mk−α is an upper Riemann sumfor∞∫mdt t−α, while being a lower Riemann sum for∞∫m−1dt t−α ≥∞∫m2dt t−α.Proposition ( 2.3.6 ) For every a ∈ R and b ≥ 1, there exists a constant c = c(a, b) > 0such that for all γ > 1 ∑1≤j≤bγjae−γj ≤ cγa+1.Proof. Consider first the case a ≥ 0, then∑1≤j≤bγjae−γj ≤ ∑1≤j≤bγja ≤2bγ∫0dt ta =(2bγ)a+1a+ 1= c(a, b)γa+1.If a < 0, the function t 7→ tae− γt is increasing on 0 < t < −γaand decreasing for t > −γa.Thus, by breaking the sum at −γawe can bound from above the given sum as follows∑1≤j≤bγjae−γj ≤ c2bγ∫1dt tae−γt .Introduce the change of variables s(t) =γt, to obtain∑1≤j≤bγjae−γj ≤ cγa+11∫12bds s−(a+2)e−s = c(a, b)γa+1,as claimed.Proposition ( 2.3.7 ) There exists a universal constant c > 0 such that for all systems(a, p, q, r, t) with r > 2, a, p, q > 0 and t ∈ Zd,∑‖x‖≤re−aÄ ‖x‖2p+‖x−t‖2qä≤cminÅr2,paã d2e−a4‖t‖2q if ‖t‖ > 2rcminÅr2, paã d2e−a4‖t‖2q + min(‖t‖24,qa) d2e−a4‖t‖2p if ‖t‖ ≤ 2r.Proof. We prove both items simultaneously. We will use (2.3.3). Write(∗) ∑‖x‖≤re−aÄ ‖x‖2p+‖x−t‖2qä=∑‖x‖≤r‖x−t‖≤ ‖t‖2e−aÄ ‖x‖2p+‖x−t‖2qä+∑‖x‖≤r‖x−t‖> ‖t‖2e−aÄ ‖x‖2p+‖x−t‖2qä.34Chapter 2. Green’s function boundsWhen ‖x− t‖ > ‖t‖2,e−aÄ ‖x‖2p+‖x−t‖2qä≤ e−a4 ‖t‖2q e−a‖x‖2p .Summing over ‖x‖ ≤ r leads to∑‖x‖≤r‖x−t‖> ‖t‖2e−aÄ ‖x‖2p+‖x−t‖2qä≤ e−a4 ‖t‖2q∑‖x‖≤re−a‖x‖2p ≤ cminÅr2,paã d2e−a4‖t‖2q .If the condition ‖t‖ > 2r holds, then the first sum in (∗) is empty since the relation ‖x− t‖ ≤‖t‖2implies ‖x‖ ≥ ‖t‖2, and then we have ‖x‖ > r. The bound for ‖t‖ > 2r has beenestablished. Assume then, ‖t‖ ≤ 2r. Similarly,∑‖x‖≤r‖x−t‖≤ ‖t‖2e−aÄ ‖x‖2p+‖x−t‖2qä≤ e−a4 ‖t‖2p∑‖x‖≤ ‖t‖2e−a‖x‖2q≤ cmin(‖t‖24,qa) d2e−a4‖t‖2p ,which is the desired bound.§ 2.4. Some properties of (network) random walk.These properties are well-known and are included for completeness of the exposition. Wedenote by S = (Sn) the network random walk of a network Γ satisfying the usual assumptions(A.22), and G will denote its Green’s function, recall its definition from (1.1.9). The readershould also recall the definition of continuous time network random walk, (1.1.14)Proposition ( 2.4.1 ) Consider any network Γ satisfying the usual assumptions (A.22) andits Green’s function G. Denote by S the network random walk. Suppose V0 is a subset ofvertices, S0 = o ∈ V0 and let τ = τV{0(S) be the exit time of V0 by S. We assume τ <∞ a.s.Then, for every vertex x ∈ V,G(o, x) = GV0(o, x) + Eo (G(Sτ , x)) .35Chapter 2. Green’s function boundsProof. By the strong Markov property:G(o, x) = Eo( ∞∑m=01{Sm=x})= GV0(o, x) + EoÇ ∞∑m=τ1{Sm=x}å= GV0(o, x) + Eo(ESτ( ∞∑m=01{Sm=x}))= GV0(o, x) + E0 (G(Sτ , x)) ,which is the conclusion to be reached.Proposition ( 2.4.2 ) Assume S is the network random walk on some network Γ satisfyingthe usual assumptions (A.22) and let V0 denote a subset of vertices. Let τ = τV{0(S) be theexit time of V0. For any pair of vertices z1, z2 of this network, we haveµ(z1)Pz1 (Sn = z2, τ > n) = µ(z2)Pz2 (Sn = z1, τ > n) ,and thereforeµ(z1)GV0(z1, z2) = µ(z2)GV0(z2, z1).Proof. If either z1 or z2 do not belong to V0,Pz1 (Sn = z2, τ > n) = Pz2 (Sn = z1, τ > n) = 0.Assume then z1 and z2 belong to V0. By definitionPz1 (Sn = z2, τ > n) =∑γ−=z1,γ+=z2length(γ)=nPz1 ((S0, . . . , Sn) = γ) ,for paths without vertices outside V0. From the reversibility equation (1.1.5), it follows easilythatµ(z1)Pz1 ((S0, . . . , Sn) = γ)) = µ(z2)Pz2 ((S0, . . . , Sn) = −γ) ,and hence the first claim. The second claim is immediate upon summing over n ∈ Z+.The reader should recall the definition of continuous time network random walk (1.1.14).Proposition ( 2.4.3 ) For any two vertices o and v of Γ,G(o, v) =∞∫0dt qSt (o, v).36Chapter 2. Green’s function boundsProof. By Lebesgue-Fubini’s Theorem,∞∫0dt qSt (o, v) =∞∑k=0∞∫0dt Po (Sk = v) e−t tkk!=∞∑k=0Po (Sk = v) Ik, Ik =∞∫0dt e−ttkk!.A quick integration by parts shows that Ik = Ik−1, and it easily follows that I0 = 1.Proposition ( 2.4.4 ) Suppose Γ is transitive (A.23). For any vertex v of Γ, defineτv = τvÄS˜ä= inf¶t ≥ 0; S˜t = v©.Then, G(o, v) = G(o, o)Po (τv <∞) .Proof. By virtue of Proposition (2.4.3) and the strong Markov property at τv,G(o, v) =∞∫0dt qSt (o, v) =∞∫0dt PoÄS˜t = v, τv <∞ä=∞∫0dt EoÅES˜τvÅ1{S˜t=v}ã1{τv<∞}ã=∞∫0dt∞∫0Po (τv ∈ ds)EvÅ1{S˜t−s=v}ã=∞∫0dt∞∫0Po (τv ∈ ds) qSt (o, o)= G(o, o)Po (τv <∞) ,where EvÅ1{S˜t−s=v}ã= qSt (o, o) by transitivity of the network Γ.Proposition ( 2.4.5 ) Let S = (Sn)n∈Z+ be a random walk on Z whose step-lengths havedistribution q · ε−1 + r · ε0 + p · ε1 and we assume p > q > 0, p+ q + r = 1. Then, for everyn ∈ N,P0 (S ever reaches level − n) = P0 (τ−n(S) <∞) = e− log(pq)n.Proof. Use the strong Markov property applied at the times τ−j(S) (j = 1, . . . , n − 1).This shows P0 (τ−n(S) <∞) = P0 (τ−1(S) <∞)n and first-step analysis shows that x =P0 (τ−1(S) <∞) satisfies the equation x = q + rx+ px2, so thatx =p+ q −»(p+ q)2 − 4pq2p=p+ q − (p− q)2p=qp(the choice of sign came from the fact that x < 1). The result follows readily.37Chapter 2. Green’s function boundsProposition ( 2.4.6 ) Consider a random walk S on Z with step-lengths distributed asq · ε−1 + r · ε0 + p · ε1. Assume 1 > p+ q > p > q > 0. Consider the eventsCu = {for exactly one n, Sn = u} =⋃n∈Z+⋂m6=n{Sn = u, Sm 6= u}andEu = {for at least one n, Sn = u} =⋃n∈Z+{Sn = u}.Then, for u ≥ 0 and v < 0,(a) P0 (Cu) = P0Äτ+0 =∞ä= p− q,(b) P0Ñn2⋃u=0Cué≥ 1− (1− (p− q))n − nÇqpån,(c) P0 (Ev) =Çpqåv.Proof. We notice that by transience, starting at k, all points k + s (s ∈ N) will be visitedwith probability 1, so the events Cu are actually the events of reaching the state u and neverreturning to it. The first equality in item (a) is immediate from the strong Markov propertyupon applying it to the stopping time τu. The last claim was proved in (2.4.5) and usingthe same method (strong Markov property and first-step analysis) we can derive the secondequality of the item (a); the details are left to the reader.We now propose to prove the inequality; consider Du the complement of Cu and observethat the statement is equivalent to proving P0Ñn2⋂u=0Dué≤ (1− (p− q))n + nÇqpånand forthis, it suffices to showP0(n⋂u=0Dun)≤ (1− (p− q))n + nÇqpån.By definition, Du is the event where u will never be reached or else, it will be reached at leasttwice. Now, since the random walk is started at zero, almost surely, it will visit all statesu ≥ 0 and so, with respect to P0, the events Du (u ≥ 0) are the events where the state u isvisited twice or more, in particular, the entrance and first return to u are at finite times onDu. To establish the bound, consider the stopping times τt and introduce θt defined by “thetime of first return to t”, that is,θt = inf{k > τt | Sk = t} (t ∈ Z+).38Chapter 2. Green’s function boundsIt is clear that (τun)u=1,...,n is strictly increasing, that τt < θt for all t ∈ Z+, and also thatthe complement of the eventn⋂j=1{θ(j−1)n < τjn} is contained in the eventn⋃j=1{τjn < θ(j−1)n}.Therefore,P0(n⋂u=0Dun)≤ P0 (θ0 < τn < θn < . . . < τn2 < θn2 <∞) +n∑j=1P0Äτjn < θ(j−1)n <∞ä.By the strong Markov property,P0Äτjn < θ(j−1)n <∞ä= P0 (E−n) =Çqpån.Set En = F Sτn = σ(Sk; 0 ≤ k ≤ τn) andγk = P0 (θ0 < τn < θn < . . . < τkn < θkn <∞) .Observe that the strong Markov property impliesγn = P0 (θ0 < τn < θn < . . . < τn2 < θn2 <∞)= E0(1{θ0<τn<∞}E0Ä1{θn<...<τn2<θn2<∞}∣∣∣ Enä)= P0 (θ0 < τn <∞) γn−1 ≤ P0Äτ+0 <∞äγn−1,which leads at once to γn ≤ (1− (p− q))n since γ1 ≤ 1− (p− q).§ 2.5. A hitting time for the continuous time networkrandom walk.In this section we will investigate certain random times associated with the continuoustime random walk (2.2.1). These properties are arguably well-known and are included forcompleteness of the exposition. In what follows, we use (2.1.5)p =12d+ 1 + eλ, a = (eλ − 1)p, σ20 = (eλ + 1)p.Introduceτ 0n = inf¶t ≥ 0; B˜t = n©(n ∈ N),which is the time in which the zeroth coordinate of the continuous-time network randomwalk reaches level n. Then, τ 0n is an almost surely finite stopping time, it is independent of39Chapter 2. Green’s function boundsthe process X˜ and by successive applications of the strong Markov property at the times τ 0j(1 ≤ j ≤ n − 1), it is immediately seen that the law of τ 0n is the nth convolution power ofthe law of τ 01 , in other words, τ0n has the law of the sum of n independent random variablesdistributed as τ 01 .Proposition ( 2.5.1 ) We have E0 (τ 01 ) =1a.Proof. By definition, τ 01 is the hitting time of 1 by B˜. Recall the decomposition B˜t = BLt ,and letθ1 = inf{t ≥ 0;Lt > 0}the time of the first jump of the Poisson process L. It is a well-known fact that θ1 is anexponential random variable with mean1σ20and that is independent of the process B. Then,using the strong Markov property at time θ1,E0Äτ 01ä= E0(EB˜θ1Äτ 01ä+ θ1)=∞∫0dt σ20e−σ20tÇ11 + eλE−1Äτ 01äå+1σ20.Since E−1 (τ 01 ) = 2E0 (τ 01 ) , we deduce that x = E0 (τ 01 ) satisfies the equationx =2x1 + eλ+1σ20.Solving this equation yields x =1a. (To eliminate the solution x =∞ we need to show thatE0 (τ 01 ) < ∞. However, P0 (τ 01 > t) ≤ P0ÄB˜t ≤ 0äand using the central limit theorem it isstraightforward to show that for t ≥ 1, P0 ÄB˜t ≤ 0ä ≤ ct−2.)Proposition ( 2.5.2 ) Let ζ denote the Laplace transform of τ 01 , that is, ζ(t) = E0Äetτ01ä.Then ζ(t) < ∞ for t < (eλ2 − 1)2p and ζ(t) = ∞ for t > (eλ2 − 1)2p. We have the followingboundsζ(t) ≤2eλp2eλp− t for −∞ < t ≤ 0,2eλpσ20 − tfor 0 < t < (eλ2 − 1)2p.Proof. It is obvious that ζ < ∞ for t ≤ 0, so we need to prove that ζ < ∞ for 0 < t <(eλ2 − 1)2p. Suppose t > 0. Denote by τ1(B) the visit time of 1 by biased random walk Band defineϕ(z) = E0Äzτ1(B)ä=∞∑k=1P0 (τ1(B) = k) zk,40Chapter 2. Green’s function boundsthe probability generating function of τ1(B). Obviously, ϕ is a power series, convergent for|z| ≤ 1. We will show first that ϕ is actually convergent on the larger disc |z| < 1 + eλ2eλ2. Tobegin, observe that first step analysis provides the equationϕ(z) = z(eλ1 + eλ+11 + eλϕ(z)2)where ϕ(z)2 = E−1Äzτ1(B)äfollows from the strong Markov property at time τ0(B). Let usfocus on the case z > 0 so that ϕ is positive or ∞. The previous equation trivially holdswhenever ϕ(z) =∞, suppose then that ϕ(z) <∞. Then ϕ is explicitly expressible asϕ(z) =1−»1− 4q(1− q)z22(1− q)z ,where we have written q =eλ1 + eλfor simplicity (the minus sign was chosen from the trivialrelation ϕ(1) = 1). Using the binomial expansion(1 + t)α = 1 +∞∑p=1(αp)tp,(αp)=α(α− 1) · · · (α− p+ 1)p!,convergent for |t| < 1, we can easily see thatz 7→ 1−»1− 4q(1− q)z22(1− q)zcan be explicitly expressed as a power series inside the disc |z| < 12»q(1− q) =1 + eλ2eλ2andthis power series coincides with ϕ on the segment z ∈ [0, 1]. The claim then follows from theprinciple of isolated zeroes, see [Die69, (9.1.5)].We return to the Laplace transform of τ 01 , namely ζ. It is immediate from definition thatτ 01 =τ1(B)∑j=1θj,where the θj are the times between jumps of the Poisson process L. These times θj aremutually independent and independent of B, furthermore, they are are equally distributedfollowing an exponential law of mean1σ20. Therefore,ζ(t) = E0(etτ01)= E0(E(etτ01∣∣∣τ1(B))) = E0 ÅE Äetθ1äτ1(B)ã = ϕÇ σ20σ20 − tå41Chapter 2. Green’s function boundsTherefore, ζ(t) <∞ for all σ20σ20 − t<1 + eλ2eλ2; solving this inequality gives that ζ(t) <∞ forall t < (eλ2 − 1)2p. The previous equation shows at once that if t > (eλ2 − 1)2p then ζ(t) =∞sinceσ20σ20 − tfalls outside the disc of convergence of ϕ.Suppose now that t < (eλ2 − 1)2p. Thenζ(t) = ϕÇσ20σ20 − tå.Using q =eλpσ20and 1− q = pσ20, it easily follows thatζ(t) =σ20 − t−»(σ20 − t)2 − 4eλp22p=2eλpσ20 − t+»(σ20 − t)2 − 4eλp2.Since»(σ20 − t)2 − 4eλp2 ≥ 0, it follows immediately that ζ(t) ≤2eλpσ20 − t, which is the upperbound for t > 0. Suppose now that t ≤ 0. Then,»(σ20 − t)2 − 4eλp2 ≥»σ40 − 4eλp2 = a.Thus,ζ(t) ≤ 2eλpσ20 + a− t=2eλp2eλp− t ,as claimed.Proposition ( 2.5.3 ) (a) For α ≤ n2eλp,P0Äτ 0n ≤ αä ≤ exp(−2eλpα + n log(2eλ+1pαn)).(b) For 0 < β <12eλpthere exists a constant u(β) > 0 such that for all n,P0Äτ 0n ≤ βnä ≤ e−u(β)n,(c) There exists two constants T0, K0 > 0 such thatP0Äτ 0n ≥ αä ≤ e−T0(α−na )valid for α ≥ (K0 + 1a)n.42Chapter 2. Green’s function boundsProof. The bound in (c) is a Corollary of Theorem 1 and Lemma 5, in Chapter III of [Pet75],and observe we can apply these results since the previous proposition proved that the Laplacetransform of τ 01 is absolutely convergent on an interval around zero. We now prove the boundfor P0 (τ 0n ≤ α) . Markov’s inequality shows, for t < 0,P0Äτ 0n ≤ αä= P0(etτ0n ≥ etα)≤ e−tαζ(t)n,where ζ is the Laplace transform of τ 01 . From the previous proposition, we know ζ(t) ≤2eλp2eλp− t for t ≤ 0, and soP0Äτ 0n ≤ αä ≤ e−tα+n logÄ 2eλp2eλp−tä, t < 0.Minimising gives the minimiser t = 2eλp − nα, which is valid only when t < 0, that is,only when α <n2eλp, the bound follows readily upon evaluation at the minimiser. Whenα =n2eλp, we reach the trivial bound P0 (τ 0n ≤ α) ≤ 1, and so it is also valid in this case.Finally, the substitution α = βn impliesP0Äτ 0n ≤ βnä ≤ e−u(β)n,where u(β) = 2eλpβ − log(2eλ+1pβ). A standard minimisation shows u(β) > uÇ12eλpå= 0for 0 < β <12eλp.Proposition ( 2.5.4 ) Recall from (2.4.4) the definition of τz for z = (n, x) ∈ Zd+1. Thereexists c1, c2 > 0 such that, for all x ∈ Zd, P0Äτ(0,x) <∞ä ≤ c1e−c2‖x‖.Proof. Let r = rx be the smallest integer k ≥ ‖x‖2. ThenP0Äτ(0,x) <∞ä ≤ P0 Äτ(0,x) ≤ τ 0r ä+ P0 Äτ 0r < τ(0,x) <∞ä≤ P0 Äτ(0,x) ≤ τ 0r ≤ crä+ P0 Äτ(0,x) ≤ τ 0r , τ 0r > crä+ P0 Äτ 0r < τ(0,x) <∞ä ,where c ≥ K0 + 1ais chosen so that crx ≥ (K0 + 1a)rx for all x ∈ Zd, x 6= 0 and K0 is as in(2.5.3). Then, recalling S˜ =ÄB˜, X˜äand item (f) in §2.2,P0Äτ(0,x) ≤ τ 0r ≤ crä ≤ P0 Ç sup0≤s≤cr∥∥∥X˜s∥∥∥ ≥ ‖x‖å ≤ c′e−c′′ ‖x‖2c‖x‖ = c′e− c′′c ‖x‖,43Chapter 2. Green’s function boundsfor two constants c′, c′′ > 0. From (2.5.3),P0Äτ(0,x) ≤ τ 0r , τ 0r > crä ≤ e−c′r,for c′ = T0(c− 1a). Applying the strong Markov property at the time τ 0r , we reach thatP0Äτ 0r < τ(0,x) <∞ä ≤ E0 ÅPS˜τ0r Äτ(0,x) <∞äã ≤ P0 ÄB˜ ever reaches − rä= P0 (B ever reaches − r) ≤ e−c′‖x‖,for some c′ > 0.Proposition ( 2.5.5 ) Recall from (2.4.4) the definition of τz for z ∈ Zd+1. If z = (n, x),we haveG(0, z) = G(0, 0)∑y∈ZdP0Äτ(0,x−y) <∞ä ∞∫0qXt (y)PÄτ 0n ∈ dtä.Proof. By (2.4.4), we haveG(0, z) = G(0, 0)P0 (τz <∞) .Apply the strong Markov property at the time τ 0n and use that τ0n and X˜ are mutuallyindependent,P0 (τz <∞) = E0ÅPS˜τ0n (τz <∞)ã=∑y∈ZdP(n,y)Äτ(n,x) <∞äP0ÄS˜τ0n = (n, y)ä=∑y∈ZdP0Äτ(0,x−y) <∞ä ∞∫0P0ÄX˜t = yäPÄτ 0n ∈ dtä.Since qXt (y) = P0ÄX˜t = yä, we reach the desired conclusion.§ 2.6. Statement and proof of the Theorem.Consider the Green’s function G of Γd(λ). By transience, the probability p0 = P0 (τ0 <∞)that the network random walk started at zero returns to zero is (strictly) less than one.Thus, the number of visits to zero, started at zero, has finite expectation since this numberfollows a geometric law with probability of success p0, in other words, G(0, 0) is some finitenumber, depending only on λ (as such, regarded as universal). The next theorem deals withthe bounds G(0, z) for z 6= 0.44Chapter 2. Green’s function boundsTheorem ( 2.6.1 ) There exists four constants ci (i = 1, . . . , 4) such that for all verticesz = (n, x) 6= 0 of the network Γd(λ),G(0, z) ≤ c1e−c2‖z‖ if ‖x‖ > n,e−c2‖x‖2n ‖z‖− d2 if ‖x‖ ≤ n,andG(0, z) ≥ c3e−c4‖z‖ if ‖x‖ > n,e−c4‖x‖2n ‖z‖− d2 if ‖x‖ ≤ n,Proof. The proof is rather lengthy and we divide it into several steps.Proof of the upper bounds.We begin with proving the upper bound when z ∈ Z− ×Zd. We start with the followingsimple observation: bearing in mind Remark (2.2.5), if z = (n, x) ∈ Z− × Zd, the upperbound will be established if we prove G(0, z) ≤ ce−c′max(|n|,‖x‖) for a pair of constants c, c′ > 0independent of z.We analyse separately the cases ‖x‖ < |n| and ‖x‖ ≥ |n|.Assume first z ∈ Z− × Zd is such that ‖x‖ < |n|, notice n ≤ 0. We claim there existsa pair of constants c, c′ > 0 (independent of z = (n, x)) such that G(0, z) ≤ ce−c′|n|. To seethis, recall (2.4.4) and and (2.4.5), so thatG(0, z) = G(0, 0)P0 (τz <∞) ≤ G(0, 0)P0 (B ever reaches level n) = G(0, 0)ec′n,where τz = τzÄS˜äis the visit time of the point z by the continuous time random walk S˜(2.2.1), proving the claim.Assume now z ∈ Z− × Zd satisfies ‖x‖ ≥ |n|, notice n ≤ 0. Here we apply (2.4.2). Wehave thatG(0, z) =µ(z)µ(0)G(0,−z),and since µ(z)  eλn, we have the following preliminary result.Lemma ( 2.6.1.1 ) If the Green’s function bounds hold on the region of z ∈ Z+×Zd such that0 ≤ n ≤ ‖x‖ , then they also hold on the region of points z ∈ Z− × Zd such that |n| ≤ ‖x‖ .Observe that (2.5.4) shows that the Green’s function bounds hold on the set {0} × Zd.Thus, with the previous lemma in mind, the case z ∈ Z− × Zd has been established.Let us focus now on the case z = (n, x) ∈ Z∗+ × Zd. By virtue of (2.5.4) and (2.5.5), wehave(∗) G(0, z) ≤ c ∑y∈Zde−c′‖x−y‖∞∫0qXt (y)PÄτ 0n ∈ dtä,45Chapter 2. Green’s function boundswhere τ 0n is the visit time of n by the walk B˜ and qXt is the continuous time transitionprobability of X˜.Lemma ( 2.6.1.1 ) There exist two constants c1, c2 > 0, independent of z, such that∞∫0qXt (y)PÄτ 0n ∈ dtä ≤ c1e−c2n + n− d2 e−c2‖y‖2n if ‖y‖ ≤ deλ + 1nc1e−c2‖y‖ if ‖y‖ > deλ + 1n.Proof of lemma. Recall that we are using p =12d+ 1 + eλ. Consider the homeomorphismt 7→ t2dpof [0,∞) onto itself, the change of measures theorem implies that∞∫0qXt (y)PÄτ 0n ∈ dtä=∞∫0qXt2dp(y)PÄ2dp τ 0n ∈ dtä.We will tackle the two cases.Case 1: ‖y‖ ≤ deλ + 1n. We split the integral as follows∞∫0qXt2dp(y)PÄ2dp τ 0n ∈ dtä=‖y‖e∫0+deλ+1n∫‖y‖e+2dp(K0+1a)n∫deλ+1n+∞∫2dp(K0+1a)n qXt2dp (y)P Ä2dp τ 0n ∈ dtä ,where K0 is as in (2.5.3). In what follows, we will use the bounds for P (τ0n ≤ α) andP (τ 0n ≤ βn) of this proposition without further notice. The first two integrals will be boundedusing the bounds (2.2.2). In the region 0 ≤ t ≤ ‖y‖e, we haveqXt2dp(y) ≤ 12de−t−‖y‖ log(‖y‖et )and the function t 7→ −t− ‖y‖ log(‖y‖et) is easily seen to be increasing on (0, ‖y‖), hence‖y‖e∫0qXt2dp(y)PÄ2dp τ 0n ∈ dtä ≤ 12de−‖y‖e PÇτ 0n ≤‖y‖2dpeå≤ 12de−‖y‖e PÇτ 0n ≤n2eλ+1på≤ e− ‖y‖+ne ≤ e−ne .46Chapter 2. Green’s function boundsSimilarly, in the region‖y‖e≤ t < ∞, we have the bound qXt2dp(y) ≤ 2de−‖y‖2e2t , and the termon the right is an increasing function of t > 0. Hence,deλ+1n∫‖y‖eqXt2dp(y)PÄ2dp τ 0n ∈ dtä ≤ 2de−Äeλ−2dä ‖y‖2n PÇτ 0n ≤n2p(eλ + 1)å≤ e−Äeλ−2dä ‖y‖2n−cn ≤ e−cnfor some constant c = uÇ12p(eλ + 1)å> 0. We use now the bounds (2.2.4). Consider theintervaldeλ + 1n ≤ t ≤ 2dpÇK0 +1aån, we have a Gaussian decayqXt2dp(y) ≤ c1t− d2 e−c2‖y‖2t .On this interval, t  n and so, there is a pair of constants c, c′ > 0 such that2dp(K0+1a)n∫deλ+1nqXt2dp(y)PÄ2dp τ 0n ∈ dtä ≤ cn− d2 e−c′ ‖y‖2n .Next, for t ≥ 2dpÇK0 +1aån, we use the factor t−d2 of the Gaussian estimate for qXt2dp(y) toconclude there exists c > 0 such thatqXt2dp(y) ≤ c1t− d2 ≤ cn− d2 .Thus,∞∫2dp(K0+1a)nqXt2dp(y)PÄ2dp τ 0n ∈ dtä= cn−d2PÇτ 0n ≥ÇK0 +1aånå= cn−d2 e−T0K0n ≤ e−T0K0n,again by (2.5.3). Putting all the bounds found, this concludes the case ‖y‖ ≤ deλ + 1n.Case 2: ‖y‖ > deλ + 1n. Here we divide∞∫0qXt2dp(y)PÄ2dp τ 0n ∈ dtä=‖y‖e∫0+2σ20(K0+1a)‖y‖∫‖y‖e+∞∫2σ20(K0+1a)‖y‖ qXt2dp(y)PÄ2dp τ 0n ∈ dtä.47Chapter 2. Green’s function boundsRecall the function t 7→ −t− ‖y‖ logÇ‖y‖etåis increasing on (0, ‖y‖). As in Case 1, we canboundqXt2dp(y) ≤e−‖y‖e if t ≤ ‖y‖e,2de−c‖y‖ if‖y‖e≤ t ≤ 2σ20(K0 + 1a) ‖y‖ ,c1 if t ≥ 2σ20(K0 + 1a) ‖y‖ ,where c1 is the constant appearing on the Gaussian bound of qXt and c =Ä2σ20e2(K0 +1a)ä−1.We may assume, should the need arise, K0 is so large so that 2σ20(K0 +1a) ≥ 1. It is obviousthen that the first two integrals decay exponentially in ‖y‖ , as for the third one, we employ(2.5.3) to obtain∞∫2σ20(K0+1a)‖y‖qXt2dp(y)PÄ2dp τ 0n ∈ dtä ≤ c1P(τ 0n ≥ σ20(K0 + 1a) ‖y‖dp ) ,the conditiondeλ + 1n < ‖y‖ guaranteesP(τ 0n ≥σ20(K0 +1a) ‖y‖dp)≤ e−T0K0 eλ+1d‖y‖.This completes Case 2 and the lemma.With the lemma proven, let us return to the proof of the upper bounds for the Green’sfunction. The reader is reminded that there only remains to prove the bounds for z = (n, x)in Z∗+ ×Zd. The previous lemma and (∗) imply that there exist two constants c, c′ > 0 suchthatG(0, z) ≤ cÑ ∑‖y‖≤bne−c′(‖x−y‖+n) +∑‖y‖≤bnn−d2 e−c′Ä‖x−y‖+ ‖y‖2nä+∑‖y‖>bne−c′(‖x−y‖+‖y‖)é,where b =deλ + 1.Bearing in mind (2.2.5), we only need to prove there exists a pair of constants c, c′ > 0such that G(0, z) ≤ ce−c′‖x‖ for all ‖x‖ > bn and that there exists a pair of constants c, c′ > 0such that G(0, z) ≤ cn− d2 e−c′ ‖x‖2n for all ‖x‖ ≤ bn.We consider first a sum of the form∑‖y‖≤bne−c(‖x−y‖+αn). Set c′ = minÄc, cα2bä, then∑‖y‖≤bne−c(‖x−y‖+αn) ≤ ∑‖y‖≤bne−c′‖x−y‖−cαn ≤ e−c′‖x‖ ∑‖y‖≤bnecα2b‖y‖−cαn≤ e−c′‖x‖Lnde− cα2 n ≤ c′′e−c′‖x‖− cα4 n,48Chapter 2. Green’s function boundsin which L is a constant, depending only in dimension, satisfying that card (BZd (0; r)) ≤ Lrdfor all r ≥ 1 and c′′ = supn∈NLnde−cα4n <∞. Thus, we have proven the following lemma.Lemma ( 2.6.1.2 ) Let α, β, γ > 0. There exist constants c, c′ > 0 such that∑‖y‖≤βne−γ(‖x−y‖+αn) ≤ ce−c′‖(n,x)‖,for all (n, x) ∈ Z× Zd.Next, we consider the sum∑‖y‖>bne−c′(‖x−y‖+‖y‖). In the case that ‖x‖ ≤ bn, then we maybound e−c′(‖x−y‖+‖y‖) ≤ e−c′bne−c′‖x−y‖, thus∑‖y‖>bne−c′(‖x−y‖+‖y‖) ≤ e−c′bn ∑y∈Zde−c′‖y‖.In the case that ‖x‖ > b|n|, we may bound ‖x− y‖ ≥ ‖x‖ − ‖y‖ , thuse−c′(‖x−y‖+‖y‖) ≤ e− c′2‖x−y‖−c′‖y‖ ≤ e− c′2‖x‖− c′2‖y‖.Therefore ∑‖y‖>bne−c′(‖x−y‖+‖y‖) ≤ e− c′2‖x‖ ∑y∈Zde−c′2‖y‖.We have established the following lemma.Lemma ( 2.6.1.3 ) Let α, β > 0. Then, there exists a pair of constants c, c′ > 0 such that∑‖y‖>βne−α(‖x−y‖+‖y‖) ≤ ce−c′‖(n,x)‖for every (n, x) ∈ Z× Zd.By virtue of the previous lemmas, it remains to show that∑‖y‖≤bnn−d2 e−c′Ä‖x−y‖+ ‖y‖2nä≤ce−c′‖x‖ if ‖x‖ > bn,ce−c′ ‖x‖2n n−d2 if ‖x‖ ≤ bn.Assume first that ‖x‖ > bn > 0. The estimate (2.3.4) shows at once∑‖y‖≤bne−cÄ‖x−y‖+ ‖y‖2nä≤ ce−c′‖x‖49Chapter 2. Green’s function boundsfor a pair of constants c, c′ > 0. This concludes the case ‖x‖ > bn > 0.Assume now that ‖x‖ ≤ bn. By (2.3.4), we also have∑‖y‖≤bnn−d2 e−c′Ä‖x−y‖+ ‖y‖2nä≤ cn− d2 e−c′ ‖x‖2nfor (possibly) another pair c, c′ > 0. This concludes the case ‖x‖ ≤ bn, and thus, the proofof the upper bounds by virtue of Lemma (2.6.1.1).Proof of lower bounds.These are much simpler. From (2.4.2), we haveG(0, z) =µ(z)µ(0)G(0,−z).If z = −(n, x) with n > 0, then µ(z)  e−λn and soG(0, z) ≥ ce−λnG(0,−z) ≥ ce−λ‖z‖G(0,−z).Therefore, if we prove lower bounds for n > 0, we will reach the lower bounds for n < 0. Forthe case n = 0 observe the following: by (2.4.4),G(0, (0, x)) = G(0, 0)P0Äτ(0,x) <∞ä ≥ G(0, 0)P0 Äτ(0,x) <∞, S˜σ1 = (1, 0)ä ,where σ1 is the time of the first jump of the Poisson process N in the definition of S˜. Now,P0Äτ(0,x) <∞, S˜Nσ1 = (1, 0)ä= eλp∞∫0dt e−tP(1,0)Äτ(0,x) <∞ä= eλpP0Äτ(−1,x) <∞ä.Thus, if the correct bounds hold for n < 0, thenG(0, (0, x)) ≥ eλpG(0, (−1, x)) ≥ ce−c′‖(1,x)‖ ‖(1, x)‖− d2 ,and it is obvious that ‖(1, x)‖  ‖(0, x)‖ (2.2.5). Whence, the proof of the lower boundsreduces to the case n > 0.When n > 0, we can apply (2.5.5). Now,∞∫0qXt (y)PÄτ 0n ∈ dtä ≥ 4(K0+ 1a )dp n∫2deλnqXt2dp(y)PÄ2dp τ 0n ∈ dtä.50Chapter 2. Green’s function boundsObserve that the lower bounds of (2.2.3) and (2.2.4) are of the same type (with possiblydifferent constants) whenever ‖y‖  t. On the interval of integration, t  n, thereforeqXt2dp(y) ≥cn− d2 e−c′ ‖y‖2n if n ≥ ‖y‖ ,c ‖y‖− d2 e−c′‖y‖ if n < ‖y‖ .Also, (2.5.3) givesPÇ2deλn ≤ 2dp τ 0n ≤ 4(K0 +1a)dp nå≥ c,for a constant c > 0 independent of n. Therefore,G(0, z)G(0, 0)≥ cÑ ∑n≥‖y‖P0Äτ(0,x−y) <∞än−d2 e−c′ ‖y‖2n +∑n<‖y‖P0Äτ(0,x−y) <∞ä ‖y‖− d2 e−c′‖y‖é .We discard all the terms except y = x, and reachG(0, z) ≥cG(0, 0)n− d2 e−c′ ‖x‖2n if n ≥ ‖x‖cG(0, 0) ‖x‖− d2 e−c′‖x‖ if n < ‖x‖ .This terminates the proof of Theorem (2.6.1).51Chapter 3The number of trees in USFIn this chapter we will establish a number of basic properties of the uniform spanning forest.We will prove that Γd(λ) has the Liouville property, that both measures WSF and FSFcoincide in this network and finally, establish when the samples are trees or forests.§ 3.1. The transition operator and harmonic functions.Let G = (V,E) be a graph, assume there is an assignment of conductances making theresulting network to possess the usual assumptions (A.22) and we assume further that thisassignment is strictly positive. The reader should recall the definition of harmonic function(1.1.8). We introduce the following “operator.” Consider the function P : RV → RV givenby(3.1.1) Pf : x 7→ Pf(x) = ∑y∈Vp(x, y)f(y),where p is the transition probability of the network random walk (1.1.4). This P is sometimesreferred to as the “transition operator.” In fact, when A is a subset of vertices, then P canbe defined, in the same manner as before, using functions A→ R, but this time, Pf is onlydefined at the vertices in A˚. We summarise important properties of P now. Recall that, fors ≥ 1, `s(V;µ) is the set of all functions f : V → R with sth absolutely integrable powerwith respect to µ. The set `∞(V;µ) is the set of real-valued functions that are bounded (sinceµ is a purely atomic measure on V with no null sets other than the empty set).Proposition ( 3.1.2 ) On a network satisfying the usual assumptions, the transition oper-ator P has the following properties.52Chapter 3. The number of trees in USF(a) Pf(x) = E (f(S1) | X0 = x) = Ex (f(S1)) , where (Sn)n∈Z+ is the network randomwalk.(b) Define P n recursively as P n+1 = P ◦ P n and set P 0 to be the identity function. ThenP nf(x) = Ex (f(Sn)) .(c) If f ≤ g, then Pf ≤ Pg.(d) |Pf |p ≤ P |f |p for all p ∈ [1,∞).(e) ‖Pf‖`p(V;µ) ≤ ‖f‖`p(V;µ) for all p ∈ [1,∞].(f) P : `2(V;µ) → `2(V;µ) is self-adjoint, that is to say, (Pf |g)`2(V;µ) = (f |Pg)`2(V;µ) forall f and g in `2(V;µ).(g) For a function f : A → R to be harmonic on A˚ it is necessary and sufficient thatPf = f on A˚.For the proof, see [Bar17].For the next proposition, the reader should recall the definition of the interior, closureand boundary of a set of vertices, see (A.10).Proposition ( 3.1.3 ) Consider any network Γ = (G, µ) such that µ(e) > 0 for every edgee, and let h, h1, h2 : V→ R be three functions and A a subset of vertices.(a) If h1 and h2 are harmonic functions on A and λ is any scalar, h1 + λh2 is a harmonicfunction on A (the “superposition principle.”)(b) If A is connected, h is harmonic on A and there exists a vertex v ∈ A for whichh(v) = supw∈Ah(w), then h is constant on A (the “maximum principle”).(c) If A is finite and ∂A 6= ∅, h1 and h2 coincide in ∂A and they both are harmonic on A,then h1 = h2 on A (the “uniqueness principle”); uniqueness may fail if A is not finite.(d) If ϕ : V \ A → R is a real valued bounded function, then there exists a functionu : V→ R such that u(x) = ϕ(x) for x ∈ V \A and u is harmonic at every x ∈ A (the“existence principle”).For the proof, see [Bar17].Proposition ( 3.1.4 ) Consider the network random walk S (started at any non-randomvertex S0 = o) on a locally finite connected network and a harmonic function h. Then, theprocess h(S) = (h(Sn))n∈Z+ is a martingale.53Chapter 3. The number of trees in USFProof. Observe first that Sn at graph-distance from S0 = o at most n, locally finitenessshows that the set B (o;n) of vertices at graph-distance at most n from S0 = o is finite and,therefore, h is bounded there. This shows that h(Sn) is integrable. In the following equalitiesthe first equal sign is definition of conditional expectation in a Markov chain and the secondis the harmonicity of h,E (h(Sn+1) | Sn) =∑y∼Snp(Sn, y)h(y) = h(Sn).So, h(S) is a martingale.Recall from (1.1.12) the definition of the lazy version of a Markov chain.Proposition ( 3.1.5 ) If h is a harmonic function on the network (G, µ), then it is alsoharmonic for any lazy version of the transition probability of the network.Proof. Let p be the transition probability of the network random walk of (G, µ) and pλ alazy version for some lazy factor λ ∈ (0, 1). By harmonicity of h,h(x) = λh(x) + (1− λ)h(x) = λh(x) + (1− λ)∑y∼xp(x, y)h(y)= λh(x) + (1− λ)p(x, x)h(x) + ∑y∼xy 6=x(1− λ)p(x, y)h(y)= pλ(x, x)h(x) +∑y∼xy 6=xpλ(x, y)h(y) =∑y∼xpλ(x, y)h(y).From this, h is also harmonic for pλ.Proposition ( 3.1.6 ) Consider a Markov chain and assume that it is:(a) irreducible; or(b) recurrent.Then any lazy version will possess the same property.Proof. Let X = (Xn)n∈Z+ be Markov chain. Let (ξn)n∈N be independent and identicallydistributed random variables following Bernoulli law of parameter λ (the lazy factor), thatis, the law of each of the ξn is λ · ε0 + (1− λ) · ε1. Set Tn =n∑p=1ξp and Xλn = XTn . It is easyto deduce that Xλ is the λ-lazy Markov chain X. The strong law of large numbers impliesthat Tn → ∞ a.s., thus Xλ and X will have the same paths (when viewed as sets) up to anull event. The result is now clear.54Chapter 3. The number of trees in USFRecall that a measure µ on Rd (or on any vector space) is called symmetric if µ(A) =µ(−A). To say that a random walk has symmetric step-lengths will mean that the distributionof the step-lengths is symmetric.Proposition ( 3.1.7 ) Let (Sn)n∈Z+ be a random walk on Z whose step-lengths are sym-metric, aperiodic and bounded by some constant M. Then (Sn) is irreducible and recurrent.We omit the proof of this theorem as it can be found in [Tho00, Chap. 2, sect. 5.3].The following proposition decomposes a random walk on Zd in the following manner: (1)first one chooses which factor to move and (2) given that the random walk is moving on thatfactor, choose the vertex where to move. Special attention needs to be placed at zero sinceit can be considered to belong to all axes (thus, it can be thought to come uniformly fromany of them). Bearing this in mind, distributions ϕ and νi in the following proposition comenaturally to mind.Proposition ( 3.1.8 ) Denote by ui (1 ≤ i ≤ d) the canonical unit vectors of Zd. Let ρ bea probability measure on Zd such that(a) the relation ρ(x) > 0 implies x is an integer multiple of some ui;(b) ρ(Z∗ui) > 0 for each i = 1, . . . , d.Introduce two probability measures, first ϕ(i) = ρ(Z∗ui) +ρ(0)d(1 ≤ i ≤ d) and thenνi(k) =ρ(kui)ϕ(i)for k ∈ Z∗ and νi(0) = ρ(0)dϕ(i). Suppose Fj ∼ ϕ and X(i)j ∼ νi are independentrandom elements (1 ≤ i ≤ d and j ∈ Z+). Set Yj = X(Fj)j =d∑i=1X(i)j 1{Fj=i}. Then, for anyS0 independent of the families X(i)j and Fj, Sn = S0 + Y1 + . . .+ Yn is a random walk on Zdstarted at S0 with transition probability ρ : P (Yj = x) = ρ(x) (x ∈ Zd).Proof. By independence of the elements involved, the family (Yj)j∈Z+ is independent andthus, S = (Sn) is a random walk on Zd. Now, a simple calculation (bearing in mind inde-pendence), shows thatP(X(Fj)j = 0)=d∑i=1ϕ(i)νi(0) = ρ(0),and for k ∈ Z∗ and i = 1, . . . , d,P(X(Fj)j = kui)= P(Fj = i,X(i)j = kui)= ρ(kui).Thus, Yj ∼ ρ as required.55Chapter 3. The number of trees in USFRemark ( 3.1.9 ) When ρ of (3.1.8) is given by (2.1.3), the appropriate distributions areϕ(i) =1 + eλ2(2d+ 1 + eλ)+12(d+ 1)if i = 022(2d+ 1 + eλ)+12(d+ 1)if i = 1, . . . , d;and the νi are given asν0(−1) = 12(2d+ 1 + eλ)ϕ(0), ν0(0) =12(d+ 1)ϕ(0), ν0(1) =eλ2(2d+ 1 + eλ)ϕ(0)and for i = 1, . . . , dνi(1) = νi(−1) = 12(2d+ 1 + eλ)ϕ(i), νi(0) =12(d+ 1)ϕ(i).§ 3.2. Coupling of random walks.Definition ( 3.2.1 ) Coupling. Consider two probability measures χ and ν on measurablespaces (X,X ) and (Y,Y ), respectively. A coupling of χ and ν is a probability measure ρ onthe product measurable space (X×Y,X ⊗Y ) such that the marginal probability measures(or marginals) of ρ, defined by ρY(B) = ρ(X × B) for B ∈ Y and ρX(A) = ρ(A × Y) forA ∈ X , are equal to χ and ν, respectively, that is ρX = χ and ρY = ν. (If X = Y, we writeρ1 and ρ2 in lieu of ρX and ρY, respectively.)Consider now random elements R and U, defined on respective probability spaces (E,E , µ)and (F,F , ν), with values on measurable spaces (X,X ) and (Y,Y ), respectively. A randomelement T defined on a probability space (D,D , ρ) and with values in measurable space(X×Y,X ⊗Y ) is a coupling of the random elements R and U if the law of T is a couplingof the laws of R and U.Even though the definition of coupling is very general, when we couple two random walks(3.2.3) we will consider a very particular class of coupling.Consider two vertices x and y in the network (G, µ). We will say that the network randomwalk started at x can be classically coupled with the network random walk started at y ifthe following condition holds:(CC) There exists a probability space (Ω,F ,P), a filtration (Fn)n∈Z+ in this probabilityspace, two Markov processes with respect to this filtration and defined on this prob-ability space, denoted by (Sn,Fn) and (S ′n,Fn) , both of them having for transition56Chapter 3. The number of trees in USFprobability that defined by the conductances of the network (1.1.4), S0 = x, S′0 = yand there exists a stopping time τ relative to the filtration (Fn)n∈Z+ , such that τ <∞P-a.e. and for those ω ∈ Ω for which τ(ω) < ∞, Sτ(ω)+n(ω) = S ′τ(ω)+n(ω) for anyn ∈ N.When this happens, we will call τ the coupling time and the condition τ < ∞ P-a.e. isexpressed by saying that “τ is successful.” Obviously, for a classical coupling to exist, it isnecessary that x and y should be connected in (G, µ), but this is far from sufficient.Remark ( 3.2.2 ) Observe that by definition, a classical coupling is a coupling T = (S, S ′)of the network random walk S started at x and the network random walk S ′ started aty satisfying strong conditions and it does not necessarily exists; for instance, the standardlattice Z1 with x = 0 and y = 1, any two standard random walks will always be at distanceat least one and so, no classical coupling can exist (the “parity problem”).We are going to show in this section that for reasonably general transition probabilitieson Zd, one can always classically couple two random walks started at any two vertices. Themain hypothesis for this transition density is that we can decompose the jumps of the walkinto selecting the axis used and then the jump on this axis. We need to assume a technicalcondition on the probabilities of jumps on each axis which will become apparent during theproof.We describe here the gist of the coupling idea. The idea is to decompose the randomwalk as in (3.1.8); we take two sequences of independent step-lengths, we will let the randomwalks imitate each other when they do not move on the first axis, and they will behave inde-pendently on this axis. Appropriate conditions of aperiodicity, irreducibility and recurrencewill show that the difference of the first coordinate of these random walk will hit 0 ∈ Z a.s.and thus, the first coordinate of the two walks will eventually be the same. We can thenuse induction to show that one by one all coordinates will be the same. These will createa series of walks Sα = (Sα,n)n∈Z+ and Sˆα =ÄSˆα,nän∈Z+ such that S1 and Sˆ1 will couple thefirst coordinate, then S2 and Sˆ2, the second, and continue in this manner. This will finishwith Sd and Sˆd with all coordinates coupled. We will finally construct a random walk S′by gluing Sˆ1 with Sˆ2 at the time the first coordinate is coupled, then Sˆ2 with Sˆ3 when thesecond coordinate is couples and so on. Once the construction of S ′ is finished, it will remainto show that S and S ′ is the desired classical coupling, which is done by showing they followthe same transition kernel ρ.Theorem ( 3.2.3 ) Let ρ be a probability measure on Zd such that(a) the relation ρ(x) > 0 implies x = kui for some k ∈ Z and some canonical unitaryvector ui;57Chapter 3. The number of trees in USF(b) ρ(Z∗ui) > 0 for each i = 1, . . . , d;(c) for every i = 1, . . . , d, the probability measures νi, defined in (3.1.8), and ν¯i definedby ν¯i : k 7→ νi(−k) (k ∈ Z) are such that their convolution νi ∗ ν¯i defines a transitionprobability kernel for a Markov chain on Z that is irreducible and recurrent.Then, for any pair of vertices x and y in Zd, there exists a classical coupling of random walksstarted at x and y, respectively.Proof. Recall the definition of ϕ in (3.1.8). Let (Ω,F ,P) be a probability space supportingindependent random elements Fj, X(i)j and X¯(i)j (j ∈ Z+, i = 1, . . . , d) such that each Fjfollow law ϕ while X(i)j , X¯(i)j follow law νi (i = 1, . . . , d).Define Yj =d∑i=1X(i)j 1{Fj=i} and Yˆj = X¯(1)j 1{Fj=1} +d∑i=2X(i)j 1{Fj=i} (j ∈ N). By (3.1.8)Yj, Yˆj ∼ ρ (j ∈ N). Now, each of the families (Yj) andÄYˆjäis independent, this follows easilyfrom the fact that for different j all the random elements defining Y and Yˆ are independent.Define (n ∈ N) the processes S1 : S1,n = x+Y1 + . . .+Yn and Sˆ1 : Sˆ1,n = y+ Yˆ1 + . . .+ Yˆn,which are random walks started at x and y, respectively, with transition kernel ρ. Considerthe difference processD1 : D1,n = S1,n − Sˆ1,n = x− y +n∑j=1(X(1)j − X¯(1)j)1{Fj=1}.This is a random walk onZd such that only its first coordinate changes value and its transitionkernel is the (1 − ϕ(1))-lazy version of ν1 ∗ ν¯1. Hence, pr1(D1) is an irreducible, recurrentand aperiodic walk on Z (3.1.6), and thus it will reach 0 ∈ Z at a finite time a.s. Whence,τ1 = τ{0}×Zd−1(D1) <∞ a.s.Consider now the random variables F2,j = Fτ1+j and X(i)2,j = X(i)τ1+j (i = 1, . . . , d andj ∈ Z+). Consider similarly the process S2,n = S1,τ1+n (n ∈ Z+). Let (Fn)n∈Z+ denote thefiltration generated by S0 when n = 0 and by S0, Fj, X(i)j , X¯(i)j where the indices run on1 ≤ j ≤ n and 1 ≤ i ≤ d for other values of n. By the (strong) Markov property, the process(S2,n − S2,0)n∈N is independent of Fτ1 .The elements F2,j have law ϕ and X(i)2,j, νi (i = 1, . . . , d and j ∈ Z+). Indeed, since{τ1 = n} ∈ Fn, it follows that {τ1 = n},(X(i)n+j)j∈N and (Fn+j)j∈N (i = 1, . . . , d) are allmutually independent. This entails implicitly that X(i)2,j and F2,j are independent of the58Chapter 3. The number of trees in USFσ-field Fτ1 . As the family (Fj) is identically distributed,P (F2,j = i) =∑n∈Z+P (F0,n+j = i | τ1 = n)P (τ1 = n)= ϕ(i)∑n∈Z+P (τ1 = n) = ϕ(i),so F2,j has law ϕ as claimed. A similar argument as in the one just given shows X(i)2,j haslaw νi. An obvious modification of the same argument shows independence between all theelements X(i)2,j and F2,j (i = 1, . . . , d and j ∈ Z+). Thus, if we defineYˆ2,j = X¯(2)j 1{F2,j=2} +d∑i=1i 6=2X(i)2,j1{F2,j=i},thenÄYˆ2,jäj∈N is an independent family of identically distributed random elements accordingto the law ρ. By setting Sˆ2 : Sˆ2,n = Yˆ2,1 + . . . + Yˆ2,n, it follows thatÄSˆ2,n,Fτ1+nän∈Z+ is arandom walk, started at zero with transition kernel ρ. In particular, we can consider thedifferenceD2 : D2,n = S2,n − Sˆ2,n = Sτ1 +n∑j=1(X(2)j − X¯(2)j)1{Fj=2}.Again, pr2(D2) is a recurrent, irreducible and aperiodic walk on Z, thus hitting zero at afinite time a.s. We can conclude that τ2 = τZ×{0}×Zd−2(D2) <∞ a.s.Proceeding in this manner, we can define, for every α = 1, . . . , d, two independent families(independence within and between families)(X(i)α,j)j∈N (i = 1, . . . , d) and (Fα,j)j∈N followinglaws νi (i = 1, . . . , d) and ϕ, respectively. The random elementsYα,j =d∑i=1X(i)α,j1{Fα,j=i}, Yˆα,j = X¯(α)j 1{Fα,j=α} +d∑i=1i 6=αX(i)j 1{Fα,j=i}have law ρ and each of the families Yα and Yˆα is independent. The recursive relation Sα :Sα,n = Sα−1,τα−1+n, holds, and we have, by setting Sˆα : Sˆα,n = Yˆα,1 + . . . + Yˆα,n, that therandom walkDα : Dα,n = Sα,n − Sˆα,n = Sτ1+...+τα−1 +(X(α)α,j − X¯(α)α,j)1{Fα,j=α}is such that prα(Dα) is an irreducible, recurrent and aperiodic walk on Z, hence the hittingtime τα of the set Zα−1×{0}×Zd−α by Dα is finite a.s. Finally, the processes (Sα,n − Sα,0)andÄSˆα,näare independent of Fτ1+...+τα−1 .59Chapter 3. The number of trees in USFDefine the random times Tα = τ1 + . . . + τα for α = 1, . . . , d. Consider the process(S ′n)n∈Z+ , started at y, and given byS ′n = Sˆ1,n1{n≤T1} +ÄSˆ1,τ1 + Sˆ1,n−T1ä1{T1<n≤T2} + . . .+ÄSˆ1,τ1 + . . .+ Sˆd−1,τd−1 + Sˆd,n−Td−1ä1{Td−1<n≤Td} + Sn1{Td<n}.We will show (S, S ′) is the desired classical coupling, with successful coupling time Td. Definethe events (n ∈ Z+)Eα(n) ={n ≤ T1} if α = 1,{Tα−1 < n ≤ Tα} for α = 2, . . . , d,{Td < n} if α = d+ 1.Notice that, no matter what n ∈ Z+ may be, the family (Eα(n))α=1,...,d+1, is disjoint and thecomplement of the union of its members is negligible; for if ω ∈ Ω is such that Td(ω) < ∞then either n is larger than Td(ω) or else, there is a minimum index 1 ≤ α ≤ d such thatn ≤ Tα(ω) and then ω ∈ Eα(n). This showed that the set {Td = ∞}, which is negligible,contains the complement ofd+1⋃α=1Eα(n). For every α = 1, . . . , d and n ∈ Z+, we claim thatEα(n) ∈ Fmin(n−1,Tα). For if k ∈ Z+ the intersection Eα(n) ∩ {min(n− 1, Tα) = k} becomesEα(n) if n− 1 = k and ∅ for other values of k, thus belonging to Fk and proving the claim.Notice Ed+1(n) ∈ FTd .Assume s, t ∈ Zd+1 are such that ρ(t− s) > 0. Then,PÄS ′m+1 = t∣∣∣S ′m = sä = d+1∑α=1PÄ¶S ′m+1 = t, S′m = s© ∩ Eα(m+ 1)äP (S ′m = s).For α = 2, . . . , d, on the event Eα(m+ 1),S ′m+1 = Sˆ1,τ1 + . . .+ Sˆα−1,τα−1 + Sˆα,m+1−Tα−1andS ′m =Sˆ1,τ1 + . . .+ Sˆα−1,τα−1 + Sˆα,m−Tα−1 , Tα−1 < mSˆ1,τ1 + . . .+ Sˆα−2,τα−2 + Sˆα−1,m−Tα−2 , Tα−1 = m ,= Sˆ1,τ1 + . . .+ Sˆα−1,τα−1 + Sˆα,m−Tα−1 ,since Sˆα,0 = 0 and Tα−1 − Tα−2 = τα−1; therefore,¶S ′m+1 = t, S′m = s© ∩ Eα(m+ 1) = ¶Sˆα,m+1−Tα−1 − Sˆα,m−Tα−1 = t− s, S ′m = s© ∩ Eα(m+ 1)=¶Sˆα,m+1−Tα−1 − Sˆα,m−Tα−1 = t− s© ∩ {S ′m = s} ∩ Eα(m+ 1)︸ ︷︷ ︸∈Fmin(m,Tα)⊂FTα.60Chapter 3. The number of trees in USFThe case α = 1 is easy since¶S ′m+1 = t, S′m = s© ∩ E1(m+ 1) = ¶Sˆ1,m+1 − Sˆ1,m = t− s© ∩ {S ′m = s} ∩ E1(m+ 1)︸ ︷︷ ︸∈Fmin(m,T1)⊂FT1;whilst for α = d+ 1,¶S ′m+1 = t, S′m = s,Ed+1(m+ 1)©= {Sm+1 − Sm = t− s} ∩ {S ′m = s} ∩ Ed+1(m+ 1)︸ ︷︷ ︸∈Fm.Using these identities and independence, one getsPÄS ′m+1 = t | S ′m = sä=d+1∑α=0ρ(t− s)P ({S′m = s} ∩ Eα(m+ 1))P (S ′m = s)= ρ(t− s);then (S ′n)n∈Z+ is a random walk with transition density ρ; this substantiates the couplingand terminates the proof.§ 3.3. The Liouville property.Recall the definitions of harmonic function (1.1.8) and Dirichlet function (C.2.3). We willsay that a network satisfies(LP) The Liouville property if every bounded harmonic function on the network is constant.(SLP) The strong Liouville property if every positive harmonic function on the network isconstant.(DLP) The Dirichlet Liouville property if every Dirichlet harmonic function is constant.Remark ( 3.3.1 ) (a) For any network, we have the following implications from the aboveproperties(SLP) =⇒ (LP) =⇒ (DLP).The first of these implications is trivial for if h is a bounded harmonic function, thenh+ c is a positive harmonic function for large enough constant c, then condition (SLP)implies h+ c is constant, and so is h. The second of this implications is harder and itfully proved in Annex D, see Proposition (D.6.3).(b) It is a well-known fact that the graph Zd satisfies the strong Liouville property. Oneproof is given in [Bar17]. Here, M. Barlow followed an already established strategy:showing that Zd satisfies the so-called elliptic Harnack inequality (EHI) and also show-ing that for any network (EHI) =⇒ (SLP).61Chapter 3. The number of trees in USF(c) The network Γd(λ) does not satisfy (SLP). Consider the function h(z) = e−λn (withz = (n, x) as usual). Then, with p = (2d+ 1 + eλ)−1,∑z′∼zp(z, z′)h(z′) = pe−λ(n−1) + 2dpe−λn + eλpe−λ(n+1) = e−λn = h(z),so that h is harmonic, and proving the claim.Proposition ( 3.3.2 ) For the network (G, µ) to satisfy the Liouville property, it sufficesthat, for any pair of vertices, two lazy network random walks (1.1.13), one started at eachvertex, can be classically coupled. In other words,(CC) =⇒ (LP).Proof. Let h be any bounded harmonic function on the network. If x and y are two verticesand S, S ′ are two network random walks started at x and at y, respectively, with successfulcoupling time τ, then the martingale convergence theorem implies that h(x) = E (h (S0)) =E (h (Sτ )) and, similarly, E (h (S′τ )) = h(y). By definition, Sτ = S′τ , so h(x) = h(y). As xand y were any two vertices, the proof is complete.Theorem ( 3.3.3 ) The network Γd(λ) has the Liouville property.Proof. By virtue of the previous proposition, it suffices to prove that any two lazy networkrandom walks (started at any two vertices) of Γd(λ) can be classically coupled. In this case,the hypothesis of theorem (3.2.3) are satisfied: the measure νi ∗ ν¯i, where νi is as in (3.1.9)is clearly symmetric and bounded, it is aperiodic by lazyness, thus (3.1.7) shows that νi ∗ ν¯iis also recurrent and irreducible.§ 3.4. Liouville property and the uniform spanning for-est measure.The Liouville property is relevant to the context of uniform spanning forests due to thefollowing theorem.Theorem ( 3.4.1 ) Let Γ be a network satisfying the usual assumptions (A.22) and suchthat the underlying graph is simple (A.8). A sufficient condition for the measures WSF andFSF, of the wired and free uniform spanning forest on G, to be the same, is that this networkshould satisfy the Liouville property.Proof. From annex D, if a network has the Liouville property, then all harmonic Dirichletfunction are constant (D.6.3), and this is equivalent to WSF = FSF by (C.4.2).62Chapter 3. The number of trees in USFAs a corollary of the foregoing.Theorem ( 3.4.2 ) WSF = FSF on Γd(λ).By virtue of the equality between WSF and FSF on Γd(λ), we will denote this measureas USF from now onwards.§ 3.5. The number of components.To count the number of components, the reader should recall the definition of transitivenetwork (A.23). We know Γd(λ) is transitive by considering translations.We need the following results:[LP16], Theorem 10.24. In any transient transitive network (G, µ), with vertex set Vand Green’s function G, the number of intersections of any two independent networkrandom walks is almost surely infinity if∑z∈VG(0, z)2 = ∞ and is almost surely finiteotherwise.[BLPS01], Theorem 9.2. For the number of components in WSF (of any network) to beone, it is necessary and sufficient that, two independent network random walk startedat any two vertices, should intersect (a.s.).[BLPS01], Theorem 9.4. If there exists a vertex o such that, for almost every realisation,the number of intersections of two independent copies of the network random walkstarted at o is finite, then the number of components in WSF is infinite.Armed with these, we are ready to prove the main theorem of this section. The proofproceeds using Fourier analysis, the proof could go using the Green’s function estimatesbut Fourier analysis is simply faster and cleaner way to show this since the Green’s functionestimates are admittedly difficult to handle, specially due to the several cases they can create.Theorem ( 3.5.1 ) Consider the network Γd(λ). The USF here (3.4.2) consists of a singletree when d = 1, 2 and it has an infinite number of components when d ≥ 3. From nowonwards, we use UST in lieu of USF for d = 1, 2.Proof. By the theorems stated above, it all reduces to show that∑z∈Zd+1G(0, z)2 diverges ford = 1, 2 and that it converges otherwise. In the sake of clarity, we separate the calculationsin the next proposition, with them the theorem is proved.63Chapter 3. The number of trees in USFAs mentioned in the proof above, the next theorem states when the Green’s functionbelongs to L 2.Theorem ( 3.5.2 ) (The “bubble condition.”)(a) If d = 1, 2,∑z∈Zd+1G(0, z)2 =∞.(b) If d ≥ 3, ∑z∈Zd+1G(0, z)2 <∞.Proof. Recall the Fourier transform of the step-lengths of the network random (2.1.6). Now,if (Sn) is the network random walk started at zero and ϕn is its Fourier transform of Sn,then ϕn(h) = ϕY (h)n. Hence, for all h 6= 0,(3.5.3) “G(h) = ∞∑n=0ϕn(h) =11− ϕY (h) ,where “G denotes the Fourier transform of the Green’s function G. By Parseval’s relation,(3.5.4)∑z∈Zd+1G(0, z)2 =1(2pi)d+1∫[−pi,pi]d+1dh∣∣∣“G(h)∣∣∣2 ,where the two integrals will converge to the same value, or they will both diverge.Using second order expansion in (2.1.6) and with a as in (2.1.5) and p = (2d+ 1 + eλ)−1,we reach(3.5.5) ϕY (h) = 1 + iah0 − a2h20 − pd∑j=0h2j + oÄh20ä+ OÄ‖h‖4ä , h→ 0.Denote αj = p for j = 1, . . . , d and α0 = p+a2,ϕY (h) = 1 + iah0 −d∑j=0αjh2j + oÄh20ä+ OÄ‖h‖4ä .Observe first thatd∑j=0αjh2j  ‖h‖2 ; also, write h = (h0, h′), then‖h‖4 = h40 + ‖h′‖4 + 2h20 ‖h′‖2  h40 + ‖h′‖4 .64Chapter 3. The number of trees in USFThen,|1− ϕY (h)|2 = a2h20 + iah0d∑j=0αjh2j + oÄh20ä+ OÄh0 ‖h‖4ä(∗)− iah0d∑j=0αjh2j +Ñd∑j=0αjh2jé2+ OÄ‖h‖6ä= a2h20 +Ñd∑j=0αjh2jé2+ oÄh20ä+ OÄh0 ‖h‖4ä+ oÄ‖h‖4ä .Divide the right-hand side of (3.5.4) as the sum of Iε+Jε, where Iε has domain of integrationKε = [−ε, ε]×BRd (0; ε) and Jε has domain of integration [−pi, pi]d+1 \Kε. It follows at onceJε <∞. We will show the existence of a positive ε such that Iε =∞ for d = 1, 2 and Iε <∞for d ≥ 3.Assume first d = 1, 2. Then by (∗) for ε small enough, |1−ϕY (h)|2 ≤ c−1(h20 + ‖h′‖4) forsome c > 0. Then,Iε ≥ c∫BRd(0;ε)dh′ε∫−εdh01h20 + ‖h′‖4= 2c∫BRd(0;ε)dh′ε∫0dh01h20 + ‖h′‖4.Changing measures to surface measure on the spheres of Rd, there is a constant c > 0 suchthatIε ≥ cε∫0dρ ρd−1ε∫0dh01h20 + ρ4= cε∫0dρ ρd−3 arctanÇερ2å.Now, ρ ≤ ε implies ερ2≥ 1ε, and ε can be assumed small enough so that arctan(ε−1) ≥ pi4,thusIε ≥ cpi4ε∫0dρ ρd−3 =∞,the last equality holds true with the assumption d = 1, 2. This establishes item (a).Similarly from (∗), we can assume ε > 0 is small enough so that for some constant c > 0,|1− ϕY (h)|2 ≥ c−1(h20 + ‖h′‖4).As before, we change to surface measure of spheres and use that ρd−1 ≤ ρ2 since ε can beassumed smaller than unity and d ≥ 3, thenIε ≤ c∫Kε1h20 + ‖h′‖4d(h0, h′) ≤ c′ε∫0dρ ρ2 · 1ρ2arctanÇh0ρ2å ∣∣∣∣∣∣h0=εh0=0<∞.65Chapter 3. The number of trees in USFTherefore, we reach that the sum (3.5.4) is finite for all d ≥ 3. This completes (b) and withit, the proposition.66Chapter 4Crossings of NRWThe purpose of this chapter if to estimate how frequently two independent network randomwalks of Γd(λ) cross each other. The reader should remember the definition of crossings orintersection of two paths and that of collision, see (A.20).§ 4.1. Dimensions d = 1, 3, 4, . . . .Theorem ( 4.1.1 ) Let d ≥ 3. Consider two independent network random walks S and S ′on Γd(λ). Suppose they have arbitrary (random) initial points S0 and S′0, respectively. Then,almost surely, the paths of S and S ′ will cross finitely many times.Proof. By virtue of (3.5.2),∑z∈Zd+1G(0, z)2 < ∞ in this range of dimensions. The theoremthen follows from [LP16, Theorem 10.24].The previous theorem is not true in dimension d = 1, 2. However, there is an importantdistinction between the cases d = 1 and d = 2. The gist of the distinction lies in howfrequently the two random paths cross. In this section we consider the case d = 1.Theorem ( 4.1.2 ) Let d = 1. Consider two independent network random walks S andS ′ on Γ1(λ) with arbitrary starting points, S0 and S ′0. Then, S − S ′ is a recurrent randomwalk; in particular, for almost every realisation, there will be infinitely many n for whichSn−S ′n = S0−S ′0. Thus, if both S and S ′ start at the same point, then S and S ′ will collideinfinitely often.Proof. By means of a translation by −(S0− S ′0), we can assume S0− S ′0 = 0. The difference67Chapter 4. Crossings of NRWS − S ′ is a random walk, started at zero and has step-length distributionµ = p2eλÅε(−2,0) + ε(2,0)ã+ p2Åε(0,2) + ε(0,−2)ã+ p2(eλ + 1)Åε(−1,−1) + ε(1,1) + ε(−1,1) + ε(1,−1)ã+ p2(e2λ + 3)ε(0,0),where ε(h,k) denotes the measures of unitary total mass at (h, k) ∈ Z2 and p = 13 + eλ.Observe that S − S ′ is a symmetric random walk with bounded step lengths on the latticegenerated by u1 = (1, 1) and u2 = (−1, 1), that is to say, the set of sums su1 + tu2 with(s, t) ∈ Z2. Consider the linear transformation L that sends u1 to (1, 0) and u2 to (0, 1) (ahomothety by factor1√2and then a rotation by −pi4). With this L, L(S − S ′) = LS − LS ′is a random walk on Z2 with the following transition densityρ = p2eλÅε(−1,1) + ε(1,−1)ã+ p2Åε(1,1) + ε(−1,−1)ã+ p2(eλ + 1)Åε(−1,0) + ε(1,0) + ε(0,1) + ε(0,−1)ã+ p2(e2λ + 3)ε(0,0).Clearly, ρ is symmetric, hence it has mean zero, it has bounded support, hence finite secondmoment, is aperiodic since ρ(0, 0) > 0, and irreducible since ρ(±1, 0) > 0 and ρ(0,±1) > 0.By [LL10, Theorem 4.1.1] LS − LS ′ is recurrent, meaning this that {LSn = LS ′n i.o.} hasprobability one. Since L is an invertible linear transformation, Sn = S′n occurs for infinitelymany n with probability 1.The remainder of this chapter is devoted into estimating how frequently there will bea crossing between two network random walk paths on Γ2(λ). Whence, in the upcomingsections we will develop the necessary tools to tackle dimension d = 2.§ 4.2. Two elementary results.For the sake of reference, we write the following. Suppose that Z is a non-negative randomvariable with finite second moment. Then, Cauchy-Schwarz inequality impliesE (Z) = E(Z1{Z> 12E(Z)})+ E(Z1{Z≤ 12E(Z)})≤ E ÄZ2ä 12 PÇZ > 12E (Z)å 12+12E (Z) .Therefore, (for Z 6= 0 a non-negative random variable)(4.2.1) PÇZ >12E (Z)å≥ E (Z)24E (Z2).68Chapter 4. Crossings of NRWConsider now an increasing sequence of σ-fields (En)n∈Z+ , with E0 the σ-field generatedby ∅, on some probability space (E,E ,P). Suppose that An is an event measurable up totime n, that is An ∈ En. Set qn = P (An | En−1) (so that q1 = P (A1)). We have the followingresult.Proposition ( 4.2.2 ) The event N = {An i.o.}4ß ∞∑n=1qn =∞™is P-null (“Levy’s exten-sion of Borel-Cantelli lemma”).The proof of this result is a standard application of martingale theory to Xn =n∑j=1(1Aj −qj). See section 7.8 of [Ash72] (apply theorem 7.8.4).§ 4.3. Setup for Γ2(λ).As before, we will write z = (n, x); any affix, such as a sub or superscript, will be given to thethree symbols: if we write z′ at some point, it will be assumed that z′ = (n′, x′), similarly, ifwe talk about n1 and x1, we will be assuming z1 = (n1, x1), etc.Divide the space Z× Z2 into the two setsS1 = {z;n ≤ ‖x‖},S2 = {z;n > ‖x‖}.Notice 0 ∈ S1 and S1 ∩ S2 = ∅. To simplify notation, we define (for any y ∈ Zr, r ≥ 1)(4.3.1) 〈y〉 = max(‖y‖ , 1),and notice 〈y〉 = ‖y‖ unless y = 0; also 〈y〉  ‖y‖+ 1 »‖y‖2 + 1.Observe that by Theorem (2.6.1), the Green’s function bounds for Γ2(λ) are:G(0, z) ≤c1e−c2‖z‖ 〈z〉−1 if z ∈ S1,c1e−c2 ‖x‖2n ‖z‖−1 if z ∈ S2,andG(0, z) ≥c3e−c4‖z‖ 〈z〉−1 if z ∈ S1,c3e−c4 ‖x‖2n ‖z‖−1 if z ∈ S2.69Chapter 4. Crossings of NRWConsider the following tessellation (p, q ∈ N and i = 1, 2) of the “positive half-space”Z+ × Z2:R1(p, q) = {z : 9p < n ≤ 2 · 9p, 3q−1 < ‖x‖ ≤ 3q},R2(p, q) = {z : 2 · 9p < n ≤ 9p+1, 3q−1 < ‖x‖ ≤ 3q}R1(p, 0) = {z : 9p < n ≤ 2 · 9p, ‖x‖ ≤ 1},R2(p, 0) = {z : 2 · 9p < n ≤ 2 · 9p+1, ‖x‖ ≤ 1},R1(0, q) = {z : 0 ≤ n ≤ 2, 3q−1 < ‖x‖ ≤ 3q}R2(0, q) = {z : 3 ≤ n ≤ 9, 3q−1 < ‖x‖ ≤ 3q}R1(0, 0) = {z : 0 ≤ n ≤ 2, ‖x‖ ≤ 1}R2(0, 0) = {z : 3 ≤ n ≤ 9, ‖x‖ ≤ 1}(4.3.2)For p ∈ N, we shall consider the following region(4.3.3) Dp =p⋃q=0R1(p, q) = {z : 9p < n ≤ 2 · 9p, ‖x‖ ≤ 3p}.The following remark contains all the technical estimates that will be employed later.Remark ( 4.3.4 ) For i = 1, 2 and (p, q) ∈ Z2+, we have the following.(a) card (Ri(p, q))  9p+q.(b) For any two different multi-indices (i1, p1, q1) and (i2, p2, q2) the two regions Ri1(p1, q1)and Ri2(p2, q2) are disjoint.(c) The previous two points imply that card (Dp)  9pp∑q=09q  92p.(d) We haveRi(p, q) ⊂ S1 if 2p+ 3 ≤ q,Ri(p, q) ⊂ S2 if q ≤ 2p.The relations between p and q will imply 9p+1 ≤ 3q−1 in the first case and 3q ≤ 9p inthe second case, hence the result. Observe that the cases q = 2p+ 1 or q = 2p+ 2 donot guarantee Ri(p, q) contained solely in S1 or only in S2.(e) If z ∈ Z+ × Z2 is in S1 and c is a positive constant, then 0 ≤ n ≤ ‖x‖ , this implies‖x‖ ≤ ‖z‖ ≤ √2 ‖x‖ , and then1√2e−√2c‖x‖ 〈x〉−1 ≤ e−c‖z‖ 〈z〉−1 ≤ e−c‖x‖ 〈x〉−1 .70Chapter 4. Crossings of NRWTherefore, if 3q−1 ≤ ‖x‖ ≤ 3q,1√2e−√2c3q3−q ≤ e−c‖z‖ 〈z〉−1 ≤ 3e− c33q3−q.(f) If z ∈ Z+ × Z2 is in S2 and c is a positive constant, then ‖x‖ < n, this implies0 < n ≤ ‖z‖ ≤ √2n, and then1√2e−c‖x‖2n n−1 ≤ e−c ‖x‖2n ‖z‖−1 ≤ e−c ‖x‖2n n−1.Therefore, if 9p ≤ n ≤ 9p+1 and 3q−1 ≤ ‖x‖ ≤ 3q19√2e−c9q−p9−p ≤ e−c ‖x‖2n ‖z‖−1 ≤ e− c819q−p9−p.(g) Define the function ϕU : Z×Z2 → R given by ϕU(z) = c1e−c2‖z‖ 〈z〉−1 if z lies in regionS1 and ϕU(z) = c1e−c2 ‖x‖2n ‖z‖−1 for z within S2. ThenG(0, z) ≤ ϕU(z).Consider z any point in R1(p, q) ∪ R2(p, q). The previous estimates show thatϕU(z) ≤3c1e− c233q3−q if z ∈ S1,c1e− c2819q−p9−p if z ∈ S2.We also know that Ri(p, q) ⊂ S1 if 2p+ 3 ≤ q and Ri(p, q) ⊂ S2 if 2p ≥ q (i = 1, 2). Ifq = 2p+ 1 or q = 2p+ 2, we can guarantee the boundϕU(z) ≤ max(3c1e− c233q3−q, c1e−c2819q−p9−p)= c1e− c2819q−p9−p,where the equality follows from direct comparison. Consequently, if z ∈ Ri(p, q) (i =1, 2)ϕU(z) ≤3c1e− c233q3−q if 2p+ 3 ≤ q,c1e− c2819q−p9−p if q ≤ 2p+ 2.(h) Similarly to the previous point, define ϕL(z) = c3e−c4‖z‖ 〈z〉−1 if z ∈ S1 and ϕL(z) =c3e−c4 ‖x‖2n ‖z‖−1 if z ∈ S2, with the constants c3 and c4 chosen so that ϕL is a lowerbound of the Green’s function z 7→ G(0, z). As in the previous point, for z ∈ Ri(p, q)(i = 1, 2)ϕL(z) ≥c3√2e−√2c43q3−q if 2p+ 3 ≤ q,c39√2e−c49q−p9−p if q ≤ 2p+ 2.71Chapter 4. Crossings of NRW(i) For z = (n, x) ∈ Z × Z2 define z˙ = (−n, x) ∈ Z × Z2 (the “reflection” of z throughthe “plane” {0} × Z2). Then, z 7→ z˙ is a bijection of Z± × Z2 onto Z∓ × Z2, it is theidentity when restricted to {0} × Z2, and for every z ∈ Z− × Z2,ϕU(z) ≤ ϕU (z˙) .To prove this, there are two cases to consider:1. Both z and z˙ fall in the same region, which is necessarily S1 and then there is, infact, the equality between ϕU(z) and ϕU (z˙) since ‖z‖ = ‖z˙‖ .2. z and z˙ fall in different regions, necessarily z ∈ S1 and z˙ ∈ S2 and the latterrelation implies ‖x‖ < −n, from which we can deduce ‖z‖ ≥ ‖x‖2−n and thusproving ϕU(z) ≤ ϕU (z˙) .(j) From previous points, there are constants d1 = 3c1, d2 =c281, d3 =c318and d4 = 2c4,such that the relation z ∈ Ri(p, q) (p, q ∈ Z+, i = 1, 2) implyG(0, z) ≤ ϕU(z) ≤d1e−d23q3−q if 2p+ 1 ≤ q,d1e−d29q−p9−p if q ≤ 2p.andG(0, z) ≥ ϕL(z) ≥d3e−d43q3−q if 2p+ 1 ≤ q,d3e−d49q−p9−p if q ≤ 2p.(k) In particular, if 0 ≤ q ≤ p, thend3e−d49−p ≤ G(0, z) ≤ d19−p.Thus, G(0, z)  9−p with any constant being uniform for z ∈ Dp and p ∈ N.z 4.3.1 Some preliminary sums of the Green’s function of Γ2(λ).In what follows, the estimates in Remark (4.3.4) will be used, often without referencing tothis remark.Proposition ( 4.3.5 ) Let S and S ′ be two independent network random walks started at 0in Γ2(λ). Then, with any implicit constants being independent of p ∈ N,(a)∑z∈DpG(0, z)2  1; and72Chapter 4. Crossings of NRW(b)∑z∈Dp−DpG(0, z)G(0,−z)  1.(c) Also, there exists a constant c > 0 such that∑z∈Dp−DpG(0, z)2 ≤ cp,for all p ∈ N.Proof. We know G(0, z)2  9−2p and card (Dp)  92p for z ∈ Dp and the implicit constantsindependent of p; this proves the first assertion∑z∈DpG(0, z)2  1, with the implicit constantsindependent of p. It remains to show the other two estimates. First, it is clear that∑z∈Dp−DpG(0, z)G(0,−z) ≥ G(0, 0)2.Hence, there remains to prove the existence of constants c > 0 and c′ > 0 such that∑z∈Dp−DpG(0, z)G(0,−z) ≤ c, ∑z∈Dp−DpG(0, z)2 ≤ c′p,for all p ∈ N. Set Q(p) = Dp − Dp and Q(p)+ = Q(p) ∩ (Z+ × Z2). Consider the function ϕUintroduced in (4.3.4), we regard the constants ck and dk (k = 1, . . . , 4) of (4.3.4) to be fixedduring the argument. Then∑z∈Q(p)G(0, z)G(0,−z) ≤ 2 ∑z∈Q(p)+G(0, z)G(0,−z) ≤ 2 ∑z∈Q(p)+ϕU(z)ϕU(−z),and ∑z∈Q(p)G(0, z)2 ≤ ∑z∈Q(p)ϕU(z)2 ≤ 2 ∑z∈Q(p)+ϕU(z)2where the second inequality in the previous line follows from considering the reflection z 7→ z˙(4.3.4)(i). To simplify notation, introduce R(s, t) to be the union R1(s, t) ∪ R2(s, t), thesesets being defined in (4.3.2). We know that there exists a constant L > 0 such that for alls, t ∈ Z+, card (R(s, t)) ≤ L9s+t. Then, one can writeQ(p)+ ⊂⋃(s,t)R(s, t),where the indices run over all choices of 0 ≤ s, t ≤ p + 1. Now, the sets R(s, t) are pairwisedisjoint and, therefore, we split∑z∈Q(p)+ϕU(z)ϕU(−z) ≤∑(s,t)∑z∈R(s,t)ϕU(z)ϕU(−z) = P1 + P2,73Chapter 4. Crossings of NRWand, in a similar fashion, ∑z∈Q(p)+ϕU(z)2 ≤ T1 + T2,where P1 and T1 are the sums corresponding to t < s and P2 and T2, the sums for t ≥ s.We shall show that P1 and P2 are bounded by constants independent of p, and T1 ≤ cp andT2 ≤ c′p for a pair of constants c, c′ > 0 independent of p; having this, we will conclude∑z∈Dp−DpG(0, z)G(0,−z) ≤ c and ∑z∈Q(p)G(0, z)2 ≤ c′p for some constants c > 0 and c′ > 0, asdesired.We now prove P1 and P2 are bounded by constants independent of p. Observe from(4.3.4)(g) that for all z ∈ R(s, t), eitherϕU(z) ≤ d1e−d23t3−tor,ϕU(z) ≤ d1e−d29t−s9−s.Also, for z ∈ Q(p)+ we have ϕU(−z) = c1e−c2‖z‖ ‖z‖−1 . Therefore (use (4.3.4)(a)),P2 ≤p+1∑s=0p+1∑t=s∑z∈R(s,t)ϕU(z)ϕU(−z)≤p+1∑s=0p+1∑t=s∑z∈R(s,t)d1Äe−d23t3−t + e−d29t−s9−sä3c1e− c233t3−t≤ 3c1d1Lp+1∑s=0p+1∑t=sÄe−d23t9s + e−d29t−s3täe−c233t≤ 3c1d1L∞∑l=0e−c233l([ ∞∑k=0e−d23k]9l +[ ∞∑k=0e−d29k3k]3l)<∞.The sum P1 is easier,P1 ≤p+1∑s=0s−1∑t=0∑z∈R(s,t)ϕU(z)ϕU(−z) ≤ c1d1Lp+1∑s=0s−1∑t=09s+te−d29t−s9−se−c29s9−s≤ c1d1Lp+1∑s=0e−c29s9−ss−1∑t=09t =c1d1L8∞∑k=0e−c29k<∞.We continue with the proof of the bounds of T1 and T2, beginning with T2 ≤ cp, for someconstant c > 0. By definition of T2,T2 =p+1∑s=0p+1∑t=s∑z∈R(s,t)ϕU(z)2.74Chapter 4. Crossings of NRWAs before, we haveϕU(z)2 ≤ d21Åe−2d23t9−t + e−2d29t−s9−2sã.Therefore, with C = d21L,T2 ≤ Cp+1∑s=0p+1∑t=s(e−2d23t9−t + e−2d29t−s9−2s)9s+t = Cp+1∑s=0p+1∑t=s(e−2d23t9s + e−2d29t−s9t−s)≤ C∞∑k=0e−d23k9k∞∑k=0e−d23k+ C(p+ 2)∞∑k=0e−2d29k9k ≤ C ′p,whereC ′ = C∞∑k=0e−d23k9k∞∑k=0e−d23k+ 3C∞∑k=0e−2d29k9k <∞is independent of p. Similarly, notice one can expressT1 =p+1∑s=0s−1∑t=0∑z∈R(s,t)ϕU(z)2,bearing this in mind and that ϕU(z) ≤ d19−s for 0 ≤ t ≤ s it follows immediately that, forsome positive constants c and c′,T1 ≤ cp+1∑s=0s∑t=09s+t9−2s ≤ c′p,as desired.z 4.3.2 Estimates on the Green’s function on some regions of Γ2(λ).Denote with S and S ′ two independent network random walks of Γ2(λ).Consider the following region (p ∈ N)(4.3.6) Up =¶(n, x) ∈ Z3 : |n| ≤ 4 · 9pk0 , ‖x‖ ≤ 3(p+1)k0© ,where k0 is a positive integer to be determined later on. Define the “separating cylinders”(creating a barrier between Dpk0 and D(p+1)k0 for p ∈ N) to be the (vertex) boundary(4.3.7) Fp = ∂Up,and set F0 = {0}. Having these sets, define (p ∈ Z+)(4.3.8) Tp = τFp(S)75Chapter 4. Crossings of NRWthe hitting time of Fp by S. If z ∈ Z3, let(4.3.9) pz = min{p | z ∈ Up}.Having defined pz, observe that when S0 = o, then a.s. the sequence of random variables(Tp)p≥po is increasing and each one of them is finite (by transience).Next, by definition Fp∩Up = ∅. It will be useful to divide the set Fp into several regions.Geometrically, the set Fp can be thought as a cylinder and we need to consider the body andthe two bases separately. Analytically, if (n, x) ∈ Fp then |n| > 4 ·9pk0 or else ‖x‖ > 3(p+1)k0 .We can specialise further for either |n| = 4 · 9pk0 + 1 and ‖x‖ ≤ 3(p+1)k0 or else, |n| ≤ 4 · 9pk0and 3(p+1)k0 ≤ ‖x‖ ≤ 3(p+1)k0 + 1. Thus, the exponential scales being considered permit usto assume that ‖z‖ , for z ∈ Fp, satisfies :(a) |n| = 4 · 9pk0 and ‖x‖ ≤ 3(p+1)k0 , or(b) |n| < 4 · 9pk0 and ‖x‖ = 3(p+1)k0 .With this assumption, we introduce the following sets. The “left-hand base of the cylinderFp” is the set of z ∈ Fp such that n = −4 · 9pk0 ; the “right-hand base of the cylinder Fp” isthe set of z ∈ Fp such that n = 4 · 9pk0 . The points in Fp that belong to neither the left northe right-hand base of the cylinder will be referred to as the “body of the cylinder Fp,” inother words, the body of the cylinder is the set of points z ∈ Fp such that |n| < 4 · 9pk0 and‖x‖ = 3(p+1)k0 .Define (T ′p) as above changing the process S for the process S′. Observe that (Tp) and(T ′p) are independent families.Having the times (Tp) and (T′p) we now consider “sections” of the network random walksand the number of intersections in Dpk0 of two such sections. Thus, define (p ∈ N)(4.3.10) Mp =∑z∈Dpk0Tp∑m=Tp−1T ′p∑m′=T ′p−11{Sm=z}1{S′m′=z}.Observe that Mp counts the number of intersections of the two random walks inside theregion Dpk0 starting in the separating cylinder Fp−1. We remark that, if S0 = o, then Scannot be inside Dpk0 before the time Tp−1 provided p−1 ≥ po; similarly, S ′ cannot be insideDpko before time T′p−1 provided S′0 = o′ and p− 1 ≥ po′ .We shall use (4.2.2), hence we will need to estimate E (Mp | Ep−1) and EÄM2p | Ep−1ä, fora filtration that will be specified later. For the time being, let us estimate E (Mp) . InvokingLebesgue-Tonelli’s theorem, independence of S and S ′, that these two process have the same76Chapter 4. Crossings of NRWdistribution and then the strong Markov property one can concludeE (Mp) =∑z∈Dpk0EÖTp∑m=Tp−1T ′p∑m′=T ′p−11{Sm=z}1{S′m′=z}è=∑z∈Dpk0EÑTp∑m=Tp−11{Sm=z}é2=∑z1∈Dpk0 ∑z2∈Fp−1PÄSTp−1 = z2äEz2ÑTp−1∑m=01{Sm=z1}é2 .We will study the Green’s function restricted to the region Up (1.1.9), that is (z ∈ Z3)(4.3.11) GUp(0, z) = E0ÑTp−1∑m=01{Sm=z}é=∞∑m=0P0 (Sm = z, Tp > m) ,and for consistency we denote U0 = {0}. For the sake of simplicity, we will write Gp = GUp .Proposition ( 4.3.12 ) For every p ∈ N,Dpk0 − Up−1 ⊂ Up.Moreover, for any ε > 0 there exists an index k0 = k0(ε) such that for some p0 ∈ N therelations p ≥ p0 and z ∈ Dpk0 − Up−1 implyG(0, z) ≤ (1 + ε)Gp(0, z).Proof. The proof contains several steps. We begin by noticing that for z1 ∈ Dpk0 andz2 ∈ Up−1 one has by (4.3.3) and (4.3.6)9pk0 <n1 ≤ 2 · 9pk0 ,−4 · 9(p−1)k0 ≤n2 ≤ 4 · 9(p−1)k0 ,‖x1‖ ≤ 3pk0 ,‖x2‖ ≤ 3pk0 ,which implies( 4.3.12.1 ) Any (n, x) ∈ Dpk0 − Up−1 satisfies12· 9pk0 ≤ n ≤ 529pk0 , ‖x‖ ≤ 2 · 3pk0 .In particular, z1−z2 belongs to the region Up so this establishes the first assertion in (4.3.12).We now prove the second assertion of (4.3.12): this requires some calculations. Beginwith (2.4.1) applied to the stopping time τ = Tp. This givesG(0, z) = Gp(0, z) + E0ÄG(STp , z)ä.77Chapter 4. Crossings of NRWObserve that STp is in Fp and, therefore, we need to estimate G(z3, z) for z3 ∈ Fp. This willbe done in the upcoming three lemmas.We will pause momentarily the proof of the proposition in order to establish these lemmasand resume it once they have been proven.Lemma ( 4.3.12.2 ) (a) For every ε > 0, there exists a p0 = p0(ε) such that for all p ≥ p0,all k0 ∈ N, and all z1 ∈ Dpk0 , z2 ∈ Up−1 and z3 ∈ Fp for which n1−n2−n3 ≤ 2 ·3(p+1)k0and z3 does not lie in the left-hand base of Fp, we have G(z3, z1− z2) ≤ εG(0, z1− z2).(b) There exists a constant c > 0 such that for p ∈ N and all z1 ∈ Dpk0 , z2 ∈ Up−1 andz3 ∈ Fp for which n1 − n2 − n3 ≥ 2 · 3(p+1)k0 and z3 does not lie in the left-hand baseof Fp then G(z3, z1 − z2) ≤ c9−k0G(0, z1 − z2).Proof of lemma. Suppose the constants ck (1 ≤ k ≤ 4) of (2.6.1) are given. By (4.3.12.1)and the estimates on the Green’s function, we haveG(0, z1 − z2)  9−pk0 .We prove part (a) first. Assume z1, z2, z3 are as stated. Consider first the case n3 < 4·9pk0 ,that is to say, consider first when z3 is not in the right-hand base of Fp. Then ‖x3‖ = 3(p+1)k0 ,which in turn yields13· 3(p+1)k0 ≤ ‖x1 − x2 − x3‖ ≤ 2 · 3(p+1)k0 .Suppose first that n1 − n2 − n3 ≤ 13· 3(p+1)k0 , then z1 − z2 − z3 is in S1, implying by (2.6.1)G(z3, z1 − z2) ≤ c1e−c2‖z1−z2−z3‖ ‖z1 − z2 − z3‖−1 ≤ 3c1e−c233(p+1)k03−(p+1)k0=[3c1e− c233(p+1)k03(p−1)k0]9−pk0 ≤[3c1e− c233pk03pk0]9−pk0 .Define u1(t) = 3c1e− c233t3t, so thatG(z3, z1 − z2) ≤ u1(pk0)9−pk0 ,for z1, z2 and z3 as stated. Next, suppose that13· 3(p+1)k0 ≤ n1−n2−n3 ≤ 2 · 3(p+1)k0 . Then,G(z3, z1 − z2) ≤ c1Çe−c2‖z1−z2−z3‖ + e−c2‖x1−x2−x3‖2n1−n2−n3å‖z1 − z2 − z3‖−1≤ 3c1(e−c233(p+1)k0 + e−c2183(p+1)k0)3−(p+1)k0=[3c1(e−c233(p+1)k0 + e−c2183(p+1)k0)3(p−1)k0]9−pk0≤[3c1(e−c233pk0 + e−c2183pk0)3pk0]9−pk0 = u2(pk0)9−pk0 ,78Chapter 4. Crossings of NRWwhere u2(t) is defined to be 3c1Äe−c233t + e−c2183tä3t. Consider now the case where z3 is in theright-hand base of the cylinder, that is to say, assume n3 = 4 · 9pk0 and ‖x3‖ ≤ 3(p+1)k0 butin this case, −72· 9pk0 ≤ n1 − n2 − n3 ≤ −32· 9pk0 and again, by (2.6.1),G(z3, z1 − z2) ≤ c1e−c2|n1−n2−n3||n1 − n2 − n3|−1≤ 2c13e−2c239pk09−pk0 .Here we set u3(t) =2c13e−2c239t , so that in this case G(z3, z1 − z2) ≤ u3(pk0)9−pk0 . Noticethat in all three cases above, we bounded G(z3, z1 − z2) ≤ ui(pk0)9−pk0 , and it is clear thatui(t)→ 0 as t→∞ in each case. Then, there exists a t0 such thatsupt≥t0maxi=1,2,3ui(t) ≤ ε.We can define p0 to be the least integer greater than or equal to t0; this p0 works for all k0since pk0 ≥ p. This finishes part (a).We now prove part (b). Suppose that z1, z2, z3 are as stated in (b). This implies therelation n1 − n2 − n3 ≥ 2 · 3(p+1)k0 . By (4.3.12.1), we have n1 − n2 − n3 ≤ 7 · 9pk0 and,therefore n1 − n2 − n3 lies in an interval of the form Jα =î(α− 1)3(p+1)k0 , α3(p+1)k0ó whereα = 3, . . . , 7 · 3(p−1)k0 . Regardless the value of α, the relation n1 − n2 − n3 ∈ Jα impliesz1 − z2 − z3 ∈ S2 and these relations imply,G(z3, z1 − z2) ≤ c1e−c2‖x1−x2−x3‖2n1−n2−n3 (n1 − n2 − n3)−1≤ 2c1e−c29α−13(p+1)k0α−13−(p+1)k0 .Observe that for any constant c > 0, the function uc(t) = t−1e−ct−1for t > 0 has an absoluteand global maximum at t = c. Its maximum equals uc(c) = e−1c−1. By considering thefunction uc for the value c =c293(p+1)k0 , we deduceG(z3, z1 − z2) ≤ 2c1e−1 9c29−(p+1)k0 = c′9−k09−pk0 ,the desired conclusion since G(0, z1− z2)  9−pk0 , with any implicit constant independent ofp and of k0.Lemma ( 4.3.12.3 ) For any ε > 0, there exist indices k0 = k0(ε) ∈ N and p0 = p0(ε) ∈ Nsuch that the relations p ≥ p0 and z ∈ Dpk0 − Up−1 implysupz3∈Fpn3>−4·9pk0G(z3, z) ≤ εG(0, z).79Chapter 4. Crossings of NRWProof of lemma. By (4.3.12.2), given ε > 0 there is a p0 ∈ N such that if p ≥ p0, k0 ∈ N andfor all z1 ∈ Dpk0 , z2 ∈ Up−1 and z3 ∈ Fp with n3 > −4 · 9p, we either have G(z3, z1 − z2) ≤εG(0, z1−z2) or G(z3, z1−z2) ≤ c9−k0G(0, z1−z2), where the constant c is universal. Choosek0 ∈ N such that c9−k0 ≤ ε. Hence, G(z3, z1− z2) ≤ εG(0, z1− z2) uniformly in z3 ∈ Fp withn3 > −4 · 9p for all p ≥ p0, which is the conclusion to be reached.We need a third and final lemma before continuing with the proof of proposition (4.3.12).The previous two lemmas handled the right-hand base and body of the separating cylinderFp, so we now handle the left-hand base. Because of the Green’s function bounds, the boundsin the next lemma are coarser but are what is to be expected.Lemma ( 4.3.12.4 ) For any ε > 0, there exist indices k0 = k0(ε) ∈ N and p0 = p0(ε) ∈ Nsuch that the relation p ≥ p0 impliesP0ÄSTp ∈ {z ∈ Fp : n = −4 · 9p}ä ≤ ε.Also, there is a universal constant c > 0 such thatsupp∈Nsupz∈Dpk0−Up−1supz3∈Fpn3=−4·9pG(z3, z) ≤ cG(0, z)Proof of lemma. Set Ap = {z ∈ Fp : n = −4 · 9pk0}. Consider the network random walk S inΓ2(λ) started at zero and define B = pr0(S). By (2.4.5),PÄSTp ∈ Apä ≤ P ÄB ever reaches − 4 · 9pk0ä = e−4λ9pk0 .This proved the first claim. To prove the second claim, use (4.3.12.1), if z3 ∈ Ap andz ∈ Dpk0−Up−1 then G(z3, z)  9−pk0 with any implicit constant being universal in p, k0, z, z3.Similarly, G(0, z)  9−pk0 for z ∈ Dpk0 − Up−1 with any implicit constant universal. Thesecond claim follows readily.We are ready to continue with the proof of Proposition (4.3.12). Let us write Ap = {z ∈Fp : n3 = −4 · 9p} for the left-hand base of Fp and Bp = Fp \ Ap for the remainder of theseparating cylinder. For the time being, consider ε > 0 to be any positive number. ByLemmas (4.3.12.2) and (4.3.12.4), there are indices k0 and p0, only depending on ε, and auniversal constant c > 0 (not depending on any index) satisfying for all p ≥ p0, z1 ∈ Dpk0 ,z2 ∈ Up−1supz3∈BpG(z3, z1 − z2) ≤ εG(0, z1 − z2),P0ÄSTp ∈ Apä ≤ ε,80Chapter 4. Crossings of NRWand for all z ∈ Dpk0 − Up−1,supz3∈ApG(z3, z) ≤ cG(0, z).Write now, using (2.4.1), for z ∈ Dpk0 − Up−1,G(0, z) = Gp(0, z) + E0ÄG(STp , z)1Ap(STp)ä+ E0ÄG(STp , z)1Bp(STp)ä≤ Gp(0, z) + cG(0, z)P0ÄSTp ∈ Apä+ εG(0, z)P0ÄSTp ∈ Bpä≤ Gp(0, z) + (c+ 1)εG(0, z).Hence, for any ε > 0, there are indices k0 and p0, depending only on ε, so that therelations p ≥ p0 and z ∈ Dpk0 − Up−1 implyG(0, z) ≤ Gp(0, z) + ε1 + εG(0, z),which yields G(0, z) ≤ (1 + ε)Gp(0, z).§ 4.4. Dimension d = 2; “logarithmic scale of crossings”of Γ2(λ).We will establish a lower bound on the probability of how frequently two random walk pathsin Γ2(λ) will cross each other. The exact result is technical in nature and we state it precisely.Invoking Proposition (4.3.12), for ε = 1 there exists two indices k0 and p0, depending onlyon ε, such that the conclusions of this proposition hold. For the remainder of this chapter,we assume both k0 and p0 are fixed and given as stated.Theorem ( 4.4.1 ) Consider two independent network random walks S and S ′ in Γ2(λ),with respective starting points o and o′. Define the random variables Tp and T ′p in (4.3.8)using S and S ′, accordingly; now introduce Mp as in (4.3.10). Consider now the σ-fieldsEp = FSTp ∨F S′T ′p .Define the event Ap = {Mp > 0} and the random variable qp+1 = P (Ap+1 | Ep) . Then,Ap ∈ Ep and there exists a positive integer P = P (o, o′) and a universal constant c > 0(depending neither on o nor on o′) such that, almost surely, qp >cpfor all p ≥ P.Proof. It is clear that Ap ∈ Ep. Consider p1 = max(po, po′) (4.3.9), that is, p1 is the firstindex such that o and o′ belong to Up. Suppose p is any integer ≥ max(p0, p1). By (1.12.3),we haveqp+1 = P(Ap+1∣∣∣STp , S ′T ′p) .81Chapter 4. Crossings of NRWIt is now straightforward to check that(∗) qp+1 =∑(z1,z2)∈F2pP(Ap+1∣∣∣STp = z1, S ′T ′p = z2)1{STp=z1,S′T ′p=z2}.Since, for every pair (z1, z2) ∈ F2p, the measure A 7→ P(A∣∣∣STp = z1, S ′T ′p = z2) is a probabilitymeasure, we can apply (4.2.1) to estimate each of the terms P(Ap+1∣∣∣STp = z1, S ′T ′p = z2) forany z1, z2 ∈ Fp. To do this, we will proceed as in previous section using the estimates in theGreen’s function and details shall be omitted. Suppose z1 and z2 are any two points in Fp.Using the strong Markov property for a pair of independent Markov processes (1.12.3) gives(inside the expectation on the right of the first equality we use abuse of notation and assume(Sm) and (S′m′) are independent network random walks started at zero)E(Mp+1∣∣∣STp = z1, S ′T ′p = z2) = ∑z3∈D(p+1)k0EÖTp+1∑m=0T ′p+1∑m′=01{Sm+z1=z3}1{S′m′+z2=z3}è=∑z3∈D(p+1)k0Gp+1(0, z3 − z1)Gp+1(0, z3 − z2) ∑z3∈D(p+1)k0G(0, z3 − z1)G(0, z3 − z2), by (4.3.12) ∑z3∈D(p+1)k09−2(p+1)k0  1, by (4.3.4)(k).We shall now estimate the second moment of Mp with respect to the conditional measure.We haveM2p+1 =∑(z3,z4)∈D2(p+1)k0Tp+1∑m1,m2=TpT ′p+1∑m′1,m′2=T′p1{Sm1=z3,Sm2=z4}1{S′m′1=z3,S′m′2=z4}.Now apply the conditional expectation, by the strong Markov property (1.12.3) and inde-pendence between S and S ′ (we again employ abuse of notation and assume S and S ′ aretwo independent network random walks started at zero after having used the strong Markovproperty for two independent Markov processes)E(M2p+1∣∣∣STp = z1, S ′T ′p = z2) = ∑(z3,z4)∈D2(p+1)k0EÑTp+1∑m1,m2=01{Sm1=z3−z1,Sm2=z4−z1}é× EÖT ′p+1∑m′1,m′2=01{S′m′1=z3−z2,S′m′2=z4−z2}è≤ B1 +B2 +B3 +B4,82Chapter 4. Crossings of NRWin whichB1 =∑m1≤m2m′1≤m′2(z3,z4)∈D2(p+1)k0P (Sm1 = z3 − z1, Sm2 = z4 − z1)P(S ′m′1 = z3 − z2, S′m′2= z4 − z2),B2 =∑m1≤m2m′1≥m′2(z3,z4)∈D2(p+1)k0P (Sm1 = z3 − z1, Sm2 = z4 − z1)P(S ′m′1 = z3 − z2, S′m′2= z4 − z2),B3 =∑m1≥m2m′1≤m′2(z3,z4)∈D2(p+1)k0P (Sm1 = z3 − z1, Sm2 = z4 − z1)P(S ′m′1 = z3 − z2, S′m′2= z4 − z2),B4 =∑m1≥m2m′1≥m′2(z3,z4)∈D2(p+1)k0P (Sm1 = z3 − z1, Sm2 = z4 − z1)P(S ′m′1 = z3 − z2, S′m′2= z4 − z2).We handle each sum separately.(a) By independent increments and writing pk(z, z′) = P (Sk = z′ | S0 = z) ,B1 =∑m1≤m2m′1≤m′2(z3,z4)∈D2(p+1)k0pm1(0, z3 − z1)pm2−m1(0, z4 − z3)pm′1(0, z3 − z2)pm′2−m′1(0, z4 − z3)=∑(z3,z4)∈D2(p+1)k0G(0, z3 − z1)G(0, z4 − z3)2G(0, z3 − z2),the last equality being an immediate consequence of Lebesgue-Tonelli’s theorem andthe definition of the Green’s function of Γ2(λ). By virtue of (4.3.12.1) and (4.3.5) wecan proceed further,B1 ∑(z3,z4)∈D2(p+1)k09−2(p+1)k0G(0, z4 − z3)2 ≤ ck0(p+ 1)∑z3∈D(p+1)k09−2(p+1)k0  p.(b) Here we getB2 =∑(z3,z4)∈D2(p+1)k0G(0, z3 − z1)G(0, z4 − z3)G(0, z4 − z2)G(0, z3 − z4) ∑z3∈D(p+1)k0∑z4∈D(p+1)k09−2(p+1)k0G(0, z4 − z3)G(0, z3 − z4) ∑z3∈D(p+1)k09−2(p+1)k0  1,where the second  is a consequence of (4.3.5).83Chapter 4. Crossings of NRW(c) Similarly to B2, we get B3  1.(d) As with B1, B4 ≤ cp, for some universal constant c > 0.Therefore, we have shown that there exist two positive constants c and c′, such that,whenever p ≥ max(p0, p1) and (z1, z2) ∈ F2p,E(Mp+1∣∣∣STp = z1, S ′T ′p = z2) ≥ c,E(M2p+1∣∣∣STp = z1, S ′T ′p = z2) ≤ c′p.By (4.2.1), we finally reach that there exists a constant c > 0 and a P = max(p0, p1) suchthat, for all p ≥ P and all (z1, z2) ∈ F2p,P(Ap+1∣∣∣STp = z1, S ′T ′p = z2) ≥ cp,comparing with the expression for qp+1 in (∗), we have substantiated the claims.Corollary ( 4.4.1.1 ) Let S and S ′ be two independent network random walks in Γ2(λ),with starting points S0 = o and S′0 = o′. There is a universal constant c > 0 and anindex P = P (o, o′), such that for all p ≥ P, the probability that the two random walk paths(Sm)m=Tp−1,...,Tp and (S′m′)m′=T ′p−1,...,T ′p cross each other inside Dpk0 is at leastcp.The following corollary follows using (4.2.2) and (4.4.1).Corollary ( 4.4.1.2 ) Any two independent network random walks on Γ2(λ) will haveinfinitely many intersections.Remark ( 4.4.2 ) There is a much simpler way to obtain Corollary (4.4.1.2). DenoteS = (B,X), that is, Sn = (Bn, Xn) for n ∈ Z+; here B is the biased random walk on Zand X a random walk on Z2. Define ζk = τ{k}×Z2(S) for k ∈ Z+, which is the time at whichthe random walk B visits k. With these hitting times, consider Yn = Xζn . Similarly, defineY ′n using S′ = (B′, X ′), which is an independent copy of S. In a similar manner as theorem(4.1.2), it is seen that Yn = Y′n for infinitely many indices n. Having this, it is now clearthat Sn = S′n′ for infinitely many pairs (n, n′). Of course, this result is much coarser than(4.4.1.1).84Chapter 5One endedness for Γd(λ)Let G be a directed infinite tree. Recall the definition of end (A.24) in a forest. In thissection we prove the following result: for almost every realisation of the uniform spanningtree (or forest) on Γd(λ), every one of its components has one end. We have to divide theproof into different dimensions.§ 5.1. One endedness for Γ1(λ).The reader should recall the definition of planar graph (A.14) and duality (A.15). We alsorefer the reader to [LP16], see Section 6.5 , Section 9.2 and Section 10.3.Following the development in [LP16, section 10.3], we remark that there exists a canonicalcorrespondence between the set of spanning trees t in the finite network (G, µ) and that ofspanning trees t× in (G†, µ†) (which is also a finite network):e ∈ t ⇐⇒ e† /∈ t×.(It is tempting to use t† in lieu of t×, but this would be a poor choice of notation since thedual graph of a tree only contain self-loops as edges.) When one takes weak limits to definethe FSF and WSF in an infinite network (G, µ), this canonical correspondence transformsFSF in G into WSF in G† (the “duality between FSF in G and WSF in G†”). In general,when WSF 6= FSF and the graph G is not planar, the FSF is poorly understood, some recentwork is available for FSF in planar graphs [HN19].Theorem ( 5.1.1 ) UST in Γ1(λ) (3.5.1) has, almost surely, one end.Proof. It is clear that the dual graph of Z2 is isomorphic to Z2; one isomorphism is given bysending the face with vertices (n, x), (n+1, x), (n+1, x+1), (n, x+1) to the vertex (n+1, x)85Chapter 5. One endedness for Γd(λ)(where n and x run through Z). Intuitively, each face in Z2 is identified with its bottomright corner. By virtue of this isomorphism, we identify (Z2)† with Z2. Then, the edge e inΓ1(λ) will correspond to the edge e† in Γ1(λ)†, with dual conductance µ†(e†) = µ(e)−1 asfollows:e = ((n, x), (n+ 1, x)),e = ((n, x), (n, x+ 1)),e = ((n, x), (n, x− 1)),e = ((n, x), (n− 1, x)),e† = ((n+ 1, x), (n+ 1, x− 1)),e† = ((n, x), (n+ 1, x)),e† = ((n, x− 1), (n+ 1, x− 1)),e† = ((n, x), (n, x− 1)),µ†(e†) = e−λ(n+1),µ†(e†) = e−λn,µ†(e†) = e−λn,µ†(e†) = e−λn.This shows at once thatµ†((n, x), (n′, x′)) = e−λmin(n,n′) = eλmax(−n,−n′).It follows that Γ1(λ)† is the network image of Γ1(λ) under the reflection (n, x) 7→ (−n, x). Inparticular, the wired and free uniform spanning forest of Γ1(λ)† coincide and the number ofUSF trees in Γ1(λ)† is the same as that of Γ1(λ); that is to say, for almost every realisation,just one tree, by (3.5.1). Bearing this in mind and the duality between UST in Γ1(λ) withUST in Γ1(λ)†, we see at once that the tree in Γ1(λ) is almost surely one ended; for if it weretwo ended, it would split the plane into two disconnected regions making it impossible forthe dual tree to be a tree.Remark ( 5.1.2 ) The core idea of the proof of previous theorem is implicit in [LP16,Theorem 10.36].§ 5.2. One endedness for Γ2(λ).We will prove the following theorem: UST in Γ2(λ) has, almost surely, one end.The proof is rather involved and to facilitate the reading of it we are going to develop itin several steps and draw the conclusion at the end.z 5.2.1 Invariance of the chronology of Wilson’s algorithm rootedat infinity.We start with a theorem related to the study of the chronological construction of Wilson’salgorithm rooted at infinity on general networks. Recall the notations of Annex B and thatof z1.7. With these notations, we construct the following random object(Fξ,ÄLξkäk∈N)=ÄFξ, Lξ1, . . . , Lξk, . . .ä.86Chapter 5. One endedness for Γd(λ)This random object is SG ×S NG = S Z+G -valued. Notice that the sequenceÄLξkäk∈N may bereferred to as a “chronology of the WSF-forest.” We are going to show that the distributionof this chronology, once the first steps are done and fixed, is invariant under reordering offuture searches, in a sense that is made precise below. Recall that Wilson’s algorithm rootedat infinity gives Fξ ∼ Fη; in other words, Wilson’s algorithm rooted at infinity gives that thedistribution of the final tree is invariant under any reordering of the entire searches. Observethat if ξ and η are two orderings satisfying ξ(j) = η(j) for 1 ≤ j ≤M then ÄLξ1, . . . , LξMä =(Lη1, . . . , LηM) a.s.. Furthermore, in the finite case we may run Wilson’s algorithm with stacksto obtain that Fξ = Fη a.s., thus in the finite caseÄFξ, Lξ1, . . . , LξMä= (Fη, Lη1, . . . , LηM) a.s..We are going to extend this observation for the finite case, to the infinite case and in ageneral network.Theorem ( 5.2.1 ) Let Γ be a transient network satisfying the usual assumptions (A.22)such that the underlying graph is simple (A.8). With the notations from z1.7, if ξ and ηare two orderings of the vertex set of G such that ξ(j) = η(j) for 1 ≤ j ≤M, thenÄFξ, Lξ1, . . . , LξMä ∼ (Fη, Lη1, . . . , LηM) ,in other words, the two random objects have the same law.Proof. Suppose (Vq)q∈N is an increasing sequence of finite subsets of V such that⋃q∈NVq = V,where V is the vertex set of G. We may assume that all the vertices vj = ξ(j) = η(j) ∈ V1(1 ≤ j ≤ M). Denote by Γq the network induced by (G, µ) on Vq with its boundary wiredinto a vertex (A.11). On the network, Γq, we may construct the uniform spanning treeusing Wilson’s algorithm with stacks, rooted at the boundary vertex ∂Vq. This constructs afamily (Sv,q)v∈Vq of network random walks in Γq on some probability spaceÄΩ(q),F (q),P(q)ä.This creates a random tree Tq that depends solely on the stacks and the choice of root, butnot in the order in which the vertices in Vq are searched (1.6.1). We know that Tq will beconstructed using only the section of Sv,q corresponding until its visit time to the root vertex,which is ∂Vq, in other words, Tq contains at most edges in the path (Sv,qm )m=0,...,τ∂Vq (Sv,q).This portion of path has the same distribution as the section of the network random walkof (G, µ) started at v until its hitting time of the set ∂Vq. For simplicity, we will denote inwhat follows by ∂Vq both the vertex and edge set of the graph induced by G on ∂Vq. Define,for each q, and inductively, the following spanning subgraphs of G :Initialisation. Set θq1 = τ∂Vq (Sv1,q) which is the visit time of the vertex ∂Vq of Γq by Sv1,q.With this time, define Lv11,q = LE (Sv1,qm )m=0,...,θq1which may be called the first branch ofthe UST in Γq.87Chapter 5. One endedness for Γd(λ)Inductive step. For 2 ≤ j ≤M, and having defined the first j − 1 branches of UST in Γq,consider the time θqj−1 = τ∂Vq∪Lv11,q∪...∪Lvj−1j−1,q(Svj ,q) , which is the hitting time of the USTpartial tree of Γq constructed until the (j − 1)th iteration by the random walk Svj ,q.Then, define the jth branch of UST by Lvjj,q = LEÄSvj ,qmäm=0,...,θqj−1.Finalisation. The random tree Tq = Lv11,q ∪ . . . ∪ LvMM,q is the UST-tree of Γq, this followsfrom Wilson’s algorithm (1.6.1).These iterations construct a sequence of random objects, namelyÄTq, Lv11,q, . . . , LvMM,qäq∈N .Since the edge set of Γq is a subset of the edge set of Γ, we may canonically regardÄTq, Lv11,q, . . . , LvMM,qä ∈ SM+1G . We will show that, on the space of measures on SM+1G ,ÄTq, Lv11,q, . . . , LvMM,qä weakly−−−→q→∞ÄFξ, Lξ1, . . . , LξMä,ÄTq, Lv11,q, . . . , LvMM,qä weakly−−−→q→∞ (Fη, Lη1, . . . , LηM) .This will conclude the result.Consider now a probability space (Ω,F ,P) supporting an independent family (Sv)v∈Vof network random walks of Γ, with Sv started at v. The ordering ξ : N → V is a bi-jection, thus, ξ−1(Vq) is finite set. As such, ξ−1(Vq) can be thought as an increasingsequenceÄa1,q, . . . , aiq ,qä. The assumption v1, . . . , vM ∈ V1 guarantees that iq ≥ M and(a1,q, . . . , aM,q) = (1, . . . ,M). Furthermore, for any integer N ≥M, there exists an index qNsuch that ξ−1(Vq) =Ä1, . . . , N, aN+1,q, . . . , aiq ,qäfor all q ≥ qN . This observation will play akey roˆle below: it will imply that, in the Wilson’s algorithm construction of USF of Γ, thefirst N searches will always be the same for all large values of q. Define inductively, for eachq ∈ N, the following random subsets of Γ,Initialisation. Set σξ1,q = τ∂VqÄSξ(a1)ä, which is the hitting time of the set ∂Vq by Sξ(a1).Having this stopping time, consider Bξ(a1)a1,q = LEÄSξ(a1)mäm=0,...,σξ1,q, which we may callthe first partial branch.Inductive step. For every 2 ≤ j ≤ iq, having defined the first j−1 partial branches, defineσξj,q = τ∂Vq∪Bξ(a1)a1,q ∪...∪Bξ(aj−1)aj−1,qÄSξ(aj)äwhich can be regarded as the first hitting time ofany of the sets ∂Vq orBξ(ak)ak,q(1 ≤ k ≤ j−1) by the random walk Sξ(aj).With this hittingtime, we may construct the jth partial branch, namely Bξ(aj)aj ,q = LE(Sξ(aj)m)m=0,...,σξj,q.Finalisation. Return Tξq =iq⋃j=1Bξ(aj)aj ,q .88Chapter 5. One endedness for Γd(λ)Again, we may regard Tξq and the Bξ(aj)aj ,q as subsets of edges of Γ. Then, the inductiveconstructions of the random objects Lvjj,q and Bξ(aj)aj ,q show at onceÄTq, Lv11,q, . . . , LvMM,qä ∼ ÄTξq, Bv11,q, . . . , BvMM,qä ,in other words, the distribution of both SM+1G -valued random objects is the same. Noticethat, on Ω, we have τ∂Vq(Sv) → ∞ as q → ∞ for every vertex v and for P-almost everyrealisation. Also, it is key to notice that if e is an edge of Γ, then the sequence¶e ∈ Tξq©iseventually constant for P-almost every realisation (this follows easily from the key observa-tion given above and transience of the NRW). All of these entail that,limq→∞ÄTξq, Bv11,q, . . . , BvMM,qä=ÄFξ, Lξ1, . . . , LξMäP−a.s.,in the topology of SM+1G . A fortiori,ÄTξq, Bv11,q, . . . , BvMM,qä weakly−−−→q→∞ÄFξ, Lξ1, . . . , LξMä.The same argument can be repeated exchanging ξ by η everywhere, whence the theorem.Corollary ( 5.2.1.1 ) The statistical properties of events depending on WSF and its firstbranches are not affected by reordering vertices not yet searched. In other words, if ϕ :SM+1G → R is any bounded measurable function or any non negative measurable function,and ξ, η : N→ V are two orderings of the vertices such that ξ(j) = η(j) for 1 ≤ j ≤M, thenEÄϕÄFξ, Lξ1, . . . , LξMää= E (ϕ (Fη, Lη1, . . . , LηM)) .The importance of the previous corollary is that one can see a partial forest, and toestimate probabilities of successive branches, we can choose the next vertices to depend onthe partial tree. (This is no surprise, since Wilson’s algorithm with stacks allows for astronger result in the finite case: the next vertices may depend on the partial tree and thefinal tree cannot change.)z 5.2.2 Preliminary estimates in the proof of one-endedness forΓ2(λ).We denote, for r > 0,B (x; r) = {y ∈ Zd | ‖x− y‖ ≤ r}S (x; r) = {y ∈ Zd | r ≤ ‖x− y‖ ≤ r + 1}.89Chapter 5. One endedness for Γd(λ)We remark that there exists a constant L = L(d) > 0 such that for all x ∈ Zd and all r ≥ 1,maxÄcard (B (x; r)) , card (S (x; r))ä ≤ L · rd.Also, B (x; r) = x + B (0; r) and S (x; r) = x + S (0; r) . Further, we show that ∂B (x; r) ⊂S (x; r) . To see this, suffices to show this when x = 0. In this case, the relation y ∈ ∂B (0; r)signifies there exists a y′ ∈ B (0; r) such that ‖y − y′‖ = 1, and then ‖y‖ ≤ r + 1. Theinequality ‖y‖ ≥ r for y ∈ ∂B (0; r) follows from definition of the vertex boundary of a set.Proposition ( 5.2.2 ) For each p ∈ N, define the setsAp = [−p, p]× S (0; p) , Bp = {p} × B (0; p) , Cp = {−p} × B (0; p) .Then, there is a pair of constants c, c′ > 0 such that for all p ∈ N,∑z∈Ap∪BpG(z, 0) ≤ ce−c′p.Furthermore, there exists a another pair of constants c, c′ > 0 and a positive number B,obeying the following: for every b ≥ B we can split Cp = Cp,0 ∪ Cp,1, whereCp,0 = Cp,0(b) =({−p} × B(0; bp12))∩ Cp, and Cp,1 = Cp,1(b) = Cp \ Cp,0,and with this division,supp∈N∑z∈Cp,1G(z, 0) ≤ ce−c′b2 .In particular, for any ε > 0, there exists an index p1 = p1(ε) ∈ N and a positive numberb1 = b1(ε) such that, for any p ≥ p1 and b ≥ b1, we can divide Cp = Cp,0 ∪ Cp,1 as before,and with this divisionsupp≥p1b≥b1∑z∈EpG(z, 0) ≤ ε,where Ep = Ap ∪ Bp ∪ Cp,1.Proof. If z ∈ Ap ∪ Bp then the bounds on the Green’s function (2.6.1) give at onceG(z, 0) = G(0,−z) ≤ c1e−c2‖z‖‖z‖− d2 ≤ c1e−c2pp− d2 .Now, card (Ap ∪ Bp)  pd and so, for a constant c3 > 0, we have∑z∈Ap∪BpG(z, 0) ≤ c3e−c2pp d2 ≤ñc3 supt≥1e−c22ttd2ôe−c22p = c4e− c22p.90Chapter 5. One endedness for Γd(λ)Now, for any b > 0, we may construct Cp,0 and Cp,1 as in the statement; if b ≥ p 12 , thenCp,1 = ∅, which is fine. Write Cp,1 ⊂⋃bp12≤k≤p{−p} × S (0; k) , so that∑z∈Cp,1G(z, 0) ≤ ∑bp12<k≤pcard (S (0; k)) c1e−c2 k2p p−d2 ≤ ∑bp12<k≤pc5e−c2 k2p p−d2kd,for a constant c5 > 0. The function t 7→ e−ct2p td has a maximum at t =Çd2cå 12p12 and it isdecreasing for t ≥Çd2cå 12p12 ; any value b ≥Çd2cå 12allows bounding∑z∈Cp,1G(z, 0) ≤ c5∫ pbp12dt e−c2t2p p−d2 td−1 = c5√p∫bdt e−c2t2td−1 ≤ c6e−c22b2 .Hence, putting all together, we have established that, for any p ∈ N and b ≥Çd2cå 12,∑z∈Ap∪BpG(z, 0) ≤ ce−c′p and ∑z∈Cp,1G(z, 0) ≤ Ce−C′b2where all these constants c, c′, C, C ′ > 0 are independent of both p and b. The desired resultsare now clear.Proposition ( 5.2.3 ) Let ε > 0 and we maintain the notations and conclusions of (5.2.2).Then, there exists a positive integer p2 = p2(ε) and a family of sets (C′p,0)p≥p2 , such thatC′p,0 ⊂ Cp,0, cardÄC′p,0ä ≤ ε · card (Cp,0)andsupp≥p2maxz∈Cp,0∥∥∥z − C′p,0∥∥∥ <∞.Proof. There exist universal constants L = L(d) such that card (B (x; r)) ≤ Lrd for all x ∈ Zdand r ≥ 1. Observe that Cp,0 = {−p} × B(0; bp12)and so each subset of the formC′p,0(k) =Ä{−p} × {k} × Zd−1ä ∩ Cp,0(a “slice of Cp,0”) has cardinality bounded from above by L(d − 1)bd−1p d−12 , we also knowthat card (Cp,0)  bdp d2 . Hence, we havecardÄC′p,0(k)äcard (Cp,0)≤ cp− 12 , for some positive constant cindependent of p. Therefore, we can set C′p,0 to be the union of as many slices ask(p) = max{k ∈ Z+ : kcp− 12 ≤ ε}.91Chapter 5. One endedness for Γd(λ)It follows thatlimp→∞k(p)p12=εc.Let m = m(ε) be the positive integer for which1m+ 1<εc≤ 1m. Definep2 = inf®s ∈ Z+ : k(s)s12≥ 12(m+ 1)´The result is now clear, for C′p,0 is defined to be the union of k(p) evenly spaced slices, so‖z − Cp,0‖ ≤ 4(m+ 1)b for all z ∈ C′p,0 and p ≥ p2.For the next proposition, the reader should recall the definition of LE (1.4.1).Proposition ( 5.2.4 ) Let S and S ′ be two independent transient irreducible Markov chainson the same countable state space (having same transition density), with initial states o ando′, respectively. Let U be any subset of the states containing o and o′. Denote by T and T ′ theexit times of U by S and S ′, respectively. If the event HU defined by “there exists 0 ≤ m ≤ Tand 0 ≤ m′ ≤ T ′ with Sm = S ′m′” has positive probability, thenP (LE(S ′m′)m′=0,...,T ′ ∩ (Sm)m=0,...,T 6= ∅) ≥ 2−8P (HU) .(This inequality obviously also hold if HU has probability zero.)We only provide a reference for this proposition: see Remark 1.3 and Lemma 4.1 of[LPS03].In (4.4.1) we proved a lower bound for the probability that two independent sections of thenetwork random walk intersect. That was an involved theorem. The following propositionprovides an upper bound for the probability that two network random walks intersect in anarrow strip. The narrowness of the strip, relative to the scales considered, allows us to usemuch coarser estimates.Proposition ( 5.2.5 ) Let d = 2. Consider k0 and p0 as in (4.3.12) with ε = 1 and definethe “strips”J+p = [4 · 9pk0 , 4 · 9pk0 + 2 · 3pk0 ]× Z2J−p = [4 · 9pk0 − 2 · 3pk0 , 4 · 9pk0 ]× Z2.With S and S ′ being two independent network random walks in Γd(λ) started at o and 0,respectively, denote by I±p (o) the number of intersections of the paths S and S′ inside thestrip J±p . Then, there exists a constant c > 0 such that for all δ > 0, there exists an indexp3 = p3(δ) such thatsupp≥p3max‖o‖≤δPÄI±p (o) > 0ä ≤ c3−pk0 .92Chapter 5. One endedness for Γd(λ)Proof. Take p3,1 = p3,1(δ) the first integer s such that the ball B (0; δ + 1) is a subset ofUs−1 defined by (4.3.6). Assume p ≥ p3,1. Write I±p instead of I±p (o) for simplicity and wewill show the bounds are universal, provided p is large enough (depending on δ); since theargument is very similar, suffices to consider the case of I+p . We haveI+p =∑z∈J+p∞∑m,m′=01{Sm=z}1{S′m′=z},and by independence,PÄI+p > 0ä ≤ E ÄI+p ä = ∑z∈J+pG(o, z)G(0, z).Write nowJ+p,0 = [4 · 9pk0 , 4 · 9pk0 + 2 · 3pk0 ]× BÄ0; 3pk0äJ+p,t = [4 · 9pk0 , 4 · 9pk0 + 2 · 3pk0 ]×ÄBÄ0; 3(t+1)pk0ä \ B Ä0; 3tpk0ää (t ∈ N),so that J+p is the union of the pairwise disjoint sets J+p,t (t ∈ Z+).We will show G(o, z)  G(0, z) for z ∈ J+p for all p large enough with any implicitconstant universal. Suppose first that z ∈ J+p and ‖o‖ ≤ δ, denote z − o = (no, xo) andz = (nz, xz). If z ∈ J+p,0, then there exists an index p3,2 = p3,2(δ) such that if p ≥ p3,2then12nz ≤ no ≤ 2nz and ‖xo‖ ≤ 2 · 3pk0 . Then, for p ≥ p3,2 and z ∈ J+p,0, Theorem(2.6.1) shows that G(o, z)  9−pk0  G(0, z). If z ∈ J+p,t, with t ≥ 1, then there is an indexp3,3 = p3,3(δ) such that if p ≥ p3,3 then 12nz ≤ no ≤ 2nz and 12‖xz‖ ≤ ‖xo‖ ≤ 2 ‖xz‖ , andthe result is immediate from the Green’s function bounds, again Theorem (2.6.1). Then, forall p ≥ max(p3,2, p3,3), if z ∈ J+p and ‖o‖ ≤ δ,G(o, z)  G(0, z),with any implicit constant universal, as claimed.Denote now p3 = max(p3,1, p3,2, p3,3). Suppose that G(o, z) ≤ cG(0, z). Substitute thisinto the equation above,PÄI+p > 0ä ≤ ∑z∈J+pG(o, z)G(0, z) ≤ c ∑z∈J+pG(0, z)2= c∑t∈Z+∑z∈J+p,tG(0, z)2.93Chapter 5. One endedness for Γd(λ)For t = 0, by (2.6.1),∑z∈J+p,0G(0, z) ≤ c219−2pk0cardÄJ+p,0ä  3−4pk0 × 3pk0 · 32pk0 = 3−pk0 .For t = 1, and with card (B (x; r)) ≤ Lr2 for x ∈ Z2 and r ≥ 1, we have∑z∈J+p,1G(0, z)2 ≤ c214·9pk0+2·3pk0∑n=4·9pk02·3pk0∑r=1∑r3pk0≤‖x‖≤(r+1)3pk0e−2c2‖x‖2n n−2≤ 2c219−2pk0 · 3pk02·3pk0∑r=1L(r + 1)29pk0e−c2r2≤[2c21L∞∑r=1(r + 1)2e−c2r2]3−pk0 .For t ≥ 2 and z = (nz, xz) ∈ J+p,t, the condition 3tpk0 ≤ ‖xz‖ ≤ 3(t+1)pk0 guarantees the boundG(0, z) ≤ c1e−c2‖z‖, and we obtain bounds that are negligible compared with the two boundsalready obtained. Indeed,∑t≥2∑z∈J+p,0G(0, z)2 ≤ c21L∑t≥24·9pk0+2·3pk0∑n=4·9pk0e−c23tpk032(t+1)pk0≤2c21L∑t≥2e−c223tpk03(2t+3)pk0 e− c22 32pk0 .This completes the proof.Before continuing with the next proposition, we introduce the following terminology. Aset H of the form H = {h}×Zd will be called a splitting hyperplane (splitting plane if d = 2)for the path (vn) (either finite or infinite) if for one, and exactly one value of m, we havepr0vm = h. In this case, we will call h to be a splitting level for this path (vm).Proposition ( 5.2.6 ) Let S be the network random walk on Γd(λ) started at S0. Supposethat ζ ≤ η are two stopping times for S, with ζ < ∞ almost surely (η = ∞ with positiveprobability not excluded). Denote by Sh(ζ, η) the event that h ∈ Z is a splitting level for thepath S = (Sm)ζ≤m≤η (whenever η = ∞, replace m ≤ η by m < η). On Sh(ζ, η), denote bym0 to be the unique positive integer m for which pr0Sm = h. Then, on the event Sh(ζ, η),LE(S) = LE(Sm)ζ≤m≤m0 ∨ LE(Sm)m0≤m≤η.Proof. It is immediate from definition, for the two paths (Sm)m≤m0 and (Sm)m>m0 are disjointand hence, no point in the second path can create a cycle in the first path.94Chapter 5. One endedness for Γd(λ)The next proposition shows that splitting levels do occur with a universal overwhelmingprobability if one let appropriate scales to be considered. In other words, if the random walkmoves enough so that the initial and last state considered are n levels apart, at the veryleast, one in roughly every√n consecutive levels will be splitting. The gist of its proof liesin random walk in one dimension, see (2.4.6).Proposition ( 5.2.7 ) Consider the network random walk S on Γd(λ) started at zero. Forevery positive integer a, every pair ζp < ηp of stopping times for S such that, for almostevery realisation,ζp ≤ τ{a}×Z2(S), τ{a+p2}×Z2(S) ≤ ηp,and for h ∈ Z+, consider the eventSh(ζp, ηp) =¶h is a splitting level for (Sm)m=ζp,...,ηp©.There exists a universal constant c > 0 and an index p4 ∈ N such that, for all p ≥ p4, allsystems (a, ζp, ηp) as above and all integers b satisfying a ≤ b and b+ p ≤ a+ p2, we havePÑb+p⋃h=bSh(ζp, ηp)é≥ 1− e−cp.Proof. Let Kp be the event “some h = b, . . . , b+p is a splitting level for the path (Sm)m∈Z+ .”We have Kp ⊂b+p⋃h=bSh(ζp, ηp) and therefore PÑb+p⋃h=bSh(ζp, ηp)é≥ P (Kp) . Consider now therandom walk on Z given by m 7→ pr0Sm and observe that a particular value h is a splittinglevel (for S) is the same as saying that for exactly one m, pr0Sm = h; by (2.4.6) we haveP (Kp) ≥ 1− (1− (r − l))p − pÇlråp,where r =eλ2d+ 1 + eλand l =12d+ 1 + eλ, are the probabilities of jumping right and left,accordingly. Upon substitution, the result follows.Proposition ( 5.2.8 ) Let d = 2. Consider S and S ′ two independent network randomwalks in Γ2(λ) started at o and 0, respectively. Define the following stopping timeσp = τHp(S), Hp = {4 · 9pk0} × Z2,in other words, σp is the hitting time of the “plane” Hp by S; define σ′p similarly, using S′in lieu of S. DenoteL′ = LE(S ′m′)m′=σ′2(p−1),...,σ′2p ,95Chapter 5. One endedness for Γd(λ)which is the loop-erasure of the section of S ′ between the planes H2(p−2) and H2p; with this,defineβp(o) =σ2p∑m=σ2(p−1)∑z′∈L′1{Sm=z},which is the number of times the section (Sm)m=σ2(p−1),...,σ2p crosses the path L′. Then, thereexists a constant c > 0 such that for every δ > 0, there exists a p5 = p5(δ) ∈ N satisfyingmin‖o‖≤δP (βp(o) > 0) ≥ cp,for all p ≥ p5.Proof. Define Mp(o) as in (4.3.10) using S and S′. Define p5,1 = p5,1(δ) ∈ N to be the firstindex p such that B (0; δ + 1) ⊂ Up−1. Notice that for p ≥ p5,1, Tp ≤ σp and T ′p ≤ σ′p. Beingσp−1 and σ′p−1 the hitting times of the plane Hp−1, we find thatMp(o) =∑z∈Dpk0Tp∑m=σp−1T ′p∑m′=σ′p−11{Sm=z}1{S′m′=z}≤ ∑z∈Dpk0σp∑m=σp−1σ′p∑m′=σ′p−11{Sm=z}1{S′m′=z}def.= Np(o).Thus, P (Mp(o) > 0) ≤ P (Np(o) > 0) and Theorem (4.4.1) gives P (Mp(o) > 0) ≥ cp, pro-vided p ≥ P (o, 0), the constant c being universal. Define p5,2 = p5,2(δ) = max‖o‖≤δ P (o, 0), whichis finite since the ball ‖o‖ ≤ δ is finite. Finally, (5.2.4) givesP (βp(o) > 0) ≥ 2−8PÄN˜p(o) > 0ä,where N˜p(o) is the number of crossings between (Sm)m=σ2(p−1),...,σ2p and (S′m′)m′=σ′2(p−1),...,σ′2p.Clearly N˜p(o) ≥ Np(o), and the result follows with p5 = max(p5,1, p5,2).Proposition ( 5.2.9 ) Let d = 2. Continue with the hypotheses and notations of (5.2.8).Define αp(o) to be the number of intersections of LE(S′m′)m′=0,...,σ′2p and (Sm)m=0,...,σ2p . Then,for every δ > 0,limp→∞max‖o‖≤δP (αp(o) = 0) = 0.Proof. Let ε > 0. We can then find the indices p1(ε) and p2(ε), and a constant b1(ε) satisfyingthe conclusions of (5.2.2) and (5.2.3), we take b = b1(ε). The δ in the hypothesis, together with(5.2.5) and (5.2.8) give two indices p3(δ) and p5(δ), satisfying the corresponding conclusions96Chapter 5. One endedness for Γd(λ)of these results. And (5.2.7) gives yet another index p4 satisfying the outcome of thatproposition. Observe that the choice of p3(δ) guarantees that the closed ball B (0; δ + 1) iscontained in Up3(δ)−1. LetP = max(p1(ε), p2(ε), p3(δ), p4, p5(δ)).Write αp in place of αp(o) for simplicity, all the bounds will be shown to be independentof δ. Assume p ≥ P. Define the eventsL′p =4·92(p−1)k0+2·9(p−1)k0⋃h=4·92(p−1)k0ßh is a splitting level for (S ′m′)m′=σ′2(p−1),...,σ′2p™R′p =4·92pk0⋃h=4·92pk0−2·9pk0ßh is a splitting level for (S ′m′)m′=σ′2(p−1),...,σ′2p™,(A)that is, the events where one of the 2 · 9(p−1)k0 left-most levels between the planes H2(p−1)and H2p and one of the 2 · 9pk0 right-most levels in the same region is a splitting level for thestated section of S ′. Denote by R′{p and L′{p their corresponding complements; we also defineβp = βp(o) from (5.2.8). Then,P (αp = 0) = PÄ{αp = 0} ∩ L′pä+ P ({αp = 0} ∩ L′{p )≤ P Ä{αp = 0} ∩ L′p ∩ {βp = 0}ä+ P Ä{αp = 0} ∩ L′p ∩ {βp > 0}ä+ P (L′{p ) .We analyse now what happens on the intersection {αp = 0} ∩ L′p ∩ {βp > 0}. Assume thatthis event occurs. Then, by definition of the event L′p (A), there exists a splitting levelh ∈ ¶4 · 92(p−1)k0 , . . . , 4 · 92(p−1)k0 + 2 · 9(p−1)k0© for the section (S ′m′)m′=σ′2(p−1),...,σ′2p . Denoteby h′p the first of such splitting levels and by m′p the unique index for which pr0S′m′p = h′p.According to (5.2.6), we have(B) LE (S ′m′)m′=σ′2(p−1),...,σ′2p= LE(S ′m′)m′=σ′2(p−1),...,m′p ∨ LE(S ′m′)m′=m′p,...,σ′2p .Since σ′2(p−1) is the hitting time of the set H2(p−1), it is then obvious that h′p is also a splittinglevel for the section (S ′m′)m′=0,...,σ′2p and so, we also have(C) LE(S ′m′)m′=0,...,σ′2p = LE(S′m′)m′=0,...,m′p ∨ LE(S ′m′)m′=m′p,...,σ′2p .That αp = 0 entails that the section (Sm)m=0,...,σ2p does not intersect the set in (C) and therelation βp > 0 means that it does intersect the set in (B). Therefore, on the event underconsideration, the intersection of (Sm)m=0,...,σ2p with LE(S′m′)m′=σ′2(p−1),...,σ′2poccurs at some97Chapter 5. One endedness for Γd(λ)S ′m′ with m′ ∈ ¶σ′2(p−1), . . . ,m′p© ; in particular, S and S ′ intersect inside the strip J+2(p−1)(notation of (5.2.5)). In other words, with the notation I+p of (5.2.5),PÄ{αp = 0} ∩ L′p ∩ {βp > 0}ä ≤ P ÄI+2(p−1) > 0ä .Whence, by the arguments of the previous two paragraphs,(D) P (αp = 0) ≤ PÄ{αp = 0} ∩ L′p ∩ {βp = 0}ä+ P (L′{p )+ P ÄI+2(p−1) > 0ä .Now observe that{αp = 0} ∩ L′p ∩ {βp = 0} =Ä{αp = 0, αp−1 = 0} ∩ L′p ∩ {βp = 0}ä∪ Ä{αp = 0, αp−1 > 0} ∩ L′p ∩ {βp = 0}ä⊂ Ä{αp−1 = 0} ∩ L′p ∩ {βp = 0}ä ∪ {αp = 0, αp−1 > 0},and from this it follows thatPÄ{αp = 0} ∩ L′p ∩ {βp = 0}ä ≤ P Ä{αp−1 = 0} ∩ L′p ∩ {βp = 0}ä+ P (αp = 0, αp−1 > 0) .By (1.12.4), {αp−1 = 0} and L′p ∩ {βp = 0} are independent events. Write now{αp = 0, αp−1 > 0} =Ä{αp = 0, αp−1 > 0} ∩ R′p−1ä ∪ ({αp = 0, αp−1 > 0} ∩ R′{p−1) ;as done before, on the set {αp = 0, αp−1 > 0} ∩ R′p−1, the two random walks S and S ′intersect in the strip J−2(p−1), yieldingPÄ{αp = 0, αp−1 > 0} ∩ R′p−1ä ≤ P ÄI−2(p−1) > 0ä .With the aforementioned considerations and substituting into (D), we reachP (αp = 0) ≤ P (αp−1 = 0)P (βp = 0)+ PÄI−2(p−1) > 0ä+ P(R′{p−1)+ P(L′{p)+ PÄI+2(p−1) > 0ä.By virtue of propositions (5.2.5) and (5.2.7), we havePÄI−2(p−1) > 0ä+ P(R′{p−1)+ P(L′{p)+ PÄI+2(p−1) > 0ä ≤ ce−c′pand by (5.2.8)P (βp = 0) ≤ 1− c′′p,for p ≥ P, where the constants c, c′ and c′′ are all universal. We can conclude then thatP (αP+p = 0) ≤p∏q=0Ç1− c′′P + qå+ ce−c′Pp∑q=0e−c′q.The conclusion of the proposition is now clear given the last inequality and bearing in mindthat all estimates are uniform in ‖o‖ ≤ δ.98Chapter 5. One endedness for Γd(λ)Proposition ( 5.2.10 ) Let d = 2. With the hypotheses of (5.2.9), let α˜p(o) be the numberof intersections of LE(S ′m′)m′∈Z+ and (Sm)m=0,...,σ2p . Then, for every δ > 0,limp→0 max‖o‖≤δP (α˜p(o) = 0) = 0Proof. We use the notation of (5.2.7). Simply writeP (α˜p(o) = 0) = P (α˜p(o) = 0, αp(o) = 0) + P (α˜p(o) = 0, αp(o) > 0)≤ P (αp(o) = 0) + P ({α˜p(o) = 0, αp(o) > 0} ∩ Rp)+ P({α˜p(o) = 0, αp(o) > 0} ∩ R{p).Exactly as in the proof of (5.2.9), the two relations α˜p(o) = 0 and αp(o) > 0 on the eventRp imply that two independent random walks will intersect in the strip J−2p, hence the resultfollows immediately upon invoking (5.2.5), (5.2.7) and (5.2.9).z 5.2.3 The proof of the main theorem.Theorem ( 5.2.11 ) For almost every realisation of UST in Γ2(λ), the tree has one end.Proof. Denote by UST the measure of uniform spanning tree on the network Γ2(λ). Whenwe define an event on spanning trees, we use UST to refer to the tree itself. We know thatUST has at least one end, up to a UST-null event. Consider the following eventN = {UST has at least two ends}.We aim at showing UST(N ) = 0. For a vertex z ∈ Z3, considerNz = {there are two disjoint rays starting at z in UST}.Then,N =⋃z∈Z3Nz.Further, the translation invariance of the probability kernel of the network random walk ofΓ2(λ) shows that UST(Nz) is a value independent of z ∈ Z3. Hence, to show N is UST-null,it suffices to show N0 is UST-null. Whence, our aim is now to show N0 is UST-null.Consider now a set of vertices K of Z3 with the following property (a “cutset” between0 and infinity): every ray from 0 must cross K at some vertex. Recall an “edge of K” is anedge of the subgraph induced on K. Then, N0 is contained in the eventCK = {there are two disjoint paths in UST from K to 0 that use no edge of K} .99Chapter 5. One endedness for Γd(λ)In particular, we have the following:UST(N0) ≤ infK is a cutsetbetween 0 and ∞UST(CK).As in proposition (5.2.2), define the setsAp = [−p, p]× S (0; p) , Bp = {p} × B (0; p) , and Cp = {−p} × B (0; p) .Their union Kp is a cutset between 0 and ∞. In particular,(A) UST(N0) ≤ infp∈NUST(CKp).Construct T the UST of Z3 using Wilson’s algorithm rooted at infinity and following somestarting at zero predefined ordering of the vertices of Z3, and assume it is defined on someprobability space (Ω,F ,P). (For instance, we can follow the order in which we start with0, and having searched all vertices such that ‖z‖1 = k, we search the vertices satisfying‖z‖1 = k + 1 arranging them lexicographically.) Thus, we have a family of independentnetwork random walks (Sz)z∈Z3 defined on this probability space with Sz started at z. Since0 is the first vertex searched,L0 = LE(S0m)m∈Z+ ⊂ T(both trees canonically identified with their sets of edges). Up to a P-null event, L0 is aninfinite branch. Thus,{T ∈ CKp} = {some vertex of Kp is connected to 0 in T \ L0}.DenoteC(z) = {z is connected to 0 in T \ L0},so that{T ∈ CKp} =⋃z∈KpC(z).By virtue of (5.2.2) we reach the existence of an index p1(ε) and a number b1(ε) such thatwe may divide Cp = Cp,0∪Cp,1 as in this proposition (taking b = b1(ε)) and for all p ≥ p1(ε),∑z∈EpG(z, 0) ≤ ε,with the notation Ep = Ap ∪ Bp ∪ Cp,1 of the proposition (in particular, G is the Green’sfunction of Γ2(λ)). Next,UST(CKp) = PÄT ∈ CKpä ≤ ∑z∈EpP (C(z)) + PÑ ⋃z∈Cp,0C(z)é.100Chapter 5. One endedness for Γd(λ)By corollary (5.2.1.1), when calculating P (C(z)) we may assume that the order in whichT was constructed is (0, z, . . .). If the order in the construction of T were (0, z, . . .), thenC(z) would be the event where Sz hits L0 for the first time at zero, and then C(z) would becontained in the event where τ0(Sz) <∞. Thus, if the order were (0, z, . . .), we would have1C(z) ≤ 1 ⋃m∈Z+{Szm=0} ≤∑m∈Z+1{Szm=0}.The foregoing showed thatP (C(z)) ≤ G(z, 0).Substituting this inequality in the previous bounds∑z∈EpG(z, 0) ≤ ε, we obtain that for allp ≥ p1(ε),(B) UST(CKp) ≤ ε+ PÑ ⋃z∈Cp,0C(z)é.Using (5.2.3), there is an index p2(ε) and a family of sets (C′p,0)p≥p2(ε), for which theconclusions of this proposition hold. In particular, we may define δ0 <∞ as followsδ0 = supp≥p2(ε)maxz∈Cp,0∥∥∥z − C′p,0∥∥∥ .For each z′ ∈ C′p,0, consider the set Cp,0(z′) = {−p} × B (z′; δ0) . The definition of δ0 showsat onceCp,0 ⊂⋃z′∈C′p,0Cp,0(z′).Then,PÑ ⋃z∈Cp,0C(z)é≤ ∑z′∈C′p,0PÑ ⋃z∈Cp,0(z′)C(z)é.By definition,⋃z∈Cp,0(z′)C(z) is an event depending solely on T and L0. Thus, (5.2.1.1) showsthat its probability is independent of the ordering of the vertices of Z3\{0}, in particular, wemay assume that T was constructed using the ordering (0, z′, . . .). Consider now the followingeventH(z′) =ßSz′τL0(Sz′) 6= 0™,in other words, H(z′) is the event where Sz′hits L0 for the first time anywhere except zero(recall Sz′will hit L0 a.s.), which is the same as the event where z′ is not connected to 0 inT \ L0 (with the assumed order (0, z′, . . .) of Z3). Then,1− P (H(z′)) = PÅSz′τL0(Sz′) = 0ã≤ P Äτ0 ÄSz′ä <∞ä ≤ G(z′, 0).101Chapter 5. One endedness for Γd(λ)Since z′ ∈ C′p,0 ⊂ Cp,0 = {−p} × B(0; bp12), we have by Theorem (2.6.1),G(z′, 0) ≤ c1 ‖z′‖−1 ≤ c1p−1.Then,PÑ ⋃z∈Cp,0(z′)C(z)é≤ PÑ ⋃z∈Cp,0(z′)C(z) ∩H(z′)é+ c1p−1.Summing over z′ ∈ C′p,0,∑z′∈C′p,0PÑ ⋃z∈Cp,0(z′)C(z)é≤ ∑z′∈C′p,0PÑ ⋃z∈Cp,0(z′)C(z) ∩H(z′)é+ c1p−1 · card ÄC′p,0ä .By (5.2.3), we have that for p ≥ p2(ε), cardÄC′p,0ä ≤ cp · ε, where c is a universal constant.By (B), we have shown hitherto the existence of a universal constant c such that for allp ≥ max(p1(ε), p2(ε)),(C) UST(CKp) ≤ c · ε+∑z′∈C′p,0∑z∈Cp,0(z′)P (C(z) ∩H(z′)) .By virtue of (A) and (C), we reduce the proof of the theorem to showing following: thereexists a function ϕ(p) such that(D)∑z′∈C′p,0∑z∈Cp,0(z′)P (C(z) ∩H(z′)) ≤ ϕ(p)and ϕ(p)→ 0 via some subsequence.We begin by estimating P (C(z) ∩H(z′)) . Again by (5.2.1.1), to calculate the probabil-ity of the event C(z) ∩ H(z′) we may assume that T was constructed using the ordering(0, z′, z, . . .) of Z3. Assuming this ordering of Z3, and on the event C(z) ∩ H(z′), the firstthree steps of the construction of T proceed as follows:1) The first branch at zero is created, we call it L0.2) The random walk Sz′runs, it hits the first branch at zero at some non zero vertex.Denote by Lz′ this second branch; by definition,Lz′ = LEÄSz′mäm=0,...,τL0(Sz′)3) The random walk Sz runs, and it hit L0 at 0 before hitting Lz′ .102Chapter 5. One endedness for Γd(λ)Introduce the eventN(z, z′) =¶τ0(Sz) < τLz′ (Sz)©,which is the event where Sz hits L0 at zero before it touches Lz′ anywhere. The aforemen-tioned first three steps show that(E) C(z) ∩H(z′) ⊂ N(z, z′).In what follows, we will consider the planesHp =¶4 · 92pk0©× Z2,and for z, z′ ∈ Z3, the hitting timesσp(z, z′) = τHp+z′(Sz).As such, we will make a “change of scales” and consider from now onwards indices of theform 4 · 92pk0 . Assume then that z′ ∈ C′4·92pk0 ,0 and z ∈ C4·92pk0 ,0(z′). Denote ap = −4 · 92pk0 +4 · 92(p−1)k0 , so that Hp−1 + z′ = {ap} × Z2. Consider the eventsEp =¶S0 ever reaches − 9pk0© ,Sp(z′) =ap⋃h=ap−2·9(p−1)k0{h is a splitting level forÄSz′m′äm′∈Z+}.By (2.4.5) and (5.2.7) we know there exists a constant c > 0 and an index p4 ∈ N such thatfor all p ≥ p4,P (Ep) = e−λ9pk0 , P (Sp(z′)) ≥ 1− e−c9pk0 .Hence,(F) P (N(z, z′)) ≤ P(N(z, z′) ∩ Sp(z′) ∩ E{p)+ e−c9pk0 + e−λ92pk0 .We introduce the following “good event”Gp(z′) = Sp(z′) ∩ E{p;on this event S0 never goes left of the plane {−9pk0} × Z2 and Sz′ has a splitting level inthe stated range. On the good event Gp, the random walk Sz′ must have crossed the planeHp−1 + z′ before hitting the first branch. In particular,σp−1(z′, z′) < τL0ÄSz′ä<∞ on Gp(z′).103Chapter 5. One endedness for Γd(λ)Also, on the good event Gp, there is a splitting plane forÄSz′m′äm′∈Z+ ; introduce hp(z′)and mp(z′) ≤ σp−1(z′, z′) to be the last splitting level (amongst the levels h = ap − 2 ·9(p−1)k0 , . . . , ap) and the only index m′ such that pr0ÄSz′m′ä= hp(z′). By (5.2.6), on the goodevent,Lz′ = LEÄSz′m′äm′=0,...,τL0(Sz′)= LEÄSz′m′äm′=0,...,mp(z′)∨ LE ÄSz′m′äm′=mp(z′),...,τL0(Sz′).Consider now the “partial branch at z′ (until its hitting time of Hp−1 + z′)” which isLz′p−1def.= LEÄSz′m′äm′=0,...,σp−1(z′,z′).On Gp(z′),Lz′p−1 = LEÄSz′m′äm′=0,...,σp−1(z′,z′)= LEÄSz′m′äm′=0,...,mp(z′)∨ LE ÄSz′m′äm′=mp(z′),...,σp−1(z′,z′) .Defineαp−1(z, z′) =∑m=0,...,σp−1(z,z′)∑z′′∈Lz′p−11{Szm=z′′},which is the number of intersections between the partial branch at z′, Lz′p−1, and the section(Szm)m=0,...,σp−1(z,z′). Similarly, define βp−1(z, z′) to be the number of intersections betweenLz′ and the same section of Sz, that isβp−1(z, z′) =∑m=0,...,σp−1(z,z′)∑z′′∈Lz′1{Szm=z′′}.We have the following limitlimp→∞ max‖z−z′‖≤δ0P (αp−1(z, z′) = 0) = 0,where the maximum runs over all pairs (z, z′) ∈ C4·92pk0 ,0 × C′4·92pk0 ,0. This last limit is acorollary of proposition (5.2.9): the translation invariance of the probability kernel shows atonce that the maximum inside the limit does not depend on the pair (z, z′) above but solelyon the distance separating them.Lemma ( 5.2.11.1 ) limp→∞ max‖z−z′‖≤δ0P (Gp(z′) ∩ {βp−1(z, z′) = 0}) = 0, where the maximumruns over all pairs (z, z′) ∈ C4·92pk0 ,0 × C′4·92pk0 ,0.104Chapter 5. One endedness for Γd(λ)Proof of lemma. Observe thatP (Gp(z′) ∩ {βp−1(z, z′) = 0}) ≤ P (Gp(z′) ∩ {αp−1(z, z′) > 0, βp−1(z, z′) = 0})+ P (αp−1(z, z′) = 0) .The second of these two terms tends uniformly to zero for the set of stated pairs (z, z′), aswas shown before the lemma. On the good event Gp we can apply the two decompositionsabove for Lz′ and Lz′p−1, givingGp ∩ {αp−1(z, z′) > 0, βp−1(z, z′) = 0} ⊂¶Sz cross Sz′inside Jp−1©,where Jp−1 =¶ap − 2 · 9(p−1)k0 , . . . , ap©×Z2. By (5.2.5), we have that there exists a universalconstant c > 0 and an index p3 such that for p ≥ p3,PÄSz cross Sz′inside Jp−1ä ≤ c3−pk0 ,this bound being uniform for z′ ∈ C′4·92pk0,0 and z ∈ C4·92pk0 ,0(z′).We continue now with the proof of the theorem. Recall we consider z′ ∈ C′4·92pk0 ,0 andz ∈ C4·92pk0 ,0(z′). Introduce the following random timeθp−1(z, z′) = minÄσp−1(z, z′), τLz′ (Sz)äwhich is the hitting time of (Hp−1 + z′) ∪ Lz′ by Sz. Then, θp−1(z, z′) is a stopping time forthe filtration ÄX z′k (z) = σ (Szm | 0 ≤ m ≤ k) ∨F∞ ({0, z′})äk∈Z+ ,where F∞(T) = σ(Stm | t ∈ T,m ∈ Z+). Observe that Gp(z′) ∈ F∞({0, z′}). We apply thestrong Markov property to the processÄSzk ,Xz′k (z)äk∈Z+ in the third line belowEÄ1N(z,z′)∩Gp(z′)ä= E(EÄ1N(z,z′)1Gp(z′)∣∣∣ X z′θp−1(z,z′)(z)ä)(G)= E(1Gp(z′)EÄ1N(z,z′)∣∣∣ X z′θp−1(z,z′)(z)ä)= E(1Gp(z′)EÄ1N(z,z′)∣∣∣ Szθp−1(z,z′)ä)= EÄ1Gp(z′)PÄN(z, z′) | Szθp−1(z,z′)ää.The definition of the stopping time θp−1(z, z′) shows thatPÄN(z, z′)∣∣∣Szθp−1(z,z′) = z′′ä = 0105Chapter 5. One endedness for Γd(λ)for all z′′ ∈ Lz′ lying to the left or on the plane Hp−1 + z′. Therefore, we havePÄN(z, z′)∣∣∣Szθp−1(z,z′)ä = P ÄN(z, z′)∣∣∣Szθp−1(z,z′)ä1{Szθp−1(z,z′)/∈Lz′}.Clearly N(z, z′) ⊂ {τ0(Sz) <∞}, also, we have thatθp−1(z, z′) = σp−1(z, z′), on the event¶Szθp−1(z,z′) /∈ Lz′©;this implies Szθp−1(z,z′) ∈ Hp−1 + z′ on this event. Whence,(H) PÄN(z, z′)∣∣∣Szθp−1(z,z′)ä ≤ ( supz′′∈Hp−1+z′P (τ0(Sz) <∞ | Sz0 = z′′))1{Szθp−1(z,z′)/∈Lz′}.We know from the bounds of the Green’s function that(I) supz′′∈Hp−1+z′P (τ0(Sz) <∞ | Sz0 = z′′) ≤ c1|ap|−1 ≤ c9−2pk0 ,where c > 0 is a universal constant.Use the bound of (I) in (H), and then apply this to (G). Thus, we have shown up to thispoint in the argument the existence of a universal constant c > 0 such thatP (N(z, z′) ∩ Gp(z′)) ≤ c9−2pk0PÄGp(z′) ∩ ¶Szθp−1(z,z′) /∈ Lz′©ä .It is also obvious thatGp(z′) ∩ ¶Szθp−1(z,z′) /∈ Lz′© ⊂ Gp(z′) ∩ {βp−1(z, z′) = 0}.Hence,(J) P (N(z, z′) ∩ Gp(z′)) ≤ c9−2pk0∆p,with∆p = max‖z−z′‖≤δ0P (Gp(z′) ∩ {βp−1(z, z′) = 0})→ 0as p→∞, the pairs (z, z′) ∈ C4·92pk0 ,0 × C′4·92pk0 ,0.We are in conditions to conclude the proof of the theorem. By the foregoing,P (C(z) ∩H(z′)) ≤ P (N(z, z′)) , use (E)≤ P (N(z, z′) ∩ Gp(z′)) + e−c′9pk0 + e−λ9−2pk0 , use (F)≤ c9−2pk0∆p + e−c′9pk0 + e−λ9−2pk0 , use (J)106Chapter 5. One endedness for Γd(λ)with the two constants c, c′ > 0 being universal. We consider now z′ ∈ C′4·92pk0 ,0 and sumover z ∈ C4·92pk0 ,0(z′), this latter set has cardinality bounded above by c′′δ20, with c′′ > 0another universal constant. This shows, for every z′ ∈ C′4·92pk0 ,0,∑z∈C4·92pk0 ,0(z′)P (C(z) ∩H(z′)) ≤ c′′δ20(c9−2pk0∆p + e−c′9pk0 + e−λ9−2pk0),this bound being uniform for z′. We know also thatcardÄC′4·92pk0 ,0ä ≤ card ÄC4·92pk0 ,0ä ≤ c′′′ · 92pk0 ,for yet another universal constant c′′′ > 0. Whence, if we sum over z′ ∈ C′4·92pk0 ,0, we reach abound of the form∑z′∈C′4·92pk0 ,0∑z∈C4·92pk0 ,0(z′)P (C(z) ∩H(z′)) ≤ c(∆p + 92pk0(e−c′9pk0 + e−λ92pk0)),for a constant c > 0, not depending on p. The right hand side of this inequality tends tozero, and this completes the proof of Theorem (5.2.11) by (D).§ 5.3. One endedness for Γd(λ), with d ≥ 3.We finally deal with the case of “high dimensions.” Here, the proof is considerably simplerthan d = 2. We will make use of several general results of electrical networks, compiled inAnnex C for the reader’s convenience.Theorem ( 5.3.1 ) Let d ≥ 3. Then, for almost every realisation of the uniform spanningforest in Γd(λ), all the components in this realisation have one end.Proof. For every p ∈ N, consider the sets Ap, Bp, Cp, defined in proposition (5.2.2). Usingthe notation of this proposition, p14 will be larger than B starting at some index p∗. As such,we may divide Cp = Cp,0 ∪ Cp,1, with Cp,0 = {−p} × B(0; p34)and Cp,1 = Cp \ Cp,0.( 5.3.1.1 ) For every p ≥ p∗,∑z∈Ap∪BpG(z, 0) ≤ ce−c′p and ∑z∈Cp,1G(z, 0) ≤ ce−c′p12 ,where c, c′ are positive constants.107Chapter 5. One endedness for Γd(λ)Assume F is the random spanning forest of Γd(λ) as constructed by Wilson’s algorithmrooted at infinity, see sect. §1.7. To be more precise, assume that on some probability space(Ω,F ,P) there is a family of independent network random walks of Γd(λ), say (Sz)z∈Zd+1with Sz started at z, and then construct F using some predefined order starting at zero ofZd+1 following Wilson’s algorithm. (For instance, start at zero and then search the ‖·‖1-spheres by first increasing the radii one unit at a time and then lexicographically orderingeach sphere.) We know then that F is a random spanning forest following law USF of Γd(λ);this signifies thatP (F ∈ E ) = USF(E ),for all events E in the probability space of USF spanning forests of Γd(λ).Because of the order chosen to construct F, L0 = LE(S0m)m∈Z+ is (a.s.) an infinite path,we will call it “first branch of the random forest F.” Define the following set in Zd+1 (considerz ∈ Zd+1) and events in F :Ep = Ap ∪ Bp ∪ Cp,1,E(z) = {z is connected to 0 in F \ L0}Ep =⋃z∈EpE(z) = {some vertex in Ep is connected to 0 in F \ L0}.Then,P (Ep) ≤∑z∈EpP (E(z)) .To calculate P (E(z)) we may apply corollary (5.2.1.1) and assume that F was constructedusing the ordering (0, z, . . .) of Zd+1. With this ordering, the event E(z) is the event whereSz hits L0 for the first time at vertex 0, whenceP (Ep) ≤∑z∈EpP (τ0(Sz) <∞) ≤ ∑z∈EpG(z, 0) < ce−c′p12 ,where the second inequality is an obvious consequence of1{τ0(Sz)<∞} = 1⋃m∈Z+{Szm=0} ≤∑m∈Z+1{Szm=0},and the last one is a consequence of (5.3.1.1). By virtue of Borel-Cantelli lemma, we reachthe existence of a random index P : Ω→ Z+ (measurable relative to F ) such that for P-a.e.ω ∈ Ω, ω /∈ ⋃p≥P (ω)Ep.Let U denote the (random connected) component of 0 in the forest F \L0. Denote by Upthe graph induced by U in the cylinder Vp = [−p, p]× B (0; p) of Z× Zd.108Chapter 5. One endedness for Γd(λ)Lemma ( 5.3.1.2 ) If P (U is finite) = 1, then P (all the components of F have one end) = 1.Proof of lemma. For every z ∈ Zd+1, denote by T(z) the event where the component of z inF is not one ended. Then,Tdef.= {there exists a component of F that is not one ended} = ⋃z∈Zd+1T(z).We want to prove P (T) = 0. By translation invariance, P (T(z)) is independent of z, andtherefore, it suffices to prove that T(0) is a null event.Now, since 0 is a vertex of L0 ⊂ F and this is an infinite branch (a.s.), the relationthat the component of 0 in F is not one ended signifies that U does not possess a ray,and this is equivalent to U being finite since it is a locally finite tree. We established thatT(0){ = {U is finite}. The conclusion of the lemma is now clear.We know that for p ≥ P, no edge in U is adjacent to Ep, and the only way U can haveinfinitely many edges is by using edges adjacent to Cp,0. Let Kp−1 be the (random) set ofedges of Vp = [−p, p]×B (0; p) that are adjacent to Up−1 and Cp,0. We need to study the sizeof the sets Kp, the following lemma will be needed.Lemma ( 5.3.1.3 ) Let Γ be a network satisfying the usual assumptions (A.22). Supposethat A is a random spanning subgraph of G defined on some probability space (Ω,F ,P).Suppose that A satisfies the following: if v and v′ are two vertices that are end-points ofedges of A, then v and v′ are in the same connected-component relative to A. (Think of Aas a connected subgraph of plus all the vertices of G.) Denote by (Vn) an exhaustion of Gsuch that Vn ⊂ Vn+1 for every n ∈ N. Let Fn denote the σ-algebra generated by the events{e ∈ A} as e runs through the edges of the graph induced on Vn. Define An to be the spanningsubgraph of G whose edge set consists of the edges of A with both end-points belonging to Vn(with A0 = ∅). Consider the events (n ∈ N)Gn = {An 6= An−1},so that, Gn is the event where there was “growth by edges” of A in Vn. Set Yn = P (Gn | Fn−1)(n ∈ N). Then (Gn)n∈N is a decreasing sequence, if G∞ =⋂n∈NGn, then G∞ is the event whereA has infinite edges; furthermore, on G∞, Yn → 1.Proof of lemma. It is clear that G{n ⊂ G{n+1, hence (Gn) is decreasing. The events G{n ∩ Gn−1(where there growth ends exactly in Vn−1) are pairwise disjoint, implying1 ≥ P( ⋃n∈NG{n ∩ Gn−1)=∑n∈NE(EÄ1G{n1Gn−1∣∣∣Fn−1ä) .109Chapter 5. One endedness for Γd(λ)It is clear that Gn−1 ∈ Fn−1 and so,1 ≥ E(∑n∈N1Gn−1(1− Yn)).Therefore, the positive random variable∑n∈N1Gn−1(1 − Yn) is finite for P-a.e. realisationω ∈ Ω. Thus, ∑n∈N(1 − Yn) is finite P-a.s. on the event G∞, which immediately implies theconclusion to be reached, namely, that Yn → 1 on G∞.Let us return to the proof of Theorem (5.3.1). We apply the previous lemma to therandom graph U, and the exhaustion Vp = [−p, p] × B (0; p) . Then, the events constructedin the lemma are Gp = {Up 6= Up−1} and the σ-algebras areFp = σ({e ∈ Up}; e is an edge of the graph induced on Vp by Zd+1).By construction, Kp−1 is Fp−1-measurable. Set Yp = P (Gp | Fp−1) , so that P-a.s. Yp → 1on the event where U is infinite. Now,1− Yp = P (U = Up−1 | Fp−1) = P (U = Up−1, p ≥ P | Fp−1) + P (U = Up−1, p < P | Fp−1)= P (F ∩ Kp−1 = ∅, p ≥ P | Fp−1) + P (U = Up−1, p < P | Fp−1)= P (F ∩ Kp−1 = ∅ | Fp−1)− P (F ∩ Kp−1 = ∅, p < P | Fp−1)+ P (U = Up−1, p < P | Fp−1) .Observe that{U = Up−1} ⊂ {F ∩ Kp−1 = ∅},since, by definition, Kp−1 are the edges adjacent to both Cp,0 and Up−1 in Vp. Thus,1− Yp = P (F ∩ Kp−1 = ∅ | Fp−1)− P (F ∩ Kp−1 = ∅,U 6= Up−1, p < P | Fp−1) .If U grew at stage p, but it did not so through Kp−1, then U grew through some edge adjacentto Ep, henceP (F ∩ Kp−1 = ∅,U 6= Up−1, p < P | Fp−1) ≤ P (Ep | Fp−1) .We know that P (Ep)→ 0, so the sequence (P (Ep | Fp−1)) converges to zero in L 1. We mayassume, passing through a subsequence should the need arise, that (P (Ep | Fp−1)) convergesto zero P-a.s. Therefore,1− Yp ≥ P (F ∩ Kp−1 = ∅ | Fp−1)− P (Ep | Fp−1)110Chapter 5. One endedness for Γd(λ)thus, on the event where U is an infinite component,P (F ∩ Kp−1 = ∅ | Fp−1)→ 0.Denote by Gp the random graph obtained from Zd+1 where all edges e ∈ Up are shortedand the edges e in Vp that are not in Up are cut (p ∈ N), see §C.8. Elementary propertiesof wired spanning forest measures show thatP (F ∩ Kp−1 = ∅ | Fp−1) = WSFGp−1 (F ∩ Kp−1 = ∅) ,the right hand side meaning the WSF measure on the random graph Gp−1; in this graphhowever, Kp−1 is not random, while F is its WSF-distributed random object. We can apply[LP16, Lemma 10.40] and concludeWSFGp−1 (F ∩ Kp−1 = ∅) ≥∏e∈Kp−111 + µ(e)RWGp−1\Kp−1 (e−, e+),whereRWG (e−, e+) stands for the wired effective resistance between e− and e+ in the networkG, see §C.12. We know that µ(e)  e−λp for e ∈ Kp−1 and that (C.12.7)RWGp−1\Kp−1Äe−, e+ä ≤ RGp−1\Kp−1 Äe−,∞ä+RGp−1\Kp−1 Äe+,∞ä .The following lemma is crux in the proof and shows at once why d ≥ 3 is needed (which hasnot been needed so far).Lemma ( 5.3.1.4 ) Let d ≥ 3. For e ∈ Kp−1, we have RGp−1\Kp−1 (e±,∞)  eλp (any implicitconstant being universal).Proof of lemma. By Thompson’s principle (C.11.1)RΓ (v,∞) = infθE (θ) ,where θ runs on the sets of unit flows from v to ∞. Since d ≥ 3, we have that the standardrandom walk of Zd is transient and T. Lyon’s criterion (C.10.1) shows the existence of a unitflow θ0 from 0 to infinity having finite energy in the graph Zd. We know that the networkGp−1 \ Kp−1 contains the hyperplane Hp = {−p} × Zd. Observe that Hp as a subnetwork ofGp−1 \Kp−1 has constant resistances equal to eλp. Translate θ0 to Hp in the obvious way anddenote the new flow by θp. Clearly, (C.1.1)EHp (θp) = eλpEZd (θ0) ,111Chapter 5. One endedness for Γd(λ)where EG (θ) is the energy of the flow θ on the network G. Extend θp to Gp−1 \ Kp−1 bysending zero flow through the edges outside Hp. The energy of the extension of θp, denotedby θp, does not change, that is to say,EHp (θp) = EGp−1\Kp−1Äθpä.Thus, we proved that there exists a c = EZd (θ0) > 0 such that for all p ∈ N and all edgese ∈ Kp−1,RGp−1\Kp−1Äe±,∞ä ≤ ceλp.The reverse inequality is not needed but we prove it for completeness. Here we usethat C (v,∞) = R (v,∞)−1 and we bound from above the effective conductance. We applyDirichlet’s principle (D.3.9). Consider a vertex z = (−p, x) and define ϕ = 1z. Then,CGp−1\Kp−1 (z,∞) ≤ DGp−1\Kp−1 (ϕ) ≤∑e and edge of Γd(λ)e−=zµ(e) ≤ 2(d+ 1)eλe−λp.Thus, for c−1 = 2(d+ 1)eλ > 0, we haveRGp−1\Kp−1 (z,∞) ≥ ceλp.The proof of the lemma is complete.We may continue with the proof of one endedness in Γd(λ). By virtue of the previouslemma, there exists a universal δ > 0 such that P-a.s.WSFGp−1 (F ∩ Kp−1 = ∅) ≥ δcard(Kp−1).On the event U is infinite, we proved that WSFGp−1 (F ∩ Kp−1 = ∅) → 0 and so, on thisevent, card (Kp−1)→∞.Consider then the random variables Zp−1 = card (Kp−1) . Notice that (−p, x) ∈ Cp,0 isadjacent to an edge of Kp−1 if and only if (−p+ 1, x) is adjacent to an edge in Up−1. Thus,E (Zp−1) =∑z∈Cp−1,0P (z is connected to 0 in Up−1)≤ ∑z∈Cp−1,0P (z is connected to 0 in F \ L0) .Using (5.2.1.1), we may calculate P (z is connected to 0 in F \ L0) assuming that the order inwhich the vertices of Zd+1 were searched was (0, z, . . .). Then, the event where z is connectedto 0 in F \ L0 is the event where Sz visits 0 before hitting L0 anywhere else. In particular,E (Zp−1) ≤∑z∈Cp−1,0P (τ0(Sz) <∞) ≤ ∑z∈Cp−1,0G(z, 0) ≤ c112Chapter 5. One endedness for Γd(λ)by (5.3.1.1). This implies P (Zp →∞) = 0 for Fatou’s lemma allows deducingEÅlim · infp→∞ Zpã≤ lim · infp→∞ E (Zp) ≤ c.Finally, we showed thatP (card (U) =∞) ≤ P (Yp → 1) ≤ P (Zp →∞) = 0.The proof of (5.3.1) is complete by Lemma (5.3.1.2).§ 5.4. On the rays of Γd(λ).We will say that a unending path γ = (vj)j∈Z+ is ultimately inside a region R if there existsan almost surely finite random index J such that vj ∈ R for all j ≥ J.DenoteP =¶z = (n, x) ∈ Z× Zd; ‖x‖2 ≤ n©andPε =¶z; ‖x‖2 ≤ n1+ε© .Theorem ( 5.4.1 ) Let z be any vertex of Γd(λ). Then, there exists one and only one rayin USF (UST in d = 1, 2) starting at z. This ray is a random object (is USF-measurable).Furthermore, for every ε > 0, this ray is ultimately inside z + Pε, for every ε > 0.Proof. By Theorems (5.1.1), (5.2.11) and (5.3.1) there exists one and only one ray, this provesat once such ray is USF-measurable. There remains to prove that said ray is ultimately insidez + Pε for every ε > 0. We may construct UST in d = 1, 2 or USF for d ≥ 3 using Wilson’salgorithm rooted at infinity with the ordering of Zd+1 to be (z, . . .). Thus, suffices to showthat if Sz is the network random walk of Γd(λ) started at z, then Sz is ultimately insidez + Pε; furthermore, by translation invariance, we may assume z = 0.Let S denote the network random walk of Γd(λ) started at zero. To show that S isultimately inside P is the same as showing that, for every ε > 0, S will only be finitely manytimes outside Pε. We may apply Borel-Cantelli lemma to show this and as such, it all reducesto show that∑n∈Z+P (Sn /∈ Pε) <∞. The Green’s function estimates (2.6.1) give at once∑n∈Z+P (Sn /∈ Pε) ≤∑z /∈PεG(0, z) =∑z∈S1G(0, z) +∑z∈S2∩PεG(0, z),113Chapter 5. One endedness for Γd(λ)where S1 = {z;n ≤ ‖x‖} and S2 = S{1. Since∑z∈S1G(0, z) ≤ ∑z∈Zd+1e−c‖z‖ < ∞, we focus onthe second term. We have (write z = (n, x))∑z∈S2∩PεG(0, z) ≤ c1∑n∈Z+∑n1+ε≤‖x‖2≤n2e−c2‖x‖2n ≤ c ∑n∈Z+e−c2nεnd <∞.This completes the proof.114Chapter 6Probability of belonging in the samecomponentIn this chapter we will provide a function u : Zd+1 → R+ with “polynomial decay” suchthat for all z ∈ Zd+1, the probability that 0 and z are in the same tree of Γd(λ) is  u(z);the important part is that the decay depends solely on dimension. Recall that for d = 1 ord = 2, on Γd(λ) we have a tree, thus the probability that 0 and any vertex z are connectedis 1, thus no decay.§ 6.1. The probability that two vertices are connectedin the USF.As usual, we will write z = (n, x), z′ = (n′, x′), z1 = (n1, x1), etc. to denote the vertices ofZd+1 = Z× Zd. Define(6.1.1) η(z) = max(|n| 12 , ‖x‖), z = (n, x) ∈ Zd+1.This η has the following properties, all of which are obvious.(a) η(z) ≥ 0, and η(z) = 0 is equivalent to z = 0.(b) η(z) = η(−z);(c) η(z + z′) ≤ η(z) + η(z′).Thus, the function (z, z′) 7→ η(z − z′) is a metric on Zd+1.115Chapter 6. Probability of belonging in the same componentTheorem ( 6.1.2 ) Let d ≥ 3. Then,USF(0 is connected to z)  η(z)−(d−2),with any constant depending solely on dimension.Proof. We remark that η(z)−(d−2)  max(n−d2+1, ‖x‖−(d−2)), and thus it suffices to showthe bounds for any of these expressions. We also remark that for every c > 0, there existsc′ > 0 such that e−c‖z‖ ≤ c′η(z)−(d−2); thus, if at some point, we show an upper exponentialbound in an estimate, we are done with that particular estimate.Consider the event C (0, z) that 0 is connected to z in USF. Since the measure USF istranslation invariant in Γd(λ),USF(C (0, z)) = USF(C (z′, z + z′))for any z′ ∈ Zd+1. It is also clear that C (0, z) = C (z, 0). Thus, we may assume z ∈ Z+×Zd.Construct now USF using Wilson’s algorithm rooted at infinity (see sect. §1.7) using theorder (0, z, . . .) of Zd+1, denote by F the random forest constructed in this way. Thus, wehave on some probability space (Ω,F ,P) a family of independent network random walks(Sv)v∈Zd+1 , with Sv started at v. We know F ∼ USF, and by construction {F ∈ C (0, z)} =¶Sz ∩ LE(S0m)m∈Z+ 6= ∅©. Thus, and with the aid of (5.2.4),(A) USF(C (0, z)) = PÄSz ∩ LE(S0m)m∈Z+ 6= ∅ä  P ÄSz ∩ S0 6= ∅ä .Denote by Kz the number of intersections between Sz and S0. In other words,(B) Kz =∑z′∈Zd+1∑(p,q)∈Z2+1{Szp=z′,S0q=z′}.Clearly, P (Kz > 0) = P (Sz ∩ S0 6= ∅) . Independence and the Lebesgue-Tonelli theoremshow at once(C) E (Kz) =∑z′∈Zd+1G(z, z′)G(0, z′).Lemma ( 6.1.2.1 ) We have P (Kz > 0)  E (Kz) , where any implicit constant is universal(does not depend on z).Proof of Lemma. Since Kz is a Z+-valued random variable, we have P (Kz > 0) ≤ E (Kz) .There remains to prove a lower bound of the form P (Kz > 0) ≥ cE (Kz) , for a univer-sal constant c > 0. Using the second moment inequality (4.2.1), suffices to establish thatE (K2z ) ≤ cE (Kz) , for some universal constant c > 0. By definition,K2z =∑(z′,z′′)∈Zd+1×Zd+1∑(p,q,a,b)∈Z4+1{Szp=z′,Sza=z′′}1{S0q=z′,S0b=z′′}.116Chapter 6. Probability of belonging in the same componentBy adding terms corresponding to p = a or q = b, then expanding the sum as to whetherp ≤ a or p ≥ a and q ≤ b or q ≥ b, we reach a bound with four sumsK2z ≤ L1 + L2 + L3 + L4,andE (L1) =∑(z′,z′′)∈Zd+1×Zd+1G(z, z′)G(z′, z′′)2G(0, z′)E (L2) =∑(z′,z′′)∈Zd+1×Zd+1G(z, z′)G(z′, z′′)G(z′′, z′)G(0, z′′)E (L3) =∑(z′,z′′)∈Zd+1×Zd+1G(z, z′′)G(z′′, z′)G(z′, z′′)G(0, z′)E (L4) =∑(z′,z′′)∈Zd+1×Zd+1G(z, z′′)G(z′′, z′)2G(0, z′′)It is clear that E (L1) = E (L4) and E (L2) = E (L3) , thusEÄK2zä ≤ 2ïE (L1) + E (L2) òWe first handle the expectation of L1, we obtainE (L1) =∑z′∈Zd+1G(z, z′)G(0, z′)∑z′′∈Zd+1G(z′, z′′)2= ∑z′′∈Zd+1G(0, z′′)2 ∑z′∈Zd+1G(z, z′)G(0, z′)= cE (Kz) ,the last equality being obtained from the bubble condition (3.5.2).We now handle L2. First, the translation invariance of the Green’s function shows at oncethatE (L2) =∑(z′,z′′)G(z, z′)G(z′, z′′)G(z′′, z′)G(0, z′′)=∑z′G(z, z′)∑z′′G(0, z′′)G(z′′, 0)G(0, z′ + z′′).In the following calculations we are going to establish that the inner sum in the line above,when viewed as a function of z′, possesses the same type of upper bounds as the Green’sfunction’s upper bounds (2.6.1). There are two cases to consider. Write z′ = (n′, x′) andsimilarly z′′ = (n′′, x′′).117Chapter 6. Probability of belonging in the same componentCase 1. Here we assume ‖x′‖ ≤ n′. We then divide the inner sum into two, the first consistsof all z′′ satisfying |n′′| ≤ n′2and ‖x′′‖ ≤ ‖x′‖2and the second sum is over all otherz′′. In the first sum, we reach ‖x′ + x′′‖  ‖x′‖ and |n′ + n′′|  n′  ‖z′‖ , thenG(0, z′ + z′′) ≤ ce−c′ ‖x′‖2n′ ‖z′‖− d2 for some constants c, c′ > 0. This entails∑|n′′|≤n′2,‖x′′‖≤‖x′‖2G(z′′, 0)G(0, z′′)G(0, z′+z′′) ≤ ce−c′ ‖x′‖2n′ ‖z′‖− d2 ∑z′′∈Zd+1G(z′′, 0)G(0, z′′),for a pair of constants c, c′ > 0. We may apply the inequality ab ≤ a2 + b2, and obtainthat∑z′′∈Zd+1G(z′′, 0)G(0, z′′) <∞ by the bubble condition (3.5.2). This shows that∑|n′′|≤n′2,‖x′′‖≤‖x′‖2≤ ce−c′ ‖x′‖2n′ ‖z′‖− d2 ,for a pair of positive constants c, c′ > 0. Next, we are going to handle the second sum,which is over all z′′ for which either |n′′| > n′2or ‖x′′‖ > ‖x′‖2. Here, we will use thatone of the two factors G(z′′, 0), G(0, z′′) has an upper bound of the form e−c‖z′′‖. We willwrite e−c‖z′′‖ = e−c2‖z′′‖e−c2‖z′′‖ and use the fact that the Green’s function is boundedby some constant L > 0. Since ‖z′‖  n′, we may write∑|n′′|>n′2,x′′∈ZdG(z′′, 0)G(0, z′′)G(0, z′ + z′′) ≤ L2e− c4n′ ∑z′′∈Zd+1e−c2‖z′′‖ ≤ c′e−c′′‖z′‖,for positive constants c, c′, c′′ > 0. It is clear that c′e−c′′‖z′‖ ≤ Ce−C′ ‖x′‖2n′ ‖z′‖− d2 foranother pair of constants C,C ′ > 0. We finally deal with the sum corresponding to|n′′| ≤ n′2and ‖x′′‖ > ‖x′‖2. Here we decompose again e−c‖z′′‖ =Äe−c2‖z′′‖ä2 . Theassumption ‖x′‖ ≤ n′ shows that ‖x′‖ ≥ ‖x′‖2n′, and thuse−c2‖z′′‖ ≤ e− c2‖x′′‖ ≤ e− c4‖x′‖ ≤ e− c6 ‖x′‖2n′ .Also, it is clear that G(0, z′ + z′′) ≤ c′ ‖z′‖− d2 for some positive constant c′ > 0 since‖z′ + z′′‖  ‖z′‖ . Thus,∑|n′′|≤n′2,‖x′′‖>‖x′‖2G(z′′, 0)G(0, z′′)G(0, z′ + z′′) ≤ c′e− c6 ‖x′‖2n′ ‖z′‖− d2 ,for a pair of constants.118Chapter 6. Probability of belonging in the same componentCase 2. Here we assume |n′| < ‖x′‖ . We split∑z′′G(z′′, 0)G(0, z′′)G(0, z′ + z′′) = A+B,where A is the sum corresponding to all z′′ = (n′′, x′′) satisfying |n′′| ≤ ‖x′‖2and‖x′′‖ ≤ ‖x′‖2, B is the sum over all other z′′. For z′′ in the range of A, we obtain‖x′ + x′′‖  ‖x′‖  ‖z′‖ and |n′ + n′′| ≤ c ‖x′‖ ≤ c ‖z′‖ , and this shows that, on thisrange, G(0, z′ + z′′) ≤ ce−c′‖z′‖ for a pair of constants c, c′ > 0. Then, there exists twouniversal constants c, c′ > 0 such that (recall from Case 1 that G(z′′, 0)G(0, z′′) is asummable family for z′′ ∈ Zd+1)A ≤ ce−c′‖z′‖In the range of B, we have that either |n′′| > ‖x′‖2 ‖z′‖ or ‖x′′‖ > ‖x′‖2 ‖z′‖ , thus‖z′′‖ ≥ c ‖z′‖ for a universal constant c > 0. Next, one of the two factor G(z′′, 0) orG(0, z′′) is bounded from above by e−c‖z′′‖ by (2.6.1). Let L > 0 be a bound for theGreen’s function of Γd(λ). ThenB ≤ L2e−c′‖z′‖ ∑z′′∈Zd+1e−c2‖z′′‖ ≤ Ce−c′‖z′‖.By virtue of the two cases above, we have established that there exists two constants c, c′ > 0such that (write z′ = (n′, x′))∑z′′∈Zd+1G(z′′, 0)G(0, z′′)G(z′ + z′′) ≤ce−c′‖z′‖ if ‖x′‖ > n′,ce−c′ ‖x′‖2n′ ‖z′‖− d2 if ‖x′‖ ≤ n′.The proof of (6.1.2.1) is now an immediate consequence of the next lemma. Indeed, the nextlemma shows at once that E (Kz)  η(z)−(d−2) and that E (L2) ≤ cη(z)−(d−2). Therefore,E (K2z ) ≤ cη(z)−(d−2) ≤ c′E (Kz) , as wanted.Lemma ( 6.1.2.2 ) Suppose that ϕ and ψ are two bounded functions defined on Zd+1 andwith positive values. Furthermore, suppose that there exists two constants c1, c2 > 0 suchthatmax(ϕ(z), ψ(z)) ≤c1e−c2‖z‖ if ‖x‖ > n,c1e−c2 ‖x‖2n ‖z‖− d2 if ‖x‖ ≤ n.Then, there exists a universal constant c > 0 such that for any z ∈ Zd+1,∑z′∈Zd+1ϕ(z′ − z)ψ(z′) ≤ cη(z)−(d−2).119Chapter 6. Probability of belonging in the same componentLikewise, if there exists two constants c3, c4 > 0 such thatmin(ϕ(z), ψ(z)) ≥c3e−c4‖z‖ if ‖x‖ > n,c3e−c4 ‖x‖2n ‖z‖− d2 if ‖x‖ ≤ n,then there exists a universal constant c > 0 such that for any z ∈ Zd+1,∑z′∈Zd+1ϕ(z′ − z)ψ(z′) ≥ cη(z)−(d−2).Proof of Lemma. We divide the proof of the upper and lower bounds. We write z = (n, x),z′ = (n′, x′) and so on with other affixes (such as superscripts).I. Upper bound. There is no loss of generality to assume ϕ = ψ. Decompose∑z′∈Zd+1ϕ(z′ − z)ϕ(z′) = H + I + J,(D)whereH =∑n′≤0∑x′∈Zdϕ(z′ − z)ϕ(z′),I =∑1≤n′≤n−1∑x′∈Zdϕ(z′ − z)ϕ(z′),J =∑n′≥n∑x′∈Zdϕ(z′ − z)ϕ(z′)(E)We shall bound now each of the terms H, I, J and we will see, at the end, that the largestof these bounds is of the claimed form. We will use the bounds in the hypothesis and theestimates of §2.3 to handle the resulting sums.We start by bounding H. Since z ∈ Z+ × Zd and z′ ∈ Z− × Zd, we have the boundsϕ(z′ − z)ϕ(z′) ≤ ce−c′(‖z−z′‖+‖z′‖), for a pair of positive constants. Now,ce−c′(‖z−z′‖+‖z′‖) ≤ ce− c′2(‖z‖−‖z′‖)−c′‖z′‖ ≤ ce− c′2‖z‖e−c′2‖z′‖,summing over z′ ∈ Z− × Zd, we reach(F) H ≤ ce−c′‖z‖,for a pair of constants c, c′ > 0.We now handle I from (E). Divide into two parts as followsI =∑1≤n′≤n−1‖x′‖≤n′ϕ(z′ − z)ϕ(z′) + ∑1≤n′≤n−1‖x′‖>n′ϕ(z′ − z)ϕ(z′) def.= I1 + I2.120Chapter 6. Probability of belonging in the same componentThen,I1 ≤ c∑1≤n′≤n−1‖x′‖≤n′exp(−c′(‖z − z′‖+ ‖x′‖2n′))(n′)−d2Observe that‖z − z′‖2 = (n− n′)2 + ‖x− x′‖2 ≥ (|n− n′|+ ‖x− x′‖)22,thus‖z − z′‖ ≥ |n− n′|+ ‖x− x′‖√2.Hence,(G) I1 ≤ cn−1∑n′=1(n′)−d2 e− c′√2(n−n′) ∑‖x′‖≤n′exp(− c′√2(‖x− x′‖+ ‖x′‖2n′)).For convenience, we now divide into two cases. But first, by (2.3.4), we can bound∑‖x′‖≤n′exp(− c′√2(‖x− x′‖+ ‖x′‖2n′))≤ce−c′‖x‖ if ‖x‖ > n′ce−c′ ‖x‖2n′ if ‖x‖ ≤ n′.We may now proceed with the two cases.Case 1. Here we assume n ≤ ‖x‖ . Then, there exists two constants c, c′ > 0,I1 ≤ ce−c′‖x‖n−1∑n′=1(n′)−d2 e−c′(n−n′) ≤ c[ ∞∑k=1k−d2]e−c′‖x‖ ≤ Ce−C′‖z‖.Case 2. Here we assume ‖x‖ < n. Then, we divide the sum in the right of (G) into twoparts. The first part consists on all terms for which 1 ≤ n′ ≤ ‖x‖ and the second,‖x‖ < n′ ≤ n− 1. Then, for a pair of constants C,C ′ > 0,∑1≤n′≤‖x‖(n′)−d2 e− c′√2(n−n′) ∑‖x′‖≤n′e− c′√2Å‖x−x′‖+‖x′‖2n′ã≤ C ∑1≤n′≤‖x‖(n′)−d2 e− c′√2(n−n′)e−C′‖x‖.By setting c′′ = minÇc′√2, C ′å, it is clear thate− c′√2(n−n′)e−C′‖x‖ ≤ e−c′′(n−n′+‖x‖) ≤ e−c′′n,121Chapter 6. Probability of belonging in the same componentfor 1 ≤ n′ ≤ ‖x‖ . Then, the first part in the division of I1 is bounded above by ce−c′‖z‖,for a pair of constants c, c′ > 0. As for the second part,∑‖x‖<n′≤n−1(n′)−d2 e− c′√2(n−n′) ∑‖x′‖≤n′e− c′√2Å‖x−x′‖+‖x′‖2n′ã≤ C ∑‖x‖<n′≤n−1(n′)−d2 e− c′√2(n−n′)e−C′ ‖x‖2n′ ,where C,C ′ > 0 are two constants. We have shown up to this point that under theassumption ‖x‖ < n, there exists two constants c, c′ > 0, such that(H) I1 ≤ ce−c′‖z‖ + c∑‖x‖<n′≤n−1(n′)−d2 e−c′Än−n′+ ‖x‖2n′ä.We are going to consider two further cases for the sum appearing in the previous line.(a) Assume first ‖x‖ ≤ n 12 . Here, we will bound e−c ‖x‖2n′ ≤ 1. Then,∑‖x‖<n′≤n−1(n′)−d2 e−c′Än−n′+ ‖x‖2n′ä=∑‖x‖<n′≤n 12(n′)−d2 e−c′(n−n′)+∑n12<n′≤n(n′)−d2 e−c′(n−n′).In the first of these sums, e−c′(n−n′) ≤ e−c′Än−n 12ä≤ Ce−C′n12 , for two constantsC,C ′ > 0. We reach∑‖x‖<n′≤n 12(n′)−d2 e−c′(n−n′) ≤ C[ ∞∑k=1k−d2]e−C′n12 .Observe now that if n12 < n′ ≤ n, there exists a unique integer j between 1 andn12 such that jn12 < n′ ≤ (j + 1)n 12 (there are n 12 indices n′ in this range). Thus,∑n12<n′≤n(n′)−d2 e−c′(n−n′) ≤ n− d4n 12 ∑1≤j≤n 12j−d2 e−c′Än−minÄ(j+1)n12 ,nää= n−d4+ 12∑1≤j≤n 12j−d2 e−c′n 12Än12−minÄj+1,n12ää.122Chapter 6. Probability of belonging in the same componentIn the last sum, consider first the range 1 ≤ j ≤ n124to obtain j + 1 ≤ 2j ≤ n122,giving e−c′n 12Än12−minÄj+1,n12ää≤ e− c′2 n. Next, consider the range n124< j ≤ n 12 , sothat j  n 12 . Then,∑n12<n′≤n(n′)−d2 e−c′(n−n′) ≤[ ∞∑k=1k−d2]n−d4+ 12 e−c′2n + cn−d4+ 12n−d4n12 ≤ Cn− d2+1.The foregoing shows∑‖x‖≤n′<n−1(n′)−d2 e−c′Än−n′+ ‖x‖2n′ä≤ ce−c′n + Cn− d2+1 ≤ C ′′n− d2+1.(b) Now assume n12 < ‖x‖ . We bound the sum appearing in (H). We use (2.3.6),∑‖x‖<n′≤n−1(n′)−d2 e−c′Än−n′+ ‖x‖2n′ä≤ ∑n′≤‖x‖2(n′)−d2 e−c′ ‖x‖2n′ ≤ c ‖x‖−(d−2) .Bearing in mind the previous two items, and (H), we have established thatI1 ≤ cη(z)−(d−2),for a universal constant c > 0. This completes Case 2.We now need to bound I2. Again, by the bounds in the hypothesis,I2 =∑1≤n′≤n−1∑‖x′‖>n′ϕ(z′ − z)ϕ(z′) ≤ c21∑1≤n′≤n−1∑‖x′‖>n′e−c2(‖z−z′‖+‖z′‖) ≤ ce−c′‖z‖,where we proceeded as in (F). Thus, summing the upper bounds of I1 and I2 we have shownthat(I) I ≤ cη(z)−(d−2),for a universal constant c > 0, as desired.We now proceed to upper bound the sum J of (E). First we rewrite,J =∑n′≥0∑x′∈Zdϕ(0, z′)ϕ(0, z′ + z) = J1 + J2,123Chapter 6. Probability of belonging in the same componentwhereJ1 =∑n′≥0∑‖x′‖≤n′ϕ(0, z′)ϕ(0, z′ + z),J2 =∑n′≥0∑‖x′‖>n′ϕ(0, z′)ϕ(0, z′ + z).We further divide J1,J1 = J1,1 + J1,2,withJ1,1 =∑n′≥0∑‖x′‖≤n′‖x+x′‖≤n+n′ϕ(0, z′)ϕ(0, z′ + z)andJ1,2 =∑n′≥0∑‖x′‖≤n′‖x+x′‖>n+n′ϕ(0, z′)ϕ(0, z′ + z).Let us handle J1,1 first. In this case, we have the boundJ1,1 ≤ c21∑n′≥0(n′)−d2 (n+ n′)−d2∑‖x′‖≤n′‖x+x′‖≤n+n′e−c2Å‖x′‖2n′ +‖x+x′‖2n+n′ã.We finish the case ‖x‖ ≤ n 12 first. Here we may boundJ1,1 ≤ c21∑n′≥0(n′)−d2 (n+ n′)−d2∑‖x′‖≤n′e−c2‖x′‖2n′≤ c ∑n′≥0(n+ n′)−d2 ≤ c′n− d2+1,by (2.3.3) and (2.3.5) with all c, c′, c′′ positive constants. We may assume for the rest of thebounding of J1,1 that ‖x‖ > n 12 . By virtue of (2.3.7), in the range 0 ≤ n′ ≤ ‖x‖2, we maybound ∑‖x′‖≤n′‖x+x′‖≤n+n′e−c2Å‖x′‖2n′ +‖x+x′‖2n+n′ã≤ c(n′) d2 e−c ‖x‖2n+n′ ,with c a universal constant. Next,c∑0≤n′≤ ‖x‖2(n′)−d2 (n+ n′)−d2 (n′)d2 e−c′ ‖x‖2n+n′ = c∑0≤n′≤ ‖x‖2(n+ n′)−d2 e−c′ ‖x‖2n+n′= c∑n≤m≤n+ ‖x‖2m−d2 e−c′ ‖x‖2m124Chapter 6. Probability of belonging in the same componentwhere c and c′ are two positive universal constants. By the assumption ‖x‖ > n 12 , we have∑n≤m≤n+ ‖x‖2m−d2 e−c′ ‖x‖2m ≤ ∑m≤2‖x‖2m−d2 e−c′ ‖x‖2m ≤ c ‖x‖−(d−2) ,by (2.3.6). To sum up, we have established so far in the bounding of J1,1 (with the assumptionthat ‖x‖ > n 12 ),J1,1 ≤ cη(z)−(d−2) + c∑n′≥ ‖x‖2(n′)−d2 (n+ n′)−d2∑‖x′‖≤n′‖x+x′‖≤n+n′e−c2Å‖x′‖2n′ +‖x+x′‖2n+n′ãIf ‖x‖ ≤ 2n′, we obtain from (2.3.7)∑‖x′‖≤n′‖x+x′‖≤n+n′e−c2Å‖x′‖2n′ +‖x+x′‖2n+n′ã≤ cñ(n′)d2 e−c′ ‖x‖2n+n′ + (n+ n′)d2 e−c′ ‖x‖2n′ô.Substituting this result above,J1,1 ≤ cη(z)−(d−2) + c∑n′≥ ‖x‖2ñ(n+ n′)−d2 e−c′ ‖x‖2n+n′ + (n′)−d2 e−c′ ‖x‖2n′ô≤ cη(z)−(d−2) + 2c ∑n′> ‖x‖2(n′)−d2 e−c′ ‖x‖2n′ .Split the appearing sum in the line above into‖x‖2≤ n′ ≤ ‖x‖2 and n′ > ‖x‖2 ; boundthe first resulting sum by (2.3.6) to obtain an upper bound of the form c ‖x‖−(d−2) , and thesecond, by∑n′>‖x‖2(n′)−d2 e−c′ ‖x‖2n′ ≤ ∑n′>‖x‖2(n′)−d2  ‖x‖−(d−2) , the last  was obtained from(2.3.5). We have established,(J) J1,1 ≤ cη(z)−(d−2),c being a universal constant depending solely on dimension.We now will handle J1,2. The hypotheses giveJ1,2 ≤ c21∑n′≥0∑‖x′‖≤n′‖x+x′‖>n+n′(n′)−d2 e−c2Å‖z+z′‖+‖x′‖2n′ã≤ c21e−c2√2n ∑n′≥0(n′)−d2 e− c2√2n′ ∑‖x′‖≤n′e− c2√2Å‖x+x′‖+‖x′‖2n′ã.125Chapter 6. Probability of belonging in the same componentIt is now clear that we may bound J1,2 ≤ ce−c′n, with a pair of constants c, c′ > 0 dependingsolely on dimension. This establishes the case n12 ≥ ‖x‖ and thus, we may assume now‖x‖ > n 12 in what follows and bound e−c2√2n ≤ 1. Consider an integer 0 ≤ n′ < ‖x‖ , invoking(2.3.4) we reach ∑‖x′‖≤n′e− c2√2Å‖x+x′‖+‖x′‖2n′ã≤ ce−c′‖x‖,a fortioric21∑0≤n′<‖x‖(n′)−d2∑‖x′‖≤n′e− c2√2Å‖x+x′‖+‖x′‖2n′ã≤ ce−c′‖x‖,for universal constants c, c′ > 0. If n′ ≥ ‖x‖ , bound ‖x+ x′‖ ≥ 0 and use (2.3.3)∑‖x′‖≤n′e− c2√2Å‖x+x′‖+‖x′‖2n′ã≤ c(n′) d2 ,and then∑n′≥‖x‖(n′)−d2 e− c2√2n′ ∑‖x′‖≤n′e− c2√2Å‖x+x′‖+‖x′‖2n′ã≤ c ∑n′≥‖x‖e− c2√2n′  e−c2√2‖x‖.Thus, we may assert, there exists a constant c > 0 such that for all z = (n, x) satisfying‖x‖ > n 12 ,J1,2 ≤ cη(z)−(d−2);the same type of bound for ‖x‖ ≤ n 12 was established already. Bearing in mind these bounds,(J), and since J1 = J1,1 + J1,2, we have finally established(K) J1 ≤ cη(z)−(d−2),with c > 0 is a constant that does not depend on z.We will now prove the upper bound for J2. Similarly as was done with J1, we splitJ2 = J2,1 + J2,2, beingJ2,1 =∑n′≥0∑‖x′‖>n′‖x+x′‖≤n+n′ϕ(z′)ϕ(z′ + z)andJ2,2 =∑n′≥0∑‖x′‖>n′‖x+x′‖>n+n′ϕ(z′)ϕ(z′ + z).126Chapter 6. Probability of belonging in the same componentHere, J2,2 is easy for we have for we may repeat the calculations of (F) and obtain(L) J2,2 ≤ c21∑z′∈Zd+1e−c2(‖z′‖+‖z+z′‖) ≤ ce−c′‖z‖,with c1, c2 the constants in the hypothesis and c, c′ two positive universal constants. Thereremains to handle J2,1. Here we may boundJ2,1 ≤ c21∑n′≥0∑‖x+x′‖≤n+n′e−c2Å‖z′‖+‖x+x′‖2n+n′ã(n+ n′)−d2≤ c21∑n′≥0(n+ n′)−d2 e− c2√2n′ ∑‖x+x′‖≤n+n′e− c2√2Å‖x′‖+‖x+x′‖2n+n′ã= c21∑n′≥0(n+ n′)−d2 e− c2√2n′ ∑‖x′‖≤n+n′e− c2√2Å‖x′−x‖+‖x′‖2n+n′ã.Suppose first that ‖x‖ ≤ n 12 . The bounds of (2.3.4) give∑‖x′‖≤n+n′e− c2√2Å‖x′−x‖+‖x′‖2n+n′ã≤ ce−c′ ‖x‖2n+n′ ≤ c,for a pair of constants c, c′ > 0. Then,J2,1 ≤ c∑n′≥0(n+ n′)−d2 ≤ c′n− d2+1 = c′η(z)−(d−2),where the constants c, c′ are positive and depending on nothing but, perhaps, dimension. Thiswould finish the bound for J2,1 under the assumption ‖x‖ ≤ n 12 . Next, assume n 12 < ‖x‖ .Here, we further split (we remind the reader the sums are over n′ ∈ Z+ and x′ ∈ Zd asstated)J2,1 ≤ c21∑0≤n′+n≤‖x‖(n+ n′)−d2 e− c2√2n′ ∑‖x′‖≤n+n′e− c2√2Å‖x′−x‖+‖x′‖2n+n′ã+ c21∑n′+n>‖x‖(n+ n′)−d2 e− c2√2n′ ∑‖x′‖≤n+n′e− c2√2Å‖x′−x‖+‖x′‖2n+n′ã.If ‖x‖ ≥ n′ + n then the inner sum is bounded by an expression of the form ce−c′‖x‖, againby (2.3.4). This yieldsc21∑0≤n′+n≤‖x‖(n+ n′)−d2 e− c2√2n′ ∑‖x′‖≤n+n′e− c2√2Å‖x′−x‖+‖x′‖2n+n′ã≤ ce−c′‖x‖ ≤ Cη(z)−(d−2),127Chapter 6. Probability of belonging in the same componentwith c, c′, C > 0 universal constants. If ‖x‖ < n + n′, then the inner sum is bounded byan expression of the form ce−c′ ‖x‖2n+n′ , by the same point. Thus, there exist two constants c, c′such that for all z,c21∑n′+n>‖x‖(n+ n′)−d2 e− c2√2n′ ∑‖x′‖≤n+n′e− c2√2Å‖x′−x‖+‖x′‖2n+n′ã≤ c ∑n′+n>‖x‖(n+ n′)−d2 e−c′Än′+ ‖x‖2n+n′ä.We may now use (2.3.5) and (2.3.6) as follows:∑n′+n>‖x‖(n+ n′)−d2 e−c′Än′+ ‖x‖2n+n′ä≤ ∑n′+n≤‖x‖2(n+ n′)−d2 e−c′ ‖x‖2n+n′ +∑n′+n>‖x‖2(n+ n′)−d2≤ c ‖x‖−(d−2) + c′ ‖x‖−(d−2) = (c+ c′) ‖x‖−(d−2) .Thus, we showed that there exists a constant c > 0 such that for all z ∈ Zd+1,(M) J2,1 ≤ cη(z)−(d−2).Then, by (L) and (M), we reachJ2 ≤ cη(z)−(d−2),with c not depending on z. Furthermore, the previous bound on J2 together with (K) showthat there exists a c > 0 such that for all vertices z in Γd(λ),(N) J ≤ cη(z)−(d−2).Combining the results of (F), (I) and (N), we finally reach that∑z′∈Zd+1ϕ(z′ − z)ϕ(z′) ≤ cη(z)− d2+1.The upper bound in the lemma has been substantiated.II. Lower bound. Assume the hypothesis for the lower bound. Then∑z′∈Zd+1ϕ(z′ − z)ψ(z′) ≥ ∑n′>max(n,‖x‖2)∑‖x′−x‖2≤n′−nϕ(z′ − z)ψ(z′).128Chapter 6. Probability of belonging in the same componentThe two relations ‖x‖2 < n′ and ‖x′ − x‖2 ≤ n′−n imply ‖x′‖2 ≤ 2 ‖x‖2 +2 ‖x′ − x‖2 ≤ 4n′.Then, the bounds of the hypothesis give (the last  follows from (2.3.5))∑n′>max(n,‖x‖2)∑‖x′−x‖2≤n′−nϕ(z′ − z)ψ(z′)≥ c ∑n′>max(n,‖x‖2)(n′)−d2 (n′ − n)− d2 ∑‖x′−x‖2≤n′−ne−c′ ‖x′‖2n′≥ c ∑n′>max(n,‖x‖2)(n′)−d2 (n′ − n)− d2 ∑‖x′−x‖2≤n′−ne−4c′≥ c′′ ∑n′>max(n,‖x‖2)(n′)−d2  max Än, ‖x‖2ä− d2+1 .The lower bounds have been substantiated keeping in mind (A).With the two lemmas proved, we are in conditions to finish the theorem. By virtue of(A), (B), (C) and (6.1.2.1) we have thatUSF(C (0, z))  E (Kz) =∑z′∈Zd+1G(z, z′)G(0, z′).Applying (6.1.2.2) (bear in mind the Green’s function bounds (2.6.1)), we reach thatUSF(C (0, z))  η(z)−(d−2)as desired.§ 6.2. An upper bound on the probability that finitelymany points are USF-connected.We are going to generalise the previous result to any subset of vertices. We will follow closely[BKPS04]. First, we introduce the definition of spread of a set of vertices. For any vertexz ∈ Zd+1 and any finite subset W ⊂ Zd+1 (with two or more vertices) we define their spreads(relative to the metric η) as〈z〉η = max(1, η(z))〈W〉η = minE∏{z,z′}∈E〈z − z′〉η ,(6.2.1)129Chapter 6. Probability of belonging in the same componentwhere the minimum runs over all E ⊂ P(W) making (W,E) an undirected spanning tree.If W consists of two (different) vertices α and β, then 〈W〉η = η(α − β); in this case wewill write 〈αβ〉η in lieu of 〈W〉η . Observe that (4.3.1) defines the spread of a point relativeto the Euclidean metric. For simplicity, if W = {α, β, . . . , ξ}, we will write 〈αβ · · · ξ〉η ,in particular, 〈0z〉η = 〈z〉η . Observe that [BKPS04] also defined the concept of “spread ofsubset of vertices of Zd” but they use the Euclidean metric (see their Definition 2.1). WhenW = {α, β, γ}, then〈W〉η = min{〈αβ〉η 〈αγ〉η , 〈αβ〉η 〈βγ〉η , 〈αγ〉η 〈βγ〉η}.Proposition ( 6.2.2 ) [BKPS04, Lemma 2.6] For every L > 0 there exists a constant c > 0such that for every z ∈ Zd+1, and every W ⊂ Zd+1 with card (W) ≤ L, we have〈W ∪ {z}〉η ≤ 〈W〉η minw∈W 〈wz〉η ≤ c 〈W ∪ {z}〉η .Proof. The proof of this proposition proceeds word by word like that of Lemma 2.6 in[BKPS04].We may now generalise theorem (6.1.2) to any finite subset of vertices. We will onlyprove an upper bound.Theorem ( 6.2.3 ) Let d ≥ 3. For any W ⊂ Zd+1, denote by CW the event that all the pointsin W are in the same USF component. For every integer L > 0, there exists a constant c > 0,depending only on L and dimension, such that for whatever subset W with at most L vertices,USF(CW) ≤ c 〈W〉−(d−2)η .Proof. We apply the same reasoning as that used in the proof of Theorem 4.3 of [BKPS04].We will apply induction on the cardinality of W. The case of two points is precisely theorem(6.1.2). Assume then the result holds for card (W)−1 ≥ 2. Let z1, z2 be two different verticesof W and denote CW(z1, z2) the event which is the intersection of CW with the event wherethe USF-path connecting z1 and z2 is edge-disjoint from the USF-rays starting at the verticesof V = W \ {z1, z2}. Let C zW(z1, z2) denote the event which is the intersection of CW(z1, z2)and the event where the ray at z1 meets the ray at z2 at the vertex z. ThenUSF(CW(z1, z2)) ≤∑z∈Zd+1USF(C zW(z1, z2)).To calculate the probability of USF(C zW(z1, z2)) we may assume that we construct the USF-forest F following Wilson’s algorithm rooted at infinity by first ordering all the vertices in130Chapter 6. Probability of belonging in the same componentVzdef.= V ∪ {z}, followed by the vertices z1 and z2, and then the rest of the vertices of Zd+1.We see that{F ∈ C zW(z1, z2)} ⊂ {F ∈ CVz} ∩ {τz (Sz1) <∞} ∩ {τz (Sz2) <∞} ,and {F ∈ CVz} is an event depending solely in the random walks Sv for v ∈ Vz. By inde-pendence and induction,USF(C zW(z1, z2)) = P (F ∈ C zW(z1, z2))≤ P (F ∈ CVz)P (τz (Sz1) <∞)P (τz (Sz2) <∞)= USF(CVz)G(z1, z)G(z2, z)≤ c 〈Vz〉−(d−2)η G(z1, z)G(z2, z)≤ c′ 〈V〉−(d−2)η∑v∈V〈vz〉−(d−2)η G(z1, z)G(z2, z),the last inequality by virtue of (6.2.2). Thus,(∗) USF(CW(z1, z2)) ≤ c 〈V〉−(d−2)η∑v∈V∑z∈Zd+1〈vz〉−(d−2)η G(z1, z)G(z2, z).We isolate some calculations first.( 6.2.3.1 ) For any a ∈ R and r > 2, we have ∑〈z〉η≤r〈z〉aη  1 + rd+2+a, the implicit constantsdepending only on a and dimension.Proof of (6.2.3.1). First notice there exists an integer k ≥ 1 such that 2k−1 ≤ r < 2k. Then,∑〈z〉η≤r〈z〉aη ≤∑〈z〉η≤2k〈z〉aη ≤k∑j=1∑2j−1≤〈z〉η≤2j〈z〉−aη ≤ ck∑j=1card (Bj) 2ja,where Bj = {z; 〈z〉η ≤ 2j}. We know card (Bj)  2j(d+2), this implies∑〈z〉η≤r〈z〉−(d−2)η ≤ ck∑j=12j(d+2)2ja = ck∑j=12j(d+2+a)  2(d+2+a)k  rd+2+a.The lower bound is established similarly (please notice that 〈0〉η = 1.( 6.2.3.2 ) There exists a universal constant c > 0 such that for any two vertices z1, z2 ∈ Zd+1,we have ∑z∈Zd+1〈z〉−(d−2)η G(z1, z)G(z1, z) ≤ c 〈0z1z2〉−(d−2)η .131Chapter 6. Probability of belonging in the same componentProof of (6.2.3.2). By symmetry, we may assume 〈z1〉η ≤ 〈z2〉η . Write zi = (ni, xi), fori = 1, 2. Then,∑z∈Zd+1〈z〉−(d−2)η G(z1, z)G(z2, z) =∑〈z〉η> 12 〈z1〉η〈z〉−(d−2)η G(z1, z)G(z2, z) + I,and we may bound easily∑〈z〉η> 12 〈z1〉η〈z〉−(d−2)η G(z1, z)G(z2, z) ≤ 〈z1〉−(d−2)η∑z∈Zd+1G(z1, z)G(z2, z)≤ c 〈z1〉−(d−2)η 〈z1 − z2〉−(d−2)η ≤ c 〈0z1z2〉−(d−2)η ,where the penultimate inequality is obtained from (6.1.2.2) and the last inequality by defi-nition of the spread. The rest of the calculation is to bound I; this will be done in severalcases. By definitionI =∑〈z〉η≤ 12 〈z1〉η〈z〉−(d−2)η G(z1, z)G(z2, z).We will handle four cases, we will make use of the Green’s function estimates (2.6.1). Noticethat G(z′, z′′) ≤ c ‖z′ − z′′‖− d2 in all cases and that 〈x〉 ≤ 〈z〉η for every z ∈ Zd+1 (the lefthand side is the spread relative to the Euclidean metric (4.3.1)).Case 1. Here we assume that 〈z1〉η = |n1|12 and 〈z2〉η = |n2|12 . In this case, 〈z〉η ≤12〈z1〉ηimplies that |n| ≤ 14|n1| ≤ 14|n2| and ‖x− x1‖ ≤ ‖x‖ + ‖x1‖ ≤ 32‖x1‖ ≤ 32|n1| 12 , andsimilarly ‖x− x2‖ ≤ 32|n2| 12 , obtainingI ≤ c|n1|− d2 |n2|− d2∑〈z〉η≤ 12 〈z1〉η〈z〉−(d−2)ηWe may apply (6.2.3.1) to reachI ≤ c 〈z1〉−dη 〈z2〉−dη 〈z1〉4η ≤ c 〈z1〉−(d−2)η 〈z2〉−(d−2)η ≤ c 〈0z1z2〉−(d−2)η .This finishes Case 1.Case 2. Here we assume 〈z1〉η = |n1|12 and 〈z2〉η = ‖x2‖ . Here we have G(z1, z) ≤ c|n1|−d2 .We may split I into the sum over n and over x. This givesI ≤ c|n1|− d2∑‖x‖≤ 12|n1|12〈x〉−(d−2) ∑|n|≤ |n1|4G(z2, z).132Chapter 6. Probability of belonging in the same componentFor every ‖x‖ ≤ 12|n1| 12 , we have ‖x− x2‖ ≤ ‖x‖ + ‖x2‖ ≤ 32‖x2‖ and ‖x− x2‖ ≥‖x2‖ − ‖x‖ ≥ 12‖x2‖ . Next, break the inner sum into two parts, the first one beingover all n such that |n − n2| ≤ 12‖x2‖ and the second one being over all other n. Inthe first part, we have a bound of the form G(z2, z) ≤ ce−c′‖x2‖, thus∑|n|≤ |n1|4|n−n2|≤ ‖x2‖2G(z2, z) ≤ c ‖x2‖2 e−c′‖x2‖ ≤ c′′ ‖x2‖−(d−2) .We focus now in the second part of this inner sum. Here, we have12‖x2‖ ≤ |n− n2| ≤ |n|+ |n2| ≤ 54‖x2‖2 .Thus, we can bound G(z2, z) ≤ ce−c′‖x2‖2k k−d2 for some 1 ≤ k ≤ 54‖x2‖2 . Then, we maybound the second part of the inner sum of I as follows∑|n|≤ |n1|4|n−n2|> ‖x2‖2G(z2, z) ≤ c∑1≤k≤ 54‖x2‖2e−c′ ‖x2‖2k k−d2 .By virtue of (2.3.6), we have∑1≤k≤ 54‖x2‖2e−c′ ‖x2‖2k k−d2 ≤ c ‖x2‖2(−d2+1) = c ‖x2‖−(d−2) .Therefore, we may bound I as followsI ≤ c|n1|− d2∑‖x‖≤ 12|n1|12〈x〉−(d−2) ‖x2‖−(d−2) .Using (2.3.1), we have∑‖x‖≤ 12|n1|12〈x〉−(d−2) ≤ 1 + c12|n1|12∫1dt t−(d−2)td−1 = 1 + c12|n1|12∫1dt t ≤ c′|n1|.Substituting above yieldsI ≤ c|n1|− d2+1 ‖x2‖−(d−2) = c 〈z1〉−(d−2)η 〈z2〉−(d−2)η ≤ c 〈0z1z2〉−(d−2)η .This completed Case 2.133Chapter 6. Probability of belonging in the same componentCase 3. Here we assume 〈z1〉η = ‖x1‖ and 〈z2〉η = |n2|12 . Then |n − n2|  |n2| since|n| ≤ ‖x1‖2≤ |n2|2and we may bound G(z, z2) ≤ c ‖z − z2‖−d2 ≤ c′|n2|− d2 . This givesI ≤ c|n2|− d2∑〈z〉η≤ 12 〈z1〉η〈z〉−(d−2)η G(z1, z).We may now employ the same split of this sum as in Case 2, deducingI ≤ c|n2|− d2∑‖x‖≤ 12‖x1‖〈x〉−(d−2) ∑|n|≤ 14‖x1‖2G(z1, z).We may break again the inner sum over those n for which |n− n1| ≤ 12‖x1‖ and overall other n, and we will find that∑|n|≤ 12‖x1‖G(z1, z) ≤ c∑|n|≤ 12‖x1‖|n−n1|≤ 12‖x1‖e−c′‖x1‖ +∑1≤k≤ 54‖x1‖2e−c′ ‖x1‖2k k−d2 .And again, we apply (2.3.1) and (2.3.6), obtaining∑‖x‖≤ 12‖x1‖〈x〉−(d−2) ≤ c ‖x1‖2∑1≤k≤ 54‖x1‖2e−c′ ‖x1‖2k k−d2 ≤ c ‖x1‖2(−d2+1) = c ‖x1‖−(d−2) .It is also clear that ‖x1‖ e−c′‖x1‖ ≤ c ‖x1‖−(d−2) . Substituting this above yieldsI ≤ c|n2|− d2 ‖x1‖2 ‖x1‖−(d−2) .Since we assume 〈z1〉η ≤ 〈z2〉η , we have ‖x1‖2 ≤ |n2|, so that|n2|− d2 ‖x1‖2 ≤ |n2|− d−22 = 〈z2〉−(d−2)η .The proof of Case 3 is finished since 〈z1〉η = ‖x1‖ and the definition of spread showsat once 〈z1〉−(d−2)η 〈z2〉−(d−2)η ≤ 〈0z1z2〉−(d−2)η .Case 4. Here we assume 〈z1〉η = ‖x1‖ and 〈z2〉η = ‖x2‖ . Consider first the set of z ∈ Zd+1such that 〈z〉η ≤12〈z1〉η and |n − n1| ≤ ‖x1‖ . For z in this set, G(z1, z) ≤ ce−c′‖x1‖,this entails∑〈z〉η≤ 12‖x1‖|n−n1|≤‖x1‖〈z〉−(d−2)η G(z1, z)G(z2, z) ≤ ce−c′‖x1‖ ∑‖x‖≤ 12‖x1‖〈x〉−(d−2) ∑|n|≤ 14‖x1‖2G(z2, z)134Chapter 6. Probability of belonging in the same componentRecall we assume ‖x1‖ ≤ ‖x2‖ , and, as done in Case 2 and Case 3∑|n|≤ 14‖x2‖2G(z2, z) ≤ c ‖x2‖−(d−2) .Therefore,∑〈z〉η≤ 12‖z1‖|n−n1|≤‖x1‖〈z〉−(d−2)η G(z1, z)G(z2, z) ≤ c ‖x1‖2 e−c′‖x1‖ ‖x2‖−(d−2)≤ c′′ ‖x1‖−(d−2) ‖x2‖−(d−2) ≤ c′′ 〈0z1z2〉η .There remains to handle the terms defining I for which 〈z〉η ≤12‖x1‖ and |n− n1| >‖x1‖ . Let J be the least integer for which 12‖x1‖ ≤ 2J ‖x1‖12 . In other words, J is theleast integer for which ‖x1‖ ≤ 4J+1. In particular, 4J < ‖x1‖ ≤ 4J+1. Next, observethat the relation 〈z〉η ≤12‖x1‖ implies |n − n1| ≤ |n| + |n1| ≤ 54‖x1‖2 ≤ 5 · 4J ‖x1‖ .DefineAj =®z; 4j−1 ‖x1‖ < |n− n1| ≤ 4j ‖x1‖ , 〈z〉η ≤12‖x1‖´, 1 ≤ j ≤ J − 1AJ =®z; 4J−1 ‖x1‖ < |n− n1| ≤ 5 · 4J ‖x1‖ , 〈z〉η ≤12‖x1‖´.We can writeI ≤ c 〈0z1z2〉−(d−2)η +J∑j=1∑z∈Aj〈z〉−(d−2)η G(z1, z)G(z2, z).If z ∈ Aj, then G(z1, z) ≤ ce−c′‖x1‖4j (4j ‖x1‖)− d2 , and since ‖x1‖  4J , the relationz ∈ Aj implies G(z1, z) ≤ c ‖x1‖−d2 e−c′4J−j4−jd2 . Substituting in the right hand sideabove,J∑j=1∑z∈Aj〈z〉−(d−2)η G(z1, z)G(z2, z) ≤ c ‖x1‖−d2J∑j=1e−c′4J−j4−jd2∑z∈Aj〈z〉−(d−2)η G(z2, z).We may break the inner sum in terms of n and x (for details, see Case 2 and Case3) and this will give∑z∈Aj〈z〉−(d−2)η G(z2, z) ≤∑‖x‖≤ 12‖x1‖〈x〉−(d−2) ∑|n|≤ 14‖x1‖2G(z2, z) ≤ c ‖x1‖2 ‖x2‖−(d−2) .135Chapter 6. Probability of belonging in the same componentWhence,J∑j=1∑z∈Aj〈z〉−(d−2)η G(z1, z)G(z2, z) ≤ c ‖x1‖−d2 ‖x1‖2 ‖x2‖−(d−2)J∑j=1e−c′4J−j4−jd2 .Finally, in the inner sum reverse the order of summation,J∑j=1e−c′4J−j4−jd2 =J−1∑k=0e−c′4k4(k−J)d2 ≤ 4−J d2∞∑k=04kd2 e−c′4k  ‖x1‖−d2 .Thus,J∑j=1∑z∈Aj〈z〉−(d−2)η G(z1, z)G(z2, z) ≤ c ‖x1‖−d2 ‖x1‖2 ‖x2‖−(d−2) ‖x1‖−d2= c ‖x1‖−(d−2) ‖x2‖−(d−2)= c 〈z1〉−(d−2)η 〈z2〉−(d−2)η ≤ c 〈0z1z2〉−(d−2)η .This completes Case 4.With the four cases established, we have reached the conclusion of (6.2.3.2).A corollary of (6.2.3.2) is that for any vertex v ∈ Zd+1,∑z∈Zd+1〈vz〉−(d−2)η G(z1, z)G(z2, z) ≤ c 〈{0, z1 − v, z2 − v}〉−(d−2)η = c 〈vz1z2〉−(d−2)η .Substituting this inequality in (∗) allows deducing, with the aid of (6.2.2),USF(CW(z1, z2)) ≤ c 〈V〉−(d−2)η∑v∈V〈vz1z2〉−(d−2)η ≤ c′ 〈W〉−(d−2)η ,where the last inequality follows form the fact that the union of a tree on {v, z1, z2} and atree on V gives a tree on W = V ∪ {z1, z2}. We will show now thatCW =⋃z1 6=z2CW(z1, z2),where the vertices z1 and z2 run on W. For each vertex w ∈W, denote by Rw the USF-rayfrom w. For every z1 6= z2, two vertices of W, their corresponding rays Rz1 and Rz2 will meetat a vertex v(z1, z2). Furthermore, on the event CW, the set of vertices belonging to all Rw,for w ∈ W, is non empty and thus, there exists a vertex z∗ satisfying the following: z∗ isthe closest vertex in the graph-distance of the USF-tree, amongst all vertices belonging to allrays Rw (w ∈W), to each one of the vertices w ∈W. In other words, z∗ is the vertex whereall the rays meet. Consider now a pair z1 6= z2 such that the graph-distance of v(z1, z2) andz∗ is maximal. Then, CW(z1, z2) occurs. The proof of theorem (6.2.3) is complete.136Chapter 7The separation between components:transitions in dimensions d = 2, 6, 10, ...Given a vertex z of Γd(λ), denote by Tz the component of z in the USF-forest, regarded asspanning subgraph, see Annex B. Construct now Γd(λ)/USF from Γd(λ) by shorting all edgesof all Tz as z runs on the vertices of Γd(λ). Therefore, Tz is a vertex of the new networkΓd(λ)/USF. Define D : Zd → Z+ byD(z, z′) = dΓd(λ)/USF (Tz,Tz′) ,the right hand side being the graph-distance of Γd(λ)/USF. We call this the separationbetween the components of z and z′ relative to the chemical distance of the graph Zd+1. Ifγ is an optimal-length path between z and z′ in Γd(λ) then we induce canonically a path γ˙on Γd(λ)/USF connecting Tz and Tz′ by simply deleting the edges in γ that will be shortedin the construction of Γd(λ)/USF; in this case, length (γ) = length (γ˙) + m, where m is thenumber of USF-edges used in γ. Therefore, another way of describing D(z, z′) is as follow:it is the minimal number of edges of Γd(λ) that do not belong to any USF-tree in a pathconnecting z and z′. In this chapter we are going to show that, for almost every realisation,maxz,z′∈Zd+1D(z, z′) =¢d− 24•.This chapter is substantially based on [BKPS04].§ 7.1. Random relations and stochastic dimension.We will be considering relations between vertices of Γd(λ). If R is one such relation, we willuse the common notation z1Rz2 to mean (z1, z2) ∈ R. Also, the composition of two relations137Chapter 7. The separation between components: transitions in dimensionsd = 2, 6, 10, ...L and R is, by definition, the set of pairs (z1, z2) such that there exists a z satisfying z1Lzand zRz2. We write LR to denote the composition of L and R (in this order). We will saythat a random relation R on the vertices of Γd(λ) has stochastic dimension α ∈ [0, d + 1),relative to the metric η (6.1.1), and write dimη(R) = α if there exists a constant c = c(R) > 0such that for all vertices z1, z2, z3, z4 of Γd(λ)cP (z1Rz2) ≥ 〈z1z2〉−(d+1−α)η ,andP (z1Rz2, z3Rz4) ≤ c 〈z1z2〉−(d+1−α)η 〈z3z4〉−(d+1−α)η + c 〈z1z2z3z4〉−(d+1−α)η .Remark ( 7.1.1 ) That a random relation R has stochastic dimension d+ 1 is the same assaying that infz1,z2P (z1Rz2) > 0.Remark ( 7.1.2 ) Let d ≥ 3. Denote by UΓd(λ) the random relation: “z1 and z2 are in thesame USF-component.” Then, UΓd(λ) has stochastic dimension three, relative to η, this is aconsequence of the definition and of theorems (6.1.2) and (6.2.3). Indeed, write z ∼ z′ tomean zUΓd(λ)z′. Then, (6.1.2) shows that cP (z1 ∼ z2) ≥ 〈z1z2〉−(d+1−3)η . Next, observe thatP (z1 ∼ z2, z3 ∼ z4) ≤ P (z1 ∼ z2, z3 ∼ z4, z1 ∼ z3) + P (z1 ∼ z2, z3 ∼ z4, z1 6∼ z3) .The event {z1 ∼ z2, z3 ∼ z4, z1 ∼ z3} is the event where all the four vertices are in thesame USF-tree, this the probability of this event is bounded above by a universal multipleof 〈z1z2z3z4〉−(d+1−3)η by (6.2.3). To estimate the probability of the event {z1 ∼ z2, z3 ∼z4, z1 6∼ z3} we may construct the USF-forest F using Wilson’s algorithm rooted at infinity,see sect. §1.7, with ordering of vertices (z1, z2, z3, z4, . . .). If Sz denotes the random walk inthis construction of F, thenP (z1 ∼ z2, z3 ∼ z4, z1 6∼ z3) ≤ P (Sz1 ∩ Sz2 6= ∅, Sz3 ∩ Sz4 6= ∅)≤ P (Sz1 ∩ Sz2 6= ∅)P (Sz3 ∩ Sz4 6= ∅)≤ c 〈z1z2〉−(d+1−3)η 〈z3z4〉−(d+1−3)η ,the last inequality by virtue of lemmas (6.1.2.1) and (6.1.2.2).Remark ( 7.1.3 ) We are going to base the results of this chapter on those of the paper[BKPS04], making the necessary adaptations according to our conveniences. We will seethat, in a sense, the stochastic dimension relative to η is the same the stochastic dimensionthey defined (that is, relative to the Euclidean distance) minus one.138Chapter 7. The separation between components: transitions in dimensionsd = 2, 6, 10, ...We introduce the following annuli: for n < N two integers and z ∈ Zd+1,ANn (z) = {z′ ∈ Zd+1 | 2n ≤ 〈z − z′〉η < 2N}.Notice cardÄANn (z)ä  2N(d+2).Proposition ( 7.1.4 ) ([BKPS04, Lemma 2.8]) Let L and R be two independent randomrelations of Γd(λ). Suppose that dimη(L) = d + 1 − α and dimη(R) = d + 1 − β, both exist.Denote γ = α + β − d− 2. For z1, z2 ∈ Zd+1, and 1 ≤ n ≤ N, letSz1,z2 = Sz1,z2(n,N)def.=∑z∈ANn (z1)1{z1Lz}1{zRz2}.If 〈z1z2〉η < 2n−1 and N ≥ n,P (Sz1,z2 > 0) ≥ cN∑k=n2−kγN∑k=02−kγ,where c is a constant that may depend solely on L, R and dimension.Proof. We may repeat word by word, with very minor notational modifications, the proofof Lemma 2.8 of [BKPS04] bearing in mind that the only substantial thing that changes iscardÄAk+1k (z1)ä  2k(d+2). Also keep in mind that their Lemma 2.6 is our Proposition (6.2.2).Thus, their formula (2.6) changes toE (Sz1,z2) N∑k=n2k(d+2)2−kα2−kβ =N∑k=n2−kγ,the rest of the proof goes on without major modification except that we need to replace dby d+ 2 in their formula (2.7).Proposition ( 7.1.5 ) ([BKPS04, Corollary 2.9]) Let L and R be two independent randomrelations of Γd(λ). Suppose that dimη(L) = d + 1 − α and dimη(R) = d + 1 − β, both exist.Denote γ = max(0, α+β−d−2). There exists a constant c > 0 such that for all z1, z2 ∈ Zd+1,P (z1LRz2) ≥ c 〈z1z2〉−γη .Proof. The proof of Corollary 2.9 of [BKPS04] applies almost verbatim: in their secondsentence replace “lemma” with “Proposition (7.1.4).”139Chapter 7. The separation between components: transitions in dimensionsd = 2, 6, 10, ...Proposition ( 7.1.6 ) ([BKPS04, Lemma 2.10]) Let α, β ∈ [0, d+ 1) satisfy α+ β > d+ 2.Let γ = α + β − d− 2. Then, ∑z∈Zd+1〈z1z〉−αη 〈z2z〉−βη  〈z1z2〉−γηfor all vertices z1, z2 of Zd+1, and with any implicit constant being universal.Proof. The proof of Lemma 2.10 of [BKPS04] applies without changes except to notation.Proposition ( 7.1.7 ) ([BKPS04, Lemma 2.11]) Let M be a positive integer and let α, β ∈[0, d+ 1) satisfy α+β > d+ 2. Denote γ = α+β− d− 2. There exists a constant c > 0 suchthat for all subsets V and W of Zd+1 with at most M points, we have∑z∈Zd+1〈V ∪ {z}〉−αη 〈W ∪ {z}〉−βη ≤ c 〈V〉−αηÇmin(v,w)∈V×W〈vw〉ηå−γ〈W〉−βη ≤ c 〈V ∪W〉−γη .Proof. We may use the same proof as that of Lemma 2.11 of [BKPS04] keeping in mind thattheir Lemma 2.6 is our Proposition (6.2.2) and, Lemma 2.10 is Proposition (7.1.6).Proposition ( 7.1.8 ) ([BKPS04, Lemma 2.12]) Let α, β ∈ [0, d+ 1) satisfy α+ β > d+ 2.Set γ = α + β − d− 2.(a) There exists a constant c > 0 such that for all vertices z1, z2, z3 ∈ Zd+1,∑z∈Zd+1〈z1z〉−αη 〈z2z〉−βη 〈z3z〉−γη ≤ c 〈z1z2z3〉−γη(b) There exists a constant c > 0 such that for all vertices z1, z2, z3, z4 ∈ Zd+1,∑z∈Zd+1〈z1z2z〉−αη 〈z3z〉−βη 〈z4z〉−γη ≤ c 〈z1z2z3z4〉−γηProof. The proof is essentially the same as that of Lemma 2.12 of [BKPS04] and their formula(2.9), except we need to use (6.2.3.1) in some step. For convenience of the reader, we repeatthe proof.We begin by establishing the first item. By changing the indices should the need arise,we may assume that 〈z1z3〉η ≤ 〈z2z3〉η . Denote by A the set of z ∈ Zd+1 satisfying 〈z3z〉η ≤12〈z1z3〉η . When z ∈ A we have, 〈z1z〉η ≥ 〈z1z3〉η − 〈z3z〉η ≥12〈z1z3〉η and, similarly, since〈z1z3〉η ≤ 〈z2z3〉η , we also have 〈z2z〉η ≥12〈z2z3〉η . Therefore,∑z∈A〈z1z〉−αη 〈z2z〉−βη 〈z3z〉−γη ≤ 2α 〈z1z3〉−αη 2β 〈z2z3〉−βη∑z∈A〈z3z〉−γη .140Chapter 7. The separation between components: transitions in dimensionsd = 2, 6, 10, ...Apply (6.2.3.1) to obtain the bound∑z∈A〈z3z〉−γη ≤ c 〈z1z3〉d+2−γη . Substitute above,∑z∈A〈z1z〉−αη 〈z2z〉−βη 〈z3z〉−γη ≤ c 〈z1z3〉−αη 〈z2z3〉−βη 〈z1z3〉d+2−γη= c 〈z1z3〉−γη 〈z2z3〉−γη〈z1z3〉d+2−αη〈z2z3〉d+2−αη≤ c 〈z1z2z3〉−γη ,because γ > 0 and 〈z1z3〉η ≤ 〈z2z3〉η . We now estimate the sum over the complement of A.Here we apply (7.1.6), by definition of A, the relation z /∈ A implies 〈z3z〉−γη ≤ 2γ 〈z1z3〉−γη .Thus,∑z /∈A〈z1z〉−αη 〈z2z〉−βη 〈z3z〉−γη ≤ 2γ 〈z1z3〉−γη∑z∈Zd+1〈z1z〉−αη 〈z2z〉−βη≤ c 〈z1z3〉−γη 〈z1z2〉−γη ≤ c 〈z1z2z3〉−γη ,by virtue of (7.1.6). Combining these two estimates we obtain the first item of the proposi-tion.We now continue with the proof of the second item. By (6.2.2), we obtain〈z1z2z〉η ≥ c 〈z1z2〉η min{〈z1z〉η , 〈z2z〉η}.Use this, the first item of the proposition, and recall that 〈z1z2〉η 〈z2z3z4〉η ≥ 〈z1z2z3z4〉ηwhich holds by definition of the spread.Theorem ( 7.1.9 ) ([BKPS04, Theorem 2.4]) Let L and R be two independent randomrelations on the vertices of Γd(λ). Assume that their stochastic dimensions (relative to η)exist. Then, their composition LR also has stochastic dimension relative to η and it satisfiesdimη(LR) = min (dimη(L) + dimη(R) + 1, d+ 1) .Proof. Set α and β to be defined as follows dimη(L) = d+ 1− α and dimη(R) = d+ 1− β.Notice that γdef.= α + β − d − 2 = d + 1 − Ä dimη(L) + dimη(R) + 1ä. The arguments in theproof of Theorem 2.4 of [BKPS04] can be modified very easily by recalling that Corollary2.9 in their paper is (7.1.5), Lemma 2.10 is (7.1.6), Lemma 2.11 is (7.1.7) and equation (2.9)is part (b) of (7.1.8). Also, notice that their condition “dimS(L) + dimS(R) > d” changes to“dimη(L) + dimη(R) + 1 > d+ 1”, the latter being equivalent to γ > 0, and similarly whenthe symbol > is replaced by ≤ .141Chapter 7. The separation between components: transitions in dimensionsd = 2, 6, 10, ...§ 7.2. Tail triviality of random relations.Recall that an event in any probability space is said to trivial if its probability is either zeroor one, equivalently, if its probability p satisfies p = p2. It is easy to see that the set of trivialevents is a σ-field, called the trivial σ-field (relative to the given probability measure). Aσ-algebra is said to be trivial if it is contained in the trivial σ-field. If R is a random relationon the vertices of Γd(λ), we denote by FRA , where A ⊂ Zd+1×Zd+1, the σ-algebra generatedby the events {zRz′} as (z, z′) ∈ A. If z is a vertex, we define the tail σ-field based at z onthe left of R, denoted by L Rz , to be the σ-field generated by the events {zRz′} as z′ →∞,in other words, L Rz is the intersection of all FR{z}×K{ as K runs on all finite subsets of Zd+1.For simplicity, we will say “left tail at z” to refer to L Rz . In a similar manner we define thetail σ-field RRz based at z on the right of R, to be the intersection of all FRK{×{z} as K runs onall finite subsets of Zd+1; we will talk about “right tail at z” to refer to this σ-algebra. Wealso define the (proper) tail σ-algebra of R to be the intersection of all FRK{1×K{2as K1 andK2 run over all finite subsets of Zd+1, we denote it as T R. Finally, we define the restrictedcomposition L R of two relations L and R (in that order) to be the set of all pairs (z1, z2)such that there exists a z ∈ Zd+1 satisfying the relations z1Lz and zRz2 and z satisfies〈z1z2〉η ≤ min(〈z1z〉η , 〈z2z〉η).Theorem ( 7.2.1 ) ([BKPS04, Theorem 3.3]) Let L and R be two independent randomrelations on the vertices of Γd(λ).(a) If all left tail σ-fields L Lz of L are trivial and R has a trivial tail σ-field TR, then therestricted composition L R has trivial left tail σ-fields L LRz .(b) If dimη(L) and dimη(R) both exist, then L  R has also stochastic dimension (relativeto η) and we havedimη(LR) = min (dimη(L) + dimη(R) + 1, d+ 1) .(c) If dimη(L) and dimη(R) both exist, all left tail σ-fields L Lz of L are trivial, all righttail σ-algebras RRz of R are trivial and dimη(L) + dimη(R) ≥ d, theninfz1,z2P (z1L Rz2) = 1.Let now m ≥ 2 and (Ri)i=1,...,m be independent random relations on the vertices of Γd(λ)such that dimη(Ri) exists for each 1 ≤ i ≤ m.142Chapter 7. The separation between components: transitions in dimensionsd = 2, 6, 10, ...(d) Suppose thatm∑i=1dimη(Ri) + m ≥ d + 2, all left tails σ-fields L R1z are trivial, each ofT R2 , . . . ,T Rm−1 are trivial tail σ-algebras, and all right tail σ-algebras RRmz are alsotrivial. Then P (z1R1 · · ·Rmz2) = 1 for all z1 and z2 vertices of Γd(λ).Proof. The proofs of Theorem 3.3 and Corollary 3.4 of [BKPS04] carry over mutatis mutandisto our context.§ 7.3. The WSF connectedness relation.Let Γ be any network satisfying the usual assumptions (A.22). Assume further that theunderlying graph of Γ is simple (A.8). We will consider the following relation UΓ on thevertices of Γ : “v and v′ belong to the same WSF-component.” We have the following verygeneral result regarding tail triviality of UΓ.Theorem ( 7.3.1 ) Suppose Γ is any network satisfying the usual assumptions. Assumefurther that(a) The network random walk of Γ is transient.(b) The network possesses the Liouville property (LP) §3.3.Then, all left and right tail σ-algebras of UΓ are trivial. If, additionally to the previoushypotheses,(c) The WSF of Γ is one ended, that is to say, for almost every realisation, every tree inthe WSF has one end.Then, the tail σ-field of UΓ is also trivial.For the proof see Theorem 4.5 of [BKPS04], together with Remark 4.6.We need one last ingredient. Consider two random subsets of vertices, A and Z, of anetwork Γ. It is said that Z stochastically dominates A if there exists a coupling µ, see §3.2,of A and Z such that µ(A ⊂ Z) = 1.Theorem ( 7.3.2 ) ([BKPS04, Theorem 4.1]) Let Γ be any network satisfying the usual as-sumptions whose underlying graph is simple (A.8). Let F, F0, . . . ,Fm be independent samplesof the WSF of the network Γ. If x is a vertex of Γ, denote by VFx the vertex set of the compo-nent of x in F. Fix any vertex o of Γ and set V0 = VFo . For j ≥ 1, define inductively Vj to bethe union of all vertex components of F that are contained in, or adjacent to, Vj−1; in other143Chapter 7. The separation between components: transitions in dimensionsd = 2, 6, 10, ...words, Vj is the union of VFx as x runs on Vj−1 (closure in Γ). Let Q0 = VF0o . For j ≥ 1,define inductively Qj to be the union of all vertex components of Fj that intersect Qj−1; inother words, Qj is the union of all VFjx such that VFjx ∩ Qj−1 6= ∅. Then, Vm stochasticallydominates Qm.Proof. It is a simple adaptation of Theorem 4.1 of [BKPS04], indeed in their first sentencechange BR and replace it with BR = {v; d(v, o) < R}, where d is the graph-distance of Γ.The remainder of their proof proceeds verbatim.§ 7.4. The main theorem.Theorem ( 7.4.1 ) With the notation at the beginning of the chapter,maxz,z′∈Zd+1D(z, z′) =¢d− 24•, a.s.Proof. The proof follows the lines of that of Theorem 1.1 and Proposition 5.1 of [BKPS04].For convenience of the reader, we provide full details.When d = 1, 2, we know that USF of Γd(λ) is a tree and thus the separation betweencomponents is zero, this agrees with the formula given. Assume then that d ≥ 3.Let m =¢d− 24•. Consider m+ 1 independent copies of the UΓd(λ) relation. By (7.1.2),we know that dimηÄUΓd(λ)ä= 3. Next, by (7.3.1), bearing in mind that USF is one ended(see Chapter 5) this random relation possesses trivial tail σ-field as well as trivial left tailand right tail σ-algebras. We may apply item (d) of (7.2.1) to conclude that, if R is thecomposition of the m+ 1 copies of UΓd(λ) theninfz1,z2∈Zd+1P (z1Rz2) = 1.Finally, define S to be the relation D(z, z′) ≤ m. By virtue of (7.3.2), we know thatP (z1Sz1) ≥ P (z1Rz2) and thus, P (D(z, z′) ≤ m, ∀z, z′) = 1.The proof of the cases 3 ≤ d ≤ 6 is complete, so it only remains to prove the lower boundfor the cases d ≥ 7. To prove the lower bounds we will establish first(∗) P (D(z, z′) ≤ k) ≤ c 〈zz′〉4k−(d−2)ηfor all z, z′ ∈ Zd+1 and k ∈ N, the constant c depending on k and d, but not on z or z′.When k ≥ m, the criterion holds since then 4k ≥ d− 2 ≥ 5 and 〈zz′〉η ≥ 1. Assume k < m,so that 4k − (d − 2) ≤ −1. Consider a finite sequence (zj, z′j)j=0,...,k of pairs of vertices of144Chapter 7. The separation between components: transitions in dimensionsd = 2, 6, 10, ...Γd(λ) with z0 = z, z′k = z′ and z′j ∼ zj+1 for j = 0, . . . , k− 1. Let A = A(zj, z′j) be the eventin which z′j is a vertex of Tzj for j = 0, . . . , k and for each of these j, z′j is not a vertex ofTzi for i 6= j. Then,{D(z, z′) = k} ⊂ ⋃(zj ,z′j)A(zj, z′j)where the pairs (zj, z′j) run over all sequences (zj, z′j) as stated. To estimate the probabilityof a particular A(zj, z′j) we may construct the USF-forest using Wilson’s rooted at infinity,beginning with the random walks started at z0, z′0, . . . , zk, z′k. For the event A(zj, z′j) to hold,it is necessary that for each j = 0, . . . , k, the random walk started at zj must intersect thepath of the random walk started at z′j. Hence, if Sz denotes the random walk started at z(with the different random walks mutually independent),P (A) ≤k∏j=0PÄSzj ∩ Sz′j 6= ∅ä = k∏j=0∑z∈Zd+1G(zj, z)G(z′j, z) k∏j=0¨zjz′j∂−(d−2)η,where  came from the proof of Theorem (6.1.2). Since z′j ∼ zj+1, we can boundP (D(z, z′) = k) ≤ c ∑(zj)k+1j=0k∏j=0〈zjzj+1〉−(d−2)η ,the sum extending over all finite paths (zj) starting at z and ending at z′. We may finallybreak up this sum, starting with the sum over all zk and applying (7.1.6) with α = β = d−2,so that γ = α+ β − d− 2 = d− 6 > 0, and then continue with the sum over zk−1, and so onuntil z1, this gives the desired bound (∗). Having establish (∗), we can terminate Theorem(7.4.1). Indeed, consider z ∈ Zd+1, the eventßmaxz′∈Zd+1D(z, z′) = k™belongs to the left-tailat z relative to the relation UΓd(λ), as such it is a trivial event (7.3.1). It is clear from (∗)that this event has probability zero, therefore the event®maxz,z′∈Zd+1D(z, z′) = k´also haveprobability zero.145Chapter 8ConclusionThis thesis introduced the network Γd(λ) and studied several properties of the networkrandom walk here, as well as the uniform spanning forest. Some fundamentals were given,most importantly are the shape of the Green’s function, the frequency of crossings, thenumber of ends and the separation between the different components of the USF. There are,of course, many interesting questions regarding the USF.Problem 1. Consider the set Hd(n) = {n}×Zd where n ∈ N; the “vertical plane” at distancen to the “right” of the origin. For almost every realisation, the NRW on Γd(λ) will crossHd(n) at some point, in other terms, the hitting time τHd(n) of the SRW started at zero isfinite almost surely. What is the length of the LE that arises from the path (Sk)0≤k≤τHd(n)? Isthere an exponent δ > 0 such that this length satisfies to be proportional to n1+δ? Can thisbe improved to n log(n) or other expression involving logarithm?Problem 2. Consider a tree T in USF on Γd(λ). Let o be a vertex in T and r > 0. Denote byBT (o; r) the ball in T with centre o and radius r for the graph distance of T. What is the size(cardinality) of BT (o; r)? Since T ⊂ Rd, one can consider the Euclidean ball BRd (o;R) ofcentre o and radius R. For what (deterministic) R > 0 does one have BT (o; r) ⊂ BRd (o;R)with high probability?Problem 3. In Proposition (5.2.7) we showed that, except in an event of exponentially smallprobability, one in every√n levels the network random walk of Γd(λ) will have a splittinglevel. Can√n be replaced by nδ (for some δ > 0) while the conclusion remains the same?Problem 4. Given a vertex z of Γd(λ), we call it a leaf (of the USF-forest) if it is adjacentto one and only one edge of the USF-forest. Denote by Rz the USF-ray started at z and by Tzthe component of z. By one-endedness, each tree has at least one leaf. Does every USF-treehave infinite leaves? If not, how many leaves does the “typical” USF-tree have?146Chapter 8. ConclusionProblem 5. Related to the previous problem. By one-endeness, Tz \ Rz consists of finiteconnected subsets of edges. Suppose z is a leaf. Call each of the finite connected subsets ofTz \Rz a “branch.” How many leaves does each branch have? What is the size (cardinality)of each branch?Problem 6. What is the behaviour of NRW of a tree T in the USF-forest on Γd(λ)? Thiscan be studied when T is regarded as a graph or as a subnetwork of Γd(λ).Problem 7. What are the scaling limits of NRW on Γd(λ)? That is, consider S = (Sn) =(Bn, Xn) the NRW on Γd(λ) and the space DRd+1 [0,∞) of continuous functions on the rightwith left limits defined on [0,∞) and with values in Rd+1 endowed with a complete distancethat defines the Skorohod topology. One can induce naturally an element here using NRWby defining the mapping t 7→ S[nt], where [ζ] denotes the whole part of ζ ∈ R. Then, we areasking for transformations fn of the space, where only translations and rescaling of the factorsare allowed, such that fn(B,X) has a weak limit; explicitly, for what fn : Rd+1 → Rd+1 ofthe form fn(t, x) = (ηnt − an, ζnx − bn), does the process t 7→ fnÄS[nt]ähas a weak limit inDRd+1 [0,∞) with non-trivial components?Problem 8. If 0 ≤ λ < λ′, how different are the USF-forests of Γd(λ) and Γd(λ′)?Problem 9. In sect. 5.4 we proved that the USF-ray started at z is ultimately inside z + Pεfor every ε > 0. If this ray ultimately inside z + P? 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In Proceedings of the FifthBerkeley Symposium on Mathematical Statistics and Probability, volume 2, Pt. 2.University of California Press, 1967.[Tho00] Hermann Thorisson. Coupling, Stationarity, and Regeneration. Probability andits Applications, A Series of the Applied Probability Trust. Springer, New York,2000.151Bibliography[Wil96] David Wilson. Generating random spanning trees more quickly than the covertime. Proceeding of the Twenty-Eighth Annual ACM Symposium on Theory ofComputing, pages 296–303, 1996.152Appendices153Appendix AGraphs: definitions and assumptionsWe write this annex to assemble most of the definitions used throughout the text. Effortshave been placed to provide mathematically correct definitions, as opposed to describe themin plain English expecting the reader to fill in what the definition ought to be. We followclosely the definitions, nomenclature and notations appearing in [LP16].Definition ( A.1 ) Graphs: directed and undirected. A graphoid is a pair of setsG = (V,E) where V is the vertex set and E is the edge set. The elements of V will be calledvertices (of the graphoid) and the elements of E edges (of the graphoid). A graphoid canbe directed or undirected. A directed graphoid is called a directed multigraph and it is agraphoid with two functions from E into V : for any edge e, e− will denote the tail and e+will denote the head of e. The two vertices e± are known as end-points of e. If G is a directedmultigraph such that there are not two different edges with the same tail and head, then wewill call it a directed graph. A directed multigraph that is not a directed graph is often saidto be a graph with multiple edges. The direction of an edge is from tail towards head and inthis case, the edge is said to join the tail with the head. If v = e− = e+, we will say e is aself-loop based at v (in the directed multigraph).We will assume from now onwards that whenever we are dealing with a directed multigraphthere exists a function e 7→ −e which is a bijection of the edge set onto itself satisfyinge = −(−e), (−e)+ = e− and (−e)− = e+.If G is a directed multigraph, we will call −e the opposite edge of e.An undirected graphoid is a graphoid with a function n from E into the power set of Vsuch that, for any edge e, n(e) is not empty; all the vertices in n(e) are said to be joinedtogether by e and are known as the end-points of e. If n(e) has at most two elements for everyedge e, then the graphoid is called an undirected multigraph; if the undirected multigraph is154Appendix A. Graphs: definitions and assumptionssuch that n is injective, we will talk about undirected graphs, in the contrary case, undirectedgraphs with multiple edges. When the edge e satisfies n(e) = {v} for some vertex v, it issaid that e is a self-loop based at v (in the undirected multigraph).From now onwards, when we talk about multigraphs, we will omit to refer explicitly, unlessnecessary from context, to the functions e 7→ e± or e 7→ n(e). We will also define conceptsusing the language of graphs and assume we defined them for multigraphs as well, unless thedefinition for multigraphs is not a straightforward generalisation of that for graphs.Definition ( A.2 ) Finite and infinite graphs; denumerable graphs. A graph G =(V,E) is said to be finite if both V and E are finite sets. In the contrary case, G is infinite.If G is an graph such that both V and E are at most countably infinite, then we will saythat G is denumerable.All graphs appearing in this document are either finite or infinite denumerable.Definition ( A.3 ) Canonical construction of graphs. There is a canonical way toconstruct directed and undirected graphs, which we mention as this is usually the definitionprovided. Consider any set V, which is non empty. To make it an undirected graph, wecan take E any non empty subset of the power set of V such that the elements of E are ofthe form {v, w}, with v and w any elements of V, the function n is just the identity on E;to make it directed, we take E ⊂ V2 non empty and the tail and head functions are therestrictions to E of the projections (v, w) 7→ v and (v, w) 7→ w, respectively.Definition ( A.4 ) Canonical loss of direction; creation of direction. There isa natural way to “loss the direction” in a given directed graph; in other words, there isa canonical way to induce an undirected graph out of a directed one. If G is a directedgraph, we set n(e) = {e−, e+}. The opposite is also true, albeit the construction is no longercanonical. If G is undirected, then divide the edge set into two, E = E0∪E′, where E0 is theset of edges such that n(e) has only one element (E0 consists of all self-loops) and E′ = E\E0.Next, set E±1 = E′ × {±1}. If e ∈ E0, then set e± both to be the only element of n(e); ife ∈ E′, then n(e) = {u, v} with u 6= v and set (e, 1)+ = v, (e, 1)− = u and (e,−1)+ = u,(e,−1)− = v with u, v chosen in whatever way. Now, define Edir = E0∪E1∪E−1. Finally, thefunction e 7→ −e is given as e 7→ e if e ∈ E0 and if (e,±1) 7→ (e,∓1) for edges in E1 ∪ E−1.Definition ( A.5 ) Subgraphs and exhaustions; spanning subgraphs. A directed(resp. undirected) subgraphoid of a directed (resp. undirected) graphoid G is a pair G′ =(V′,E′) such that V′ ⊂ V, E′ ⊂ E and when the functions n or e 7→ e± are restricted toE′, the resulting structure is that of a directed (resp. undirected) graphoid; in particular, ife ∈ E′, then −e ∈ E′. It is easy to see that if G is a multigraph or a graph, then so is G′ and155Appendix A. Graphs: definitions and assumptionswe will refer to G′ as a submultigraph or subgraph, accordingly. The subgraphoid G′ of G issaid to be spanning if V′ = V. Thus, the set of spanning subgraphoids of G is canonicallyidentified with a product space {0, 1}D, see Annex B for more.An exhaustion of a graph G = (V,E) is sequence Gn = (Vn,En) (n ∈ N) of subgraphssuch that Vn ⊂ Vn+1, En ⊂ En+1, V =⋃n∈NVn and E =⋃n∈NEn. We say that an exhaustion isfinite if Gn is finite for each n. Most exhaustions we will work with are, however, not finite,they only possess a finite vertex set.Definition ( A.6 ) Induced subgraphs. Given a subset of vertices A ⊂ V, consider E|Athe subset of E consisting of all edges e joining vertices in A (that is, e± ∈ A or n(e) ⊂ A).Clearly (A,E|A) is a (directed or undirected) subgraph of G and this structure is said to beinduced on the set of vertices A by that of G. The elements of E|A are called “edges of A.”Similarly, if D ⊂ E, we can consider V|D to be set set of all vertices joined by edges of D(that is, the set of all e± as e runs through D or the union of all n(e) as e runs through D).Once again, (V|D,D) is a subgraph of G that is said to be induced on the set of edges D bythat of G. The elements of V|D are called “vertices of D.”With the notion of subgraphs, there is a natural way to construct exhaustions. Thatis given by the induced subgraphs on an increasing family (Vn) of finite subsets of verticeswhose union is all of V.Definition ( A.7 ) Adjacency, locally finiteness and paths. Two vertices in a graphare neighbours or adjacent if there is an edge joining them. A graph is locally finite if everyvertex is the end-point of finitely many edges. An edge e joining two vertices will be calledadjacent to these vertices; e is said to be adjacent to a given set if there is a vertex in thisset adjacent to e. Two edges are adjacent if they are adjacent to a common vertex. Two setsare adjacent if there are corresponding vertices of each set that are adjacent, equivalently, ifthere is an edge that is adjacent to both sets.A (directed) path in a directed graph is a finite sequence of edges γ = (ej)1≤j≤m satisfyinge+j = e−j+1 for j = 1, . . . ,m − 1. Both the ej and their opposites are called edges of thepath. The path γ = (ej)1≤j≤m generates a family of vertices (vj)0≤j≤m, where v0 = e−1 ,vj = e+j = e−j+1 for j = 1, . . . ,m− 1, and vm = e+m, these are called vertices of the path. Thevertices of a path are said to be connected by a path in G. A cycle in a directed graph is, bydefinition, a path γ = (ej)1≤j≤m, such that e−1 = e+m. If γ = (ej)1≤j≤m is a path in a directedgraph, we will call e−1 to be the tail or initial vertex of the path; we will call e+m to be thehead or terminal vertex. In the case of a graph (where multiple edges are banned), the edgesof a path can be recovered from the vertices of the path.In a graph (where multiple edges are banned), paths of vertices are canonically identified156Appendix A. Graphs: definitions and assumptionswith paths of edges.An (undirected) path in an undirected graph is a family of edges γ = (ej)1≤j≤m such thatn(ej)∩n(ej+1) 6= ∅ for j = 1, . . . ,m−1. The vertices in n(e1)∪ . . .∪n(em) are called verticesof the path. An undirected path γ = (ej)1≤j≤m is said to be a cycle if there exists a family(vj)0≤j≤m of vertices such that n(ej) = {vj−1, vj} for 1 ≤ j ≤ m and v0 = vm.If γ = (ej)1≤j≤m is a path in a graph (either directed or undirected), we will say that thelength of γ is m and we will denote length (γ) = m.Definition ( A.8 ) Simple graphs. We will say that a graph G is simple if it does nothave multiple edges and it possesses no loops.Definition ( A.9 ) Infinite paths; rays and forests. An infinite directed (resp. undi-rected) path or unending directed (resp. undirected) path is an infinite family γ = (ej)j∈Z+such that for every N ∈ N, the subfamily (ej)0≤j≤N is a directed (resp. undirected) path. Aninfinite directed path γ = (ej)j∈Z+ is said to have e−1 for its initial, first or starting vertex.An infinite directed path in a graph (where multiple edges are banned) is canonicallyidentified with a path of vertices.A path is called edge-simple if every edge of the path is used only once, this is the sameas saying that j 7→ ej is injective; a directed path is vertex-simple if j 7→ vj is injective.Observe that every vertex-simple path is also edge-simple. An unending, edge-simple pathis called a ray. A forest is a graph with no cycles (also said to be cycle-free or acyclic), thusa spanning forest of a graph is a subgraph that possesses all the vertices of the graph andthat is cycle-free. In a general graph, rays can be not vertex-simple, in a forest, every ray isvertex-simple. Every forest is a simple graph.Definition ( A.10 ) Interior, closure and boundary of a set of vertices. Let Abe a subset of vertices. The (vertex exterior) frontier or boundary of A is the set ∂A ofvertices not in A that are neighbour with a vertex in A; the (vertex) interior of A is the setA˚ of vertices in A such that all their neighbours belong to A; the (vertex) closure of A isA = A ∪ ∂A. It is clear that A˚ ⊂ A ⊂ A and A ∩ ∂A = ∅.From this point onwards we assume known the definition of electrical network (1.1.1).Definition ( A.11 ) Contracted and wired networks. Consider a network (G, µ), withG = (V,E), and a subset of its vertices, A. We define a new network (G/A, µ/A) as follows:• The vertex set of G/A is (V \ A) ∪ {A}.157Appendix A. Graphs: definitions and assumptions• The edge set of G/A is E, but we change connectedness by making e± ∈ A in thegraph G to mean e± = A in the graph G/A. (This description is for directed graphs,for undirected graphs, we make n(e) ∩ A 6= ∅ in G to mean A ∈ n(e) in G/A.)• µ/A = µ and we will write µ for both networks.The network (G/A, µ) is said to be obtained by contracting the edges of A or by collapsingA into a point. Clearly, A is a vertex of this network.In general, G/A is a multigraph, even if G is a simple graph. Observe that if G is locallyfinite and A is finite, then G/A is locally finite as well. With this definition, we introduce yetanother construction. We define the network (GWA , µWA ) to be the contraction of the graphinduced on A = A∪ ∂A by the edges of ∂A and deletion of all self-loops based at ∂A. Here,µWA is the restriction of µ to the edge set of GWA . For simplicity, and unless necessary fromcontext, we still will use µ to refer to the conductances of GWA . We call (GWA , µ) the networkinduced on A with its boundary wired into a vertex.Definition ( A.12 ) Connectedness. A graph is connected if every pair of vertices canbe connected via some path. A connected forest is a tree. Thus, a spanning tree of a givengraph is a connected subgraph such that it contains all the vertices of the given graph andit contains no cycles.Any locally finite connected graph with countably many vertices necessarily has countablymany edges. To see this, consider o to be any vertex of the graph. The hypothesis impliesthat every other vertex v is connected to o at a minimum-length path. Therefore, V can bewritten as a denumerable union over n of the set of vertices whose minimal path length too is n and it easily follows that each of this sets is finite. (This proof works verbatim forlocally finite connected multigraphs.)Definition ( A.13 ) Morphisms of graphs and networks; automorphisms. SupposeG = (V,E) is an undirected graph; let n be the function in the definition of undirectedness.Consider a pair of functions (ϕ, ψ) with ϕ : V→ A and ψ : E→ T (where A and T are sets).This pair of functions is said to have undirected consistency if the following implication holdsψ(e1) = ψ(e2) =⇒ ϕ(n(e1)) = ϕ(n(e2)),(the right-hand side being images of a set of edges by the function ϕ). When the pair (ϕ, ψ)has undirected consistency, we define the undirected image of G by the pair (ϕ, ψ) to be theundirected multigraph H = (ϕ(V), ψ(E)) in which two vertices a1 = ϕ(v1), a2 = ϕ(v2) ofH are joined by the edge ψ(e) if v1 and v2 were already joined in G by e. In other words,we define m from ψ(E) into the power set of ϕ(V) by means of m(t) = ϕ(n(e)) where e is158Appendix A. Graphs: definitions and assumptionsany edge in E such that ψ(e) = t, undirected consistency shows at once m is well-defined.Furthermore, since n(e) has at most two vertices, is also a set with ϕ(n(e)) has also at mosttwo vertices.Suppose now that G = (V,E) is a directed graph and (ϕ, ψ) is a pair of functions justas in the previous paragraph. This pair (ϕ, ψ) is said to have directed consistency if thefollowing relation holdsψ(e1) = ψ(e2) =⇒ ϕ(e−1 ) = ϕ(e−2 ), ϕ(e+1 ) = ϕ(e+2 ).When this condition is satisfied, we may define the directed image of G by this pair (ϕ, ψ)to be the directed multigraph H = (ϕ(V), ψ(E)) in which the direction of an edge t = ψ(e)of H is defined to be t± = ϕ(e±).Suppose now that (G, µ) is a network and a pair of function (ϕ, ψ) is given as before. Wesay that this pair (ϕ, ψ) have network consistency if the following relation holdsψ(e1) = ψ(e2) =⇒ c(e1) = c(e2).When (ϕ, ψ) has both undirected and network, or directed and network consistencies, wedefine on H = (ϕ(V), ψ(E)) the structure of a network by setting the conductances of theedges t in H to be κ(t) = c(e), for whatever t = ψ(e). This H is said to be the (directed orundirected) network image of (G, µ) by (ϕ, ψ).Suppose G′ = (V′,E′) is another (directed or undirected) graph. The pair of functions(ϕ, ψ), ϕ : V → V′ and ψ : E→ E′ is said to be a (directed or undirected) graph homomor-phism if the image H of G by this pair can be defined and this graph H is the induced graphon G′ by ϕ(V). If c′ is an assignment of conductances on G′ (so that (G′, c′) is a network),the pair is said to be a network homomorphism if it is a graph homomorphism, the networkimage H of G can be defined and the conductances on H coincide with the restriction of c′to ψ(E), that is, κ(e′) = c′(e′) for all e′ ∈ ψ(E). If the (graph or network) homomorphism(ϕ, ψ) satisfies that both ϕ and ψ are bijections, then we will speak of (graph or network)isomorphism. When G = G′, we use automorphism in place of isomorphism.Definition ( A.14 ) Planar graphs. Consider a locally finite graph G = (V,E). We willsay that G is embedded in Rd if the following two conditions hold: (1) every edge is a subsetof Rd diffeomorphic to the interval [0, 1], (2) vertices joined together by the edge e are theend-points of e (in particular, V ⊂ Rd). The embedding of G is, by definition, the subset ofRd consisting of the union of all the edges of G; we will use the symbol G to refer to G andthe embedding of G. The embedding of G is said to be proper if, for every compact subset Kof Rd, the intersection of K∩V is finite. The graph G is said to be planar if it is embeddedin R2 and any two different edges will not intersect unless they share a vertex.159Appendix A. Graphs: definitions and assumptionsDefinition ( A.15 ) Duality. We only explain what duality is at an intuitive level.If G is a properly embedded planar graph, each connected component of the complementof G in R2 is called a face of G and two faces are adjacent if there exists a common edgeof G, boundary to both faces. On each face of G, pick an interior point and then the edgejoining two adjacent faces is just a simple arc in R2 joining the two selected points of therespective faces in such a way that it will cross the original edge in G, common boundary tothe faces, once; if e is the edge of G, denote by e† the edge just constructed joining the twofaces of G. We finally set the direction of e† in such a way that e and e† intersect (as simplearcs of R2) in an angle ofpi2(a right-angle). The dual graph is G† = (V†,E†) where V† isthe set of faces of G and E† is the set of edges joining two faces of G. If G is a network andµ is its conductance function, we define a function µ† : E† → (0,∞) by µ†(e†) = 1µ(e), theresistance of e. We may call µ† the dual conductance and (G†, µ†) the dual network.Definition ( A.16 ) Distances and operations between paths. Given a directed graphG, we endow it with a metric space structure given as follows(A.17) dG(v, w) = d(v, w) = infγlength (γ) ,where γ runs through all the paths satisfying γ− = v and γ+ = w. This function receivesthe name of chemical distance or simply the graph distance induced on G. To check this isa distance, introduce the following notation, given a path γ = (ej)1≤j≤m, call the path(A.18) − γ = (−em−j)j=j,...,mthe reverse path to γ; clearly γ− = (−γ)+, γ+ = (−γ)− and length (γ) = length (−γ) , thisimplies d(v, w) = d(w, v). Given two paths α = (ej)j=1,...,m and β = (fj)j=1,...,n with α+ = β−,define the path α ∨ β, called the juxtaposition or concatenation of (α, β) (in this order) tobe the path(A.19) α ∨ β = (e1, . . . , em, f1, . . . , fn).Then (α ∨ β)− = α−, (α ∨ β)+ = β+ and length (α ∨ β) = length (α) + length (β) , and thisimplies d(v, w) ≤ d(v, u) + d(u,w) for any other vertex u. The other properties of distanceare easy and routine to prove. The juxtaposition α ∨ β with α a (finite) path and β andinfinite path can also be defined as an infinite path, in a similar manner as before, providedα+ = β−.If G is an undirected graph, define dG(u, v) = infγlength (γ) , where γ runs over all pathsconnecting u, v; the definitions of −γ and α ∨ β carry over provided n(em) ∩ n(f1) 6= ∅ andis easy to see dG(u, v) is also a metric.160Appendix A. Graphs: definitions and assumptionsDefinition ( A.20 ) Intersection and collision of paths. We employ the followingterminology. We will say that two paths γ = (ej)1≤j≤s and ξ = (fk)1≤k≤t of the graph Gintersect each other if they have vertices in common. If γ = (vj)0≤j≤s and ξ = (wk)0≤k≤t aretwo directed paths, we will say that γ collides with ξ at time m if vm = wm. In a similarfashion, we adopt the following common and convenient convention: we will use the samesymbols to denote paths and sets, that is to say, if γ is a path of vertices, then we will denoteby γ the set of the vertices in the path. (Then, γ and ξ intersect each other is a relationequivalent to γ ∩ ξ 6= ∅.)Definition ( A.21 ) Nearest-neighbour integer lattice. In what follows, N will denotethe set of natural numbers (the positive integers), Z+ is shorthand for N ∪ {0} and Z willbe the set of whole numbers; Z− is the set of negative whole numbers or zero; Z∗ is the setof non zero whole numbers and the definition of Z∗− is apparent. The “Euclidean lattice”,“nearest-neighbour integer lattice” or “standard lattice of dimension d” is the canonicalgraph with vertex set Zd and whose edges are, in the directed case, those pairs (x, y) forwhich ‖x− y‖ = 1 (the Euclidean distance) or, in the undirected case, those sets {x, y} forwhich ‖x− y‖ = 1. By abuse of language, the vertices in Zd will be called vectors, pointsand, of course, vertices. Every standard lattice is an infinite locally finite graph. To avoidclumsiness, 0 will denote the point zero in all of these networks Zd (d ∈ N). We sometimeswill use the following notation: given a pair of sets A and B in Zd and λ ∈ Z, we defineA + λB to be the set of points z = a + λb ∈ Zd with a ∈ A and b ∈ B. It is clear that thegraphs Zd are properly embedded in Rd (d ∈ N) and Z2 is also a planar graph.Definition ( A.22 ) Usual assumptions. Let Γ = (G, µ) be a network satisfying thefollowing:(a) G is connected;(b) G is denumerable;(c) G is directed;(d) for every vertex v of G, the conductance at v (1.1.3) is positive and finite, that is tosay, the family (c(e))e−=v is summable and not all its members are zero;(e) the network random walk is irreducible.To refer to a network which satisfies these hypotheses, we will simply say that Γ satisfies theusual assumptions. Observe we do not ban multiple edges in the usual assumptions, or eventhe possibility of infinite edges connecting two given nodes.161Appendix A. Graphs: definitions and assumptionsDefinition ( A.23 ) Transitive networks. A network Γ, satisfying the usual assumptions,is said to be transitive if for every pair of vertices o and o′, there exists an network auto-morphism of Γ such that the image of o is o′ under such automorphism. It is clear from thisdefinition that the Euclidean lattices Zd are transitive by simply considering translations.Definition ( A.24 ) Merging and ends. Let G be a forest. Consider two rays of G,γ and γ′. We shall say that γ merges with γ′ if γ ∩ γ′ is infinite. Since a tree does notcontain cycles, that γ = (vn) and γ′ = (v′n) merge signifies that there exist a pair of indicesN = N(γ, γ′) and M = M(γ, γ′) such that for all n ≥ N, vn = v′M+n.Straightforwardly, the relation “γ merges with γ′” on the set of rays of G is an equivalencerelation. Each of the classes for this equivalence relation is called an end. Therefore, oneendedness signifies that any two rays of G will eventually merge, this is the same as sayingthat from any vertex there exists one, and only one, ray.162Appendix BRandom spanning subgraphsIn this annex we will discuss Wilson’s algorithm for finite and infinite graphs, as well as themeasurable space of spanning subgraphs and its weak convergence.§ B.1. Random spanning subgraphs of a network.Suppose G is a directed or undirected graph. Suppose E is the edge set of G. If G is directeddenote by D a subset of edges chosen so that(a) for every e ∈ E, either e ∈ D or −e ∈ D;(b) if e ∈ D, then −e /∈ D.If G is undirected, set D = E. Denote by SG the set of spanning subgraphs of G. In otherwords, SG is the set of subgraphs of G of the form (V,E′). Then, SG is canonically identifiedwith the set {0, 1}D; in fact, if x = (xe)e∈D ∈ [0, 1]D, then xe = 1 signifies that e (and itsopposite if applicable) is present in x, while xe = 0 means that e is not present (neither e norits opposite in the directed case). This set is endowed with the product topology generatedby that of the factors {0, 1}. Given an edge e, denote by pre : {0, 1}D → {0, 1} the functionx 7→ x(e) (the “projection at e” or “evaluation at e”). It is elementary mathematicalanalysis to show that {0, 1}D is a metrisable compact space, its topology is the smallesttopology making all projections continuous, its Borel σ-algebra satisfiesB (SG) =⊗e∈D{∅, {0}, {1}, {0, 1}},163Appendix B. Random spanning subgraphsand it is the smallest σ-field making all the projections measurable. Introduce the followingtwo types of sets, consider K,H ⊂ D non empty,PK =⋂e∈Kpr−1e (1), AH =⋂e∈Hpr−1e (0);in other words, PK is the set of spanning subgraphs of G such that all edges in K arepresent, and AH that set of spanning subgraphs where the edges in H are absent. We setP∅ = A∅ = {0, 1}E. By definition of the product topology, the subsets PK ∩AH of {0, 1}E,with K,H ⊂ E any two finite subsets, are basic open sets, and thus the generate the Borelσ-field of {0, 1}E. But in fact, AH already generate the Borel sets for we havePK =⋂e∈KA {{e}.The same remark applies when we interchange the roˆles of the AH andPK. Observe also that⋂ζPKζ = P⋃ζKζ , for any family of subset (Kζ)ζ of E; the same is true for the AH, namely⋂ζAHζ = A⋃ζHζ . This implies that the sets of all subsets of {0, 1}E of the form PK or theset of all subset of the form AH, as K or H run over all finite subsets of E, are pi-systems.Then, Dynkin’s theorem [Ash72, 4.1.2] shows that if ν and ν ′ are two probability measureson SG such that ν(E ) = ν ′(E ) for all E = PK or all E = AH, then ν = ν ′. We add also thefollowing remarkPK ∩AH ⊂PK′ ∩AH′ ⇐⇒ K′ ⊂ K,H′ ⊂ H;the fact that K′ ⊂ K and H′ ⊂ H implies PK ∩AH ⊂PK′ ∩AH′ is obvious, for the reverse,suppose that there exists e ∈ K′ \ K, say, then there exists x ∈ {0, 1}E such that x|K = 1,x|H = 0 and x(e) = 0, then x ∈PK ∩AH but x /∈PK′ ∩AH′ , violating the hypothesis.Denote by B0 the set of unions of pairwise disjoint finite familiesÄPKj ∩AHjäj=1,...,m, inother words, an element of B0 is of the formm⋃j=1PKj ∩AHj ,with the sets in the union pairwise disjoint. It is elementary to see that B0 is a field thatgenerated the Borel σ-field B(SG). Therefore, Carathe´odory extension theorem [Ash72,1.3.10] states that any σ-finite measure on B0 can be extended in one, and only one way,to a measure on B (SG) . It is because of this that, when WSF or FSF are defined, sufficesto construct them on B0. See [LP16], sect. 10.1 for the construction of WSF and FSF ina network satisfying the usual assumptions whose underlying graph is simple. (Notice thatthe developments in Annex C do not assume locally finiteness of the underlying graph, sothe results of sect. 10.1 in the given reference can be extended verbatim to the case whereG is a simple graph that is not locally finite.)164Appendix B. Random spanning subgraphs§ B.2. Weak convergence on SG.Since SG is a metrisable compact space, any finite Borel measure on SG is identified, byvirtue of Daniell’s theorem [Ash72, 4.2.9], with a continuous linear form on SG, in otherwords, with an element of the topological dualS ′G ofSG. Then,the definition that a sequenceϕn of continuous linear forms in SG converges weakly to another such linear form ϕ is thatϕn(f)→ ϕ(f) for every real-valued continuous function f defined on SG.Suppose now that An is a sequence of SG-valued random objects, and let A be anothersuch random object. Suppose that An is defined on (Ωn,Fn,Pn) and that A is defined on(Ω,F ,P). Then, P(An ∈ dx) and P(A ∈ dx) are measures on SG. In this case, we say thatthe sequence An converges weakly to A if P(An ∈ dx) n→∞−−−→weaklyP(A ∈ dx), in other words,the weak convergence of An to A is that for every continuous function f : SG → R, thesequence of real numbers∫Ωnf(An)dPn converges to the real number∫Ωf(A)dP. We maycharacterise weak convergence in SG more simply using Stone-Weierstrass density theorem[Die69, (7.3.1)], as explained in the next paragraph.Consider the algebra a generated by the projections pre (e ∈ D) and the constant func-tions. A typical element of a is of the form(∗)m∑j=1αj∏e∈Kjpre∏e∈Hj(1− pre),where (Kj,Hj) are pairs of finite disjoint subsets of edges and αj are real scalars. It iseasy to see that a satisfies the hypothesis of Stone-Weierstrass density theorem, thus everycontinuous function on SG is a uniform limit of functions of the form (∗). We can now easilysee that weak convergence of An to A is equivalent to the following: for any k, p ∈ N, andany edges e1, . . . , ek+p ∈ D,∫Ωnk∏j=1prej(An)p∏j=1(1− prek+j)(An)dPn →∫Ω(A)p∏j=1(1− prek+j)(A)dP.And it is easily seen that, with K = {e1, . . . , ek} and H = {ek+1, . . . , ek+p},Pn (An ∈PK ∩AH) =∫Ωnk∏j=1prej(An)p∏j=1(1− prek+j)(An)dPn,P (A ∈PK ∩AH) =∫Ωk∏j=1prej(A)p∏j=1(1− prek+j)(A)dPn.165Appendix B. Random spanning subgraphsWe can go further and in fact, only check convergence of the sets PK or AH. Observe thatPK ∩AH = PK ∩⋂e∈HP{{e},and if we order H = (e1, . . . , eq), thenPK ∩⋂e∈HP{{e} = PK ∩q⋂j=1P{{ej} =ÑPK ∩q−1⋂j=1P{{ej}é\ÑPK∪{eq} ∩q−1⋂j=1P{{ej}é.If now µ is any measure on SG, thenµ(PK ∩AH) = µÑPK ∩q−1⋂j=1P{{ej}é− µÑPK∪{eq} ∩q−1⋂j=1P{{ej}é.Continuing in this manner, we will find that µ(PK ∩AH) is expressible as sums and rests ofterms of the form µ(PKs) for a finite family of Ks, which depends solely on PK ∩AH andnot on µ.Proposition ( B.2.1 ) Let (An) denote a sequence of SG-valued random objects definedon corresponding probability spaces (Ωn,Fn,Pn) and let A be another such random object,defined on the probability space (Ω,F ,P). A necessary and sufficient condition for An toconverge weakly to A is that for every finite subset of edges K,Pn (An ∈PK)→ P (A ∈PK) .The same result holds true if instead of PK we use AH.166Appendix CElectrical networks and the space offlowsThe purpose of this annex is to investigate properties of networks and show interesting anduseful interaction they have with network random walk. We follow closely the developmentin [LP16] and use [Bar17] and [Soa94] as major references as well. We will do the necessaryadaptations according to our conveniences. We remark that we will not assume locallyfiniteness of the network. Most of this Annex will be continuous usage of results of linearalgebra. Finally, we mention that most of the material presented in this Annex is arguablywell-known, however we could not find a presentation suitable for this thesis. For instance,[Bar17] does not study the decomposition `2−(E; r) = ♦ ⊕ F ⊕ ∇HD (notation originallyintroduced in [BLPS01])and [LP16] does not provide the level of details we wanted overall.§ C.1. Flows and energy.We will define some Hilbert spaces of functions associated with this network. To preventambiguity, we will use subscripts to denote the inner-product or norm of different spaces.A flow is a function θ : E→ R such that θ(e) = −θ(−e) for all edges e ∈ E. Recall thatr(e) = c(e)−1 is known as the resistance of the wire e. The term conductor is used to defineto an edge as well as its conductance; likewise, the term resistor will refer to either and edgeor its resistance. Define now `2−(E; r) to be the set of flows θ such that(C.1.1) EΓ (θ) = E (θ) =∑e∈E12r(e)θ(e)2is finite, this number is known as the energy of the flow θ, we are using the usual convention167Appendix C. Electrical networks and the space of flowsthat r(e) =∞ whenever c(e) = 0, r(e) = 0 means c(e) =∞ and that 0 ·∞ =∞· 0 = 0 (thisextension of the product is continuous on [0,∞]). Thus, whenever E (θ) <∞ and c(e) = 0,we have θ(e) = 0. Clearly `2−(E; r) is a subset of RE, the vector space of all functions E→ R;furthermore, if θ and η are two flows, then θ+aη is a flow for any scalar a ∈ R. The inequality(x+ y)2 ≤ 2(x2 + y2) shows at once that `2−(E; r) is a vector subspace of RE. It is routine tocheck that the function(C.1.2) (θ|η)`2−(E;r) =∑e∈E12r(e)θ(e)η(e),defined for flows with finite energy θ and η, is an inner product on the vector space `2−(E; r).We remark that the Cauchy-Schwarz inequality shows that the sum defining (θ|η)`2−(E;r) issummable. We will show that `2−(E; r) with this inner product is a Hilbert space. A fast wayis to observe that `2−(E; r) is the subspace of flows of the Hilbert space L2 (E; ρ) , where ρis the measure on all subsets of E (recall E is denumerable) given byρ(A) =∑e∈A12r(e),where A is any subset of edges. Then,(θ|η)`2−(E;r) =∫Edρ(e) θ(e)η(e).Proposition ( C.1.3 ) The subspace `2−(E; r) is closed in L2 (E; ρ) . Hence, it is a Hilbertspace.Proof. All that is needed is to show that if a sequence θn ∈ `2−(E; r) of flows with finite energyconverges in L 2 (ρ) to a some function θ ∈ L 2 (ρ) , then θ also belongs to `2−(E; r), that is,θ is also a flow with finite energy. Since E (θ) = ‖θ‖2L 2(ρ) , we have that the sum in (C.1.1)is finite for θ. Rest to show θ is a flow. First notice that E (θ) < ∞ implies that θ(e) = 0for all edges e such that c(e) = 0. Now, notice the following inequality, which immediatelyfollows from definitions,( C.1.3.1 ) For any θ ∈ L 2 (ρ) and any edge e, we have 12r(e)θ(e)2 ≤ ‖θ‖2L 2(ρ) . In otherwords, the evaluation function θ 7→ θ(e) from `2−(E; r) into R is a continuous linear formsatisfying |θ(e)| ≤»2c(e) ‖θ‖`2−(E;r) .To finish the proof, apply this inequality to θn − θ, to obtain|θn(e)− θ(e)| ≤»2c(e) ‖θn − θ‖L 2(ρ) ,since r(e) > 0, this leads to θn(e)→ θ(e) for all edges e, and therefore θ(e) = −θ(−e), thatis, θ is a flow.168Appendix C. Electrical networks and the space of flowsLet E 12be any subset of E chosen in such a way that it satisfies the following twoproperties:(a) For every edge e ∈ E either e or −e belongs to E 12.(b) If e ∈ E 12then −e /∈ E 12.With this, and the definition of a flow, it is obvious that (θ|η)`2−(E;r) =∑e∈E 12r(e)θ(e)η(e).Proposition ( C.1.4 ) The familyÄχe = 1{e} − 1{−e}äe∈E 12c(e)>0is an orthogonal basis for thespace `2−(E; r); in particular, `2−(E; r) is a separable Hilbert space.Proof. If c(e) > 0, χe ∈ `2−(E; r), and, in fact,( C.1.4.1 ) The flow χe has energy E (χe) = r(e).This follows immediately from definition,E (χe) =∑g∈E12r(g)χe(g)2 = r(e),and χe(g) = 0 if g 6= e while χe(e) = 1, χe(−e) = −1. Now, if e, f ∈ E 12, thenÄχe∣∣∣χfä`2−(E;r)=∑g∈E 12r(g)χe(g)χf (g)will be zero unless it happens that for some g ∈ E 12, e = f = g because either e = f orelse {e,−e} ∩ {f,−f} = ∅ as the choosing of E 12shows. When e = f = g for some g, theprevious sum equals one. This shows that (χe)e∈E 12is orthogonal. In order to see that thefamily (χe)e∈E 12spans all of `2−(E; r), it suffices to show that the orthogonal complement ofits linear spans is {0}. This is the case, first observe that( C.1.4.2 ) For every flow θ ∈ `2−(E; r) and every e ∈ E, (θ|χe)`2−(E;r) = r(e)θ(e).Which follows readily from an algebraic calculation. Hence, if θ ∈ `2−(E; r) is a flow orthog-onal to every χe (e ∈ E 12, c(e) > 0) then 0 = (θ|χe)`2−(E;r) = r(e)θ(e) and so θ(e) = 0 andtherefore θ(−e) = 0 as well. (Recall θ(e) = 0 whenever c(e) = 0.)169Appendix C. Electrical networks and the space of flowsWe aim at giving an orthogonal decomposition of `2−(E; r). We will need to define specialtypes of flows for this. If x is a vertex, the flow(C.1.5) sx =∑e−=xc(e)χe;is called the star based at x. Given a path of edges γ = (ej)1≤j≤m, we define the flow(C.1.6) θγ =m∑j=1χej ,and we will say that θγ is induced by the path γ. If γ is a cycle (as a path), we will saythat θγ is also a cycle (as a flow). With the notation of the opposite path (A.18) and thejuxtaposition of paths (A.19), it follows easily thatθ−γ = −θγ, θγ1∨γ2 = θγ1 + θγ2for paths γ, γ1 and γ2. In particular, if γ is a cycle, θγ does not depend on the initial pointof γ.Consider the spaces F defined to be the closure of the span of the stars and ♦ as theclosure of the span of the cycles (both closures in the Hilbert space `2−(E; r)). We have thefollowing proposition.Proposition ( C.1.7 ) In the Hilbert space `2−(E; r), we have F ⊥ ♦.Proof. Since the inner product is continuous and bilinear, suffices to show that any star isperpendicular to any cycle. If θ =∑e−=vc(e)χe is a star and η =m∑j=1χej is a cycle, then(θ|η)`2−(E;r) =12∑(e,j,f)r(f)χe(f)χej(f),where the sum is over all triples (e, j, f) satisfying e− = x, j = 1, . . . ,m and f ∈ E. Weshow now that if χe(f)χej(f) 6= 0 for some triple (e, j, f) then there second triple such thatthe sum of the two corresponding terms is zero. Suppose we have χe(f)χej(f) 6= 0. In thiscase, either f = e or f = −e, in the first case either ej = e = f or −ej = e = f, and inthe second case ej = f = −e or −ej = f = −e. This gives rise to four cases, however allof them are handled similarly so we will explain just the first case. Then ej = e = f andso χe(f)χej(f) = 1; consider the triple (−ej−1, f, ej) (if j = 1, consider (−em, f, e1)), whichis valid since e+j−1 = x or, equivalently, −e−j−1 = x (if j = 1, −e−m = x) and for this tripleχ−ej−1(f)χej(f) = −1. This concludes the proof.170Appendix C. Electrical networks and the space of flows§ C.2. The divergence and gradient on a network.Similarly to `2−(E; r), one can construct another space `2(V; c). This space is by definition,L 2 (V; c) where c is now the measure on all subsets of V (recall V is denumerable) givenaccording to the rulec(A) =∑x∈Ac(x),where c(x) is the conductance at x. The usual assumptions on the network provide that eachc(x) is finite and positive. Therefore, `2(V; c) is a Hilbert space with the inner product andnorm given by(f |g)`2(V;c) =∑x∈Vc(x)f(x)g(x), and ‖f‖2`2(V;c) =∑x∈Vc(x)f(x)2.We mention once and for all that if V is finite, then `2(V; c) = RV; if c(V) <∞, then `2(V; c)contains all bounded functions. If f ∈ RV, then(C.2.1) e 7→ ∇f(e) = c(e) Äf(e−)− f(e+)ädefines a flow, called the gradient of the function f. We will be interested in gradients withfinite energy. The number(C.2.2) DΓ (f) = D (f) = E (∇f)is known as Dirichlet energy of f. A function with finite Dirichlet energy is called a Dirichletfunction. We defineD = {f : V→ R | D (f) <∞}HD = {f ∈ D | f is harmonic}.(C.2.3)One way to think about D is as the inverse image of `2−(E; r) under the function∇ : RV → RE,equivalently, D is the largest subspace Z in RV for which the restriction of ∇ to Z has rangeinside `2−(E; r). The next annex will deal with the study of the spaces D and HD. In thisannex, these spaces will be used tangentially.Proposition ( C.2.4 ) We have `2(V; c) ⊂ D; in fact, if f in any function from V into R,then D (f) ≤ 2 ‖f‖2`2(V;c) .Proof. From the elementary inequality (a+ b)2 ≤ 2(a2 + b2), we deduceE (∇f) = ∑e∈E12r(e) (∇f(e))2 ≤∑e∈Ec(e)Äf(e+)2 + f(e−)2ä= 2∑e∈Ec(e)f(e−)2,171Appendix C. Electrical networks and the space of flowswhere the last equality used the fact that c(e) = c(−e) and (−e)− = e+. But now,∑e∈Ec(e)f(e−)2 =∑x∈V∑e−=xc(e)f(e−)2 = ‖f‖2`2(V;c) .This concludes the proposition.As a corollary of the previous proposition is that D = RV whenever V is finite.The function∇ : RV → RE given by∇(f) = ∇f is sometimes referred to as the “gradientoperator”; it is a linear function. By definition of D, ∇(D) ⊂ `2−(E; r) and we just showedthat ∇ : `2(V; c)→ `2−(E; r) satisfies ‖∇f‖`2−(E;r) ≤√2 ‖f‖`2(V;c) ; in other words, the linearfunction ∇ has “operator norm”, when viewed from `2(V; c) into `2−(E; r), bounded fromabove by ≤ √2.Consider now the subspace of θ ∈ RE such that for every vertex x, the family (θ(e))e−=xis summable. With this assumption on θ, we may define div θ : V→ R by(C.2.5) div θ : x 7→ c(x)−1 ∑e−=xθ(e),which is known as the divergence of θ. As with the gradient, the divergence defines a linearfunction from the stated subspace of RE into RV (the “divergence operator”). We show nowthat the divergence has domain containing `2−(E; r) and that div (`2−(E; r)) ⊂ `2(V; c), thatis, the divergence of a flow with finite energy defines a function in `2(V; c). Both of theseclaims follow from the Cauchy-Schwarz inequality. If θ ∈ `2−(E; r), then for any vertex x,∑e−=x|θ(e)| = ∑e−=x»c(e)∣∣∣»r(e)θ(e)∣∣∣≤ ∑e−=xc(e) 12  ∑e−=xr(e)θ(e)2 12 ≤ c(x) 12 ‖θ‖`2−(E;r) <∞.Next,‖div θ‖2`2(V;c) =∑x∈Vc(x)−1 ∑e−=xθ(e)2 = ∑x∈Vc(x)−1 ∑e−=x»c(e)»r(e)θ(e)2≤ ∑x∈Vc(x)−1 ∑e−=xc(e) ∑e−=xr(e)θ(e)2=∑x∈V∑e−=xr(e)θ(e)2 = 2∑e∈E12r(e)θ(e)2 = 2 ‖θ‖2`2−(E;r) .172Appendix C. Electrical networks and the space of flowsThus, it is possible to define div : `2−(E; r)→ `2(V; c) and then the divergence is continuousand has finite operator norm bounded from above by√2, just as the gradient. We summariseboth operator norms in the following point(C.2.6) ‖∇f‖`2−(E;r) ≤√2 ‖f‖`2(V;c) and ‖div θ‖`2(V;c) ≤√2 ‖θ‖`2−(E;r) .There is an important relation between the gradient and the divergence.Proposition ( C.2.7 ) The gradient and the divergence operators are adjoint of each other,that is to say, for every f ∈ `2(V; c) and every θ ∈ `2−(E; r),(div θ|f)`2(V;c) = (θ|∇f)`2−(E;r) .Proof. Recall that Cauchy-Schwarz inequality shows both sides in the previous equation areabsolutely summable. By definition,(div θ|f)`2(V;c) =∑x∈V∑e−=xθ(e)f(x) =∑e∈Eθ(e)f(e−);now e 7→ −e is a bijection of E onto itself, and so∑e∈Eθ(e)f(e−) =∑e∈Eθ(−e)f(e+) = ∑e∈E−θ(e)f(e+).Therefore,∑e∈Eθ(e)f(e−) =∑e∈E12θ(e)Äf(e−)− f(e+)ä = ∑e∈E12r(e)θ(e)∇f(e) = (θ|∇f)`2−(E;r) ,which is the desired conclusion.We continue with the following simple remark.Remark ( C.2.8 ) For a function h : V → R to be harmonic at x, it is necessary andsufficient that div ∇h(x) should be zero.Proof. To see this, simply expand,(div ∇h)(x) = ∑e−=xc(x)−1∇h(e) = ∑e−=xc(e)c(x)Äh(e−)− h(e+)äand notice that e+ is adjacent to e− = x, hence, the condition div ∇h(x) = 0, signifies thath(x) =∑e−=xc(e)c(x)h(e+),which is exactly the definition of harmonic function at x bearing in mind the transitiondensity kernel arising from a network.173Appendix C. Electrical networks and the space of flowsProposition ( C.2.9 ) Suppose θn defines a sequence of flows in `2−(E; r) converging simply(edgewise) to another such flow θ. If there exists a constant M > 0 such that E (θn) ≤ M,then θn → θ weakly, in other words, for every continuous linear form u : `2−(E; r) → R, thesequence of real numbers u(θn)→ u(θ). In particular, E (θ) ≤M as well.Proof. The hypothesis E (θn) ≤ M states that θn is a sequence defined in the closed ball Bof radius M12 in `2−(E; r), the latter being a separable Hilbert space (C.1.4) shows, invokingBanach-Alaoglu theorem, that the former is a compact metrisable space. Thus, θn is asequence defined in the metrisable compact space B, as such, θn has limit points in B. Ifη ∈ B is such a limit point, η(e) = limn→∞ θn(e) = θ(e) by (C.1.3.1). This implies that θn hasa unique limit point θ in the metrisable compact space B, thus θn → θ in B. That is to say,θn → θ weakly.§ C.3. Potentials and currents: Ohm’s law, Kirchhoff’scycle law and some consequences.A function f : V → R is a potential for θ : E → R if θ = ∇f, this is also expressed bysaying that “θ admits f as a potential” or that θ is the current defined by f. It is immediatethat the only functions θ : E → R that can have potentials are the flows, and since we areinterested in the case where θ ∈ `2−(E; r), then f will necessarily be a Dirichlet function.We mention once and for all that if θ ∈ `2−(E; r) and c(e) = 0, then θ(e) = ∇f(e) = 0 forany function f : V → R. Thus, to check that a function is a potential for a flow suffices toconsider edges with positive conductance.Remark ( C.3.1 ) By definition, the currents θ are the flows satisfying that for some functionf : V→ R and all edges e with c(e) > 0, r(e)θ(e) = f(e−)− f(e+) (“Ohm’s law”).The following is also an easy proposition.Proposition ( C.3.2 ) The relation θ ∈ ♦⊥ is equivalent to “θ admits a Dirichlet functionas a potential”, thus ♦⊥ is the set of flows in `2−(E; r) that are currents. In other words, thecurrents are the flows that are orthogonal to the cycles (“Kirchhoff’s cycle law”).Proof. To prove this remark we may assume that c(e) > 0 for every edge e. First startnoticing that if θ admits a potential f ∈ D then θ = ∇f and since the inner product iscontinuous, suffices to show that ∇f is orthogonal to every cycle (C.1.6). Bearing in mind(C.1.4.2),(∇f |θγ)`2−(E;r) =Ñ∇f∣∣∣∣∣∣m∑j=1χejé`2−(E;r)=m∑j=1r(ej)∇f(ej) =m∑j=1Äf(e−j )− f(e+j )ä= 0174Appendix C. Electrical networks and the space of flowssince γ = (ej) is a cycle. For the inverse implication, fix any vertex o in V. For any othervertex x, consider any path γ with γ− = x and γ+ = o, and set f(x) = (∇f |θγ)`2−(E;r) . Thisf is well-defined because of the condition θ ∈ ♦⊥ for if γ1, γ2 are two paths from x to o, thenγ1 ∨ (−γ2) is a cycle and therefore the inner product of f with the flow this cycle creates iszero. Then0 =Ä∇f ∣∣∣θγ1∨(−γ2)ä`2−(E;r) = (∇f |θγ1)`2−(E;r) − (∇f |θγ2)`2−(E;r) ,showing the claim. The function f is a potential for θ, for if e is any edge and γ is a pathfrom e− to o, the path −e ∨ γ starts at (−e)− = e+ and finishes at o, thereforef(e−)− f(e+) = r(e)θ(e),equivalently, ∇f(e) = θ(e) as desired.The previous remark connects the function ∇ with the space ♦⊥. In a similar fashion,we can connect div with F⊥. Recall that, by definition, div θ(x) = c(x) (θ|sx)`2−(E;r) is aconstant times the scalar product of θ with a star sx =∑e−=xc(e)χe (C.1.5). The followingproposition follows immediately.Proposition ( C.3.3 ) The condition θ ∈ F⊥ is the same as div θ = 0, in other words,F⊥ is the kernel of div : `2−(E; r)→ `2(V; c).§ C.4. The orthogonal decomposition of `2−(E; r).By virtue of (C.1.7), we have the following preliminary orthogonal decomposition`2−(E; r) =F⊕ ♦⊕ÄF⊥ ∩ ♦⊥ä .This is so since in every Hilbert space we have (U + V)⊥ = U⊥ ∩V⊥ for subspaces U and V.To understand better F⊥ ∩ ♦⊥ we need to introduce another two Hilbert spaces. Using theresults of the previous section, we can characterise of F⊥ ∩ ♦⊥.Theorem ( C.4.1 ) ((9.6), [LP16]) For any network Γ satisfying the usual assumptions, thespace of flows with finite energy has the following orthogonal decomposition`2−(E; r) =F⊕ ♦⊕∇HD.Proof. As mentioned in the paragraph preceding the proposition, the result reduces to show-ing ∇HD = ♦⊥ ∩F⊥. By propositions (C.3.2) and (C.3.3), ♦⊥ ∩F⊥ is the space of flowsthat admit a potential and whose divergence is zero, hence, it consists of the flows that canbe written as θ = ∇f and for which div ∇f = 0, that is (C.2.8) f belongs to HD.175Appendix C. Electrical networks and the space of flowsWe have done the orthogonal decomposition of `2−(E; r) but we have not mentioned yet asto why this is useful. The usefulness of this decomposition lies in the following proposition,whose proof is rather lengthy and since the result is well-known, we only provide a referenceThe result originally appeared in [BLPS01] and it can be found in [LP16, Proposition 10.14],although, the proof is done using several results from chapters 2, 4, 9 and 10 in this reference.Theorem ( C.4.2 ) (Theorem 7.4, [BLPS01]) Let Γ be a network satisfying the usual as-sumptions. Assume further that the underlying graph of this network is simple. A necessaryand sufficient condition for the free and wired spanning forest measures on Γ to coincide isthat ∇HD = {0}.The previous theorem connects intimately the measures FSF and WSF with the Dirichletharmonic functions of a network. We shall investigate further properties of these functionsin Annex D. We also remark that the condition of multiple edges being banned is actuallyirrelevant when proving ∇HD = {0}. The reason of this is proved later, see §C.8If we assume further that G has a finite vertex set, we can completely characterise `2−(E; r)(regardless of E being infinite or not).Proposition ( C.4.3 ) Any harmonic function on a network satisfying the usual assumptionand whose vertex set is finite is constant; in particular, for a such a network,`2−(E; r) =F⊕ ♦.Proof. This follows immediately from the maximum principle (3.1.3) since the finitenesshypothesis implies any harmonic functions attains a maximum.The previous proposition, together with the theorem preceding it, shows at once thatFSF = WSF for any finite graph network (where multiple edges are banned). This is hardlysurprising since there is no need to take weak limits to define these measures in this case.§ C.5. Voltages and their currents: effective conduc-tance and resistance.Consider two sets of vertices A and Z, and assume these sets share no vertex in common.A voltage (with respect to the set A ∪ Z) is a function f : V → R such that f is harmonicoutside A ∪ Z.Voltage functions always exist whatever the values on A and Z may be, provided thesevalues are bounded, as the existence principle shows (3.1.3). Having fixed values on A and176Appendix C. Electrical networks and the space of flowsZ, the voltage functions are determined on the entirety of the network provided V \ (A ∪ Z)is finite, by the uniqueness principle of the same reference.The reader should keep in mind the following scenario whenever we are studying voltagefunctions. The nodes of A are connected to a battery which is exercising some energy (energyenters through A) and the nodes at Z are grounded (the energy leaves the network throughZ).For this section, we assume A and Z are two subsets of vertices such that the complementof their union is finite.Proposition ( C.5.1 ) Let f be any voltage function relative to A∪ Z assuming value zeroon Z and otherwise values on A bounded by some constant α > 0. A sufficient condition forf to belong to `2(V; c) is that A should be an integrable set (meaning that c(A) < ∞); thiscondition is also necessary if f is bounded below by another positive constant β. When Ais integrable, ∇f ∈ `2−(E; r), that is, the currents defined by voltages (as stated) have finiteenergy provided c(A) <∞.Proof. The conclusion ∇f ∈ `2−(E; r) follows from the fact that ∇(`2(V; c)) ⊂ `2−(E; r)(C.2.4). Now, by definition,∑x∈Vc(x)f(x)2 =∑x∈Ac(x)f(x)2 +∑x∈(A∪Z){c(x)f(x)2 ≤ α2c(A) + ∑x∈(A∪Z){c(x)f(x)2.The term∑x∈(A∪Z){c(x)f(x)2 is always a finite sum, thus the sum∑x∈Vc(x)f(x)2 is finite when-ever c(A) <∞. In a similar manner, if f ≥ β > 0 on A, then∑x∈Vc(x)f(x)2 ≥ ∑x∈Ac(x)f(x)2 ≥ β2c(A),showing that A is integrable whenever f ∈ `2(V; c)z C.5.1 Voltages from a point to a set.Consider then the case A = {a} and Z any set of vertices such that a /∈ Z and the complementon V of {a} ∪ Z is finite. If f is any voltage function such that f(x) = 0 for each x ∈ Z,then P (τa < τZ | S0 = x) = f(x)f(a)is the probability that the network random walk will enterZ before visiting a, this follows by the superposition and uniqueness principles. A simplecalculation now shows thatPaÄτZ < τ+aä=1f(a)c(a)∑e−=a∇f(e).177Appendix C. Electrical networks and the space of flowsIn other words, since f = 0 on Z, any voltage difference between a and Z will satisfyf(a) =Ç1c(a)Pa (τZ < τ+a )å ∑e−=a∇f(e),and by definition,∑e−=a∇f(e) is the current entering at a. From an electrical network per-spective (Ohm’s law), this says that one can think of the entire network Γ = (G, c) as asingle wire between a and Z having conductance c(a)Pa (τZ < τ+a ) . We can therefore definethe effective conductance from a to Z to be(C.5.2) CΓ (a,Z) = C (a,Z) = c(a)PaÄτZ < τ+aä.From definition, we reach the following (by (C.5.1), ∇f ∈ `2−(E; r)).( C.5.2.1 ) When f is a voltage function relative to {a} ∪ Z such that f(a) = 1 and f = 0on Z, then C (a,Z) =∑e−=a∇f(e) = (∇f |sa)`2−(E;r) .Next, as usually is done, we define the effective resistance from a to Z as follows(C.5.3) RΓ (a,Z) = R (a,Z) = C (a,Z)−1 .With these definitions, we have shown thatf(a) = R (a,Z)∑e−=a∇f(e).In particular,( C.5.3.1 ) When f is a voltage relative to {a} ∪ Z such that ∑e−=a∇f(e) = 1 and f = 0 onZ, then R (a,Z) = f(a). Furthermore, f(x) = R (a,Z)Px (τa < τZ) .Reduction of the grounded set to a point.There is another natural way to express effective conductance and resistance from a pointa into a set Z. Start with a network Γ which satisfies the usual assumptions and constructΓ′ by first contracting Γ by the edges of Z and then deleting all self-loops based at Z in thecontracted network. Then, Γ′ = (G′, c′) satisfies the usual assumptions and it has a finitevertex set. Indeed, the only non obvious assumption that Γ′ satisfies is that c′(Z) <∞. Butsince we deleted all self-loops based at a vertex in Z,c′(Z) =∑e−∈Ze+ /∈Zc(e) ≤∑x′c′(x′) <∞,178Appendix C. Electrical networks and the space of flowswhere x′ 6= Z runs on the vertices of Γ′. We will show that the function x 7→ Px (τa < τZ)does not depend in the choice of network random walk between that of Γ and that of Γ′for x outside Z. Indeed, denote by ϕ(x) the number given by “the probability of visiting abefore hitting Z for a network random walk on Γ started at x” and by ϕ′(x′), “the probabilityof visiting a before visiting Z for a network random walk on Γ′ started at x′.” If p and p′denote the transition kernels of the network random walks of Γ and Γ′, respectively, thenp(x, y) = p′(x, y) for all x, y /∈ Z while p′(x,Z) = ∑z∼xp(x, z), where the sum runs on allvertices z ∈ Z adjacent to x in Γ. Thus, we may induce canonically a function ϕ˜′ : V → Rby means of ϕ′ by setting ϕ˜′(x) = ϕ′(x) for x /∈ Z and ϕ˜′(x) = 0 for x ∈ Z. It is easy to seethat ϕ˜′ is harmonic outside {a} ∪ Z and has the same boundary conditions as ϕ, uniquenessthen show that ϕ˜′ = ϕ and thus ϕ′ = ϕ outside Z. We established the following result.Proposition ( C.5.4 ) Let Γ be a network satisfying the usual assumptions and construct Γ′as before: first contract the graph underlying Γ by the edges of Z and then delete all self-loopsbased at Z in the contracted graph. The function x 7→ Px (τa < τZ) , for x /∈ Z, is the samefor the network random walks on Γ and Γ′.Observe that we have reduced the definition of C (a,Z) and R (a,Z) to a network withfinitely many vertices.Green’s function as voltage.We now show yet another way of writing voltages, this time using Green’s function of thenetwork. Recall from (2.4.1) the definition of GZ{ , which is the Green’s function restrictedto leaving the set Z{, or in this case, the Green’s function restricted to entering the set Z.By (2.4.2),c(u)GZ{(u, v) = c(v)GZ{(v, u).By definition, GZ{(u, v) is the expected number of visits to v, from a network random walkstarted at u before hitting Z. By the strong Markov property at time η = min (τZ , τv) ,GZ{(u, v) = EuÑESηÑ ∑m∈Z+1{Sm=v,τZ>m}éé= EuÑ1{η=τv}ESτvÑ ∑m∈Z+1{Sm=v,τZ>m}éé,the second equality follows from the fact that if η = τZ, the inner expected value is thenzero. Now, it is easy to see that ESτvÑ ∑m∈Z+1{Sm=v,τZ>m}éis a constant random variable179Appendix C. Electrical networks and the space of flowswith value GZ{(v, v). Thus,GZ{(u, v) = Pu (τv < τZ)GZ{(v, v).Started at S0 = v, the random number∑m∈Z+1{Sm=v,τZ>m} counts the number of times (“fail-ures to hit Z”) the random walk returns to v before hitting Z. It follows that its law isa geometric distribution with probability of success Pv (τZ < τ+v ) . Then, its expectation,namely GZ{(v, v), is Pv (τZ < τ+v )−1. In other words, we proved that(C.5.5) GZ{(u, v) =Pu (τv < τZ)Pv (τZ < τ+v ).Next, first step analysis show thatx 7→ GZ{(x, a)is a harmonic function outside {a}∪Z, assumes the value Pa (τZ < τ+a )−1 at a and it is equalto zero on Z. Therefore, it is the unique voltage with these properties. Henceforth, we haveestablished the following.Proposition ( C.5.6 ) Let f : V→ R be the functionf : x 7→ c(x)−1GZ{(a, x) = c(a)−1GZ{(x, a).Then f is the unique voltage relative to {a} ∪ Z with valuesf(a) =1c(a)Pa (τZ < τ+a )= R (a,Z)f |Z = 0.Furthermore, this function satisfies that its current entering at a is one:∑e−=a∇f(e) = 1.Proof. There only remains to show the last claim, but because of the uniqueness of thevoltage function, and by (C.5.3.1), the last claim follows.Because of the last proposition, we may call ∇f the unit current flow from a to Z.180Appendix C. Electrical networks and the space of flowsz C.5.2 Voltages between two sets.Let us pass now to a more general A, that is, A may have more than one point. We continueto assume V\(A∪Z) to be finite, so that voltages (relative to A∪Z) are still unique knowingtheir values on A∪Z. Keeping in mind (C.5.1), we assume A to be an integrable set, that isto say, c(A) < ∞. In this case, any voltage function relative to A ∪ Z, assuming value zeroon Z will be a Dirichlet function. It is easier to extend the definition (C.5.2.1). If f is thevoltage with value one on A and zero on Z, then the effective conductance from A to Z isdefined to be(C.5.7) C (A,Z) = CΓ (A,Z) =∑a∈A∑e−=a∇f(e),and, of course, the effective resistance from A to Z is R (A,Z) = RΓ (A,Z) = C (A,Z)−1 .(Observe that each value∑e−=a∇f(e) is non negative, so the sum defining the effective con-ductance is always defined.)There is another, perhaps more natural, way to express C (A,Z) and R (A,Z) . DefineΓ′ to be the network obtained from G by identifying A to a point, that is, the contractionof G by the edges of A. We claim that CΓ (A,Z) = CΓ′ (A,Z) . Denote by f ′ the functionequal to f on V \ A and f ′(A) = 1. The integrability of the set A shows at once that∑(e′)−=Ac(e) =∑a∈Ac(a) <∞, and the results for the case in which A had one point apply forΓ′. Then, f ′ is the voltage on Γ′ with value one at the vertex A of Γ′ and zero on the set Z ofthe same network. By the foregoing, ∇f ′ ∈ `2−(E; r) (recall that Γ and Γ′ possess the sameedge set and the same resistance function). Then,1 = RΓ′ (A,Z)∑(e′)−=A∇f ′(e) = 1CΓ′ (A,Z)∑e−∈A∇(f ′)(e)=1CΓ′ (A,Z)∑a∈A∑e−=a∇f(e) = CΓ (A,Z)CΓ′ (A,Z),proving the claim. Similarly,(C.5.8) RΓ (A,Z) = f′(A)where f : V→ R is the voltage function on G such that f is constant on A, f = 0 on Z andthe voltage function f ′ on G′, defined to be f ′ = f on V \A and f ′(A) is the constant valueof f on A, is such that∑(e′)−=A∇f ′(e′) = ∑a∈A∑e−=a∇f(e) = 1.As before, we may call ∇f the unit current flow from A to Z.181Appendix C. Electrical networks and the space of flowsRemark ( C.5.9 ) The argument on the last paragraph shows that, in fact, whenever A isintegrable, C (A,Z) <∞. We also mention that, up to this point in this Annex, we have notdefined C (A,Z) for c(A) =∞.z C.5.3 Flux and Kirchhoff’s node law.When defining effective conductance (C.5.2), the quantity∑e−=a∇f(e) = (∇f |sa)`2−(E;r) (wheresa is the star at a) was referred to as the “current entering at a.” In general, if θ is a flowwith finite energy, we will say that(C.5.10) c(x)div θ(x) = (θ|sx)`2−(E;r) =∑e−=xθ(e)is the the (net) flux at x; the (net) flux over a set is the sum of the flux at each point inthe set. A flow has sources at those x where the flux is positive and sinks where the flux isnegative. A flow θ is said to have conservation of flux if the family (θ(e))e∈E is absolutelysummable and∑x∈V(θ|sx)`2−(E;r) = 0.For the rest of this section we assume A and Z to be disjoint sets of vertices,A is integrable (c(A) < ∞) and the complement of A ∪ Z is finite. With thisconditions, every voltage function with constant value on A and zero on Z is aDirichlet function (C.5.1).We can now state the second law due to Kirchhoff.Proposition ( C.5.11 ) Let f be a voltage relative to A∪Z. If θ is the current arising fromthe voltage f, then at every x /∈ A ∪ Z, the flux of θ is zero (“Kirchhoff’s node law”);Proof. By definition θ = ∇f, f being harmonic at every x /∈ A∪Z. The result follows readilyfrom (C.2.8).Proposition ( C.5.12 ) Suppose that c(V) <∞ (this condition always holds true wheneverV is finite by the usual assumptions). All currents arising from voltages have conservationof flux. In fact, if θ is the current arising from a voltage relative to A ∪ Z, then∑a∈A(θ|sa)`2−(E;r) = −∑z∈Z(θ|sz)`2−(E;r) .Proof. The claim follows from Kirchhoff’s node law and an easy calculation. Since V isan integrable set, all constant functions are also integrable, this implies that any constant182Appendix C. Electrical networks and the space of flowsfunction belongs to `2(V; c). Next, Kirchhoff’s node law states that (θ|sx)`2−(E;r) = 0 forx /∈ A∪Z. With all of this in mind, (and also recalling div and ∇ are adjoint to each other)∑x∈A∪Z(θ|sx)`2−(E;r) =∑x∈V(θ|sx)`2−(E;r) = (div θ|1)`2(V;c) = (θ|∇1)`2−(E;r) = 0since the gradient of every constant function is zero.Proposition ( C.5.13 ) Let f be a voltage relative to A ∪ Z and let θ be its current. Ifg : V→ R is any function such that g = 1 on A and g = 0 on Z, then(θ|∇g)`2−(E;r) =∑a∈A(θ|sa)`2−(E;r) ,is the net flux entering through A by θ.Proof. Observe that∑x∈Vc(x)g(x)2 =∑x∈Ac(x) +∑x/∈A∪Zc(x)g(x)2 = c(A) +∑x/∈A∪Zc(x)g(x)2 <∞,since c(A) < ∞ and the sum appearing in the right hand side of the equation containsfinitely many terms. Thus, g ∈ `2(V; c). By (C.2.7),(θ|∇g)`2−(E;r) = (div ∇f |g)`2(V;c) =∑a∈Ac(a)div ∇f(a),since g = 0 on Z and div ∇f = 0 on the complement of A∪ Z by Kirchhoff’s node law. Theconclusion follows from (C.5.10).Proposition ( C.5.14 ) Let f be a voltage relative to A ∪ Z and let θ be its current.Suppose f = 0 on Z and it is constant on A chosen so that θ has unit flux over A. ThenE (θ) = R (A,Z) . In other words, the unit current flow from A to Z has energy the effectiveresistance from A to Z.Proof. Suppose λ is the constant value of f on A, by (C.5.8), R (A,Z) = λ. On the otherhand, by (C.5.13) the third equality below followsE (θ) = ‖∇f‖2`2−(E;r) = λÄ∇f ∣∣∣∇λ−1fä`2−(E;r)= λ∑a∈A(∇f |sa)`2−(E;r) = λ,as was claimed.Corollary ( C.5.14.1 ) The function f : V→ R given byf(x) = c(x)−1GZ{(x, o) = c(o)−1GZ{(o, x),which is the unique voltage relative to A ∪ Z, having constant value on A, being zero on Zand with unit flux on A, satisfies D (f) = E (∇f) = f(o) = R (o,Z) .183Appendix C. Electrical networks and the space of flows§ C.6. Dirichlet’s and Thompson’s principles.In this section we assume A and Z to be disjoint sets of vertices, A is integrable(c(A) <∞) and the complement of A ∪ Z is finite. With these conditions, everyvoltage function with constant value on A and zero on Z is a Dirichlet function(C.5.1).Proposition ( C.6.1 ) Assume Γ is a network satisfying the usual assumptions. Assumefurther that the vertex set of Γ is finite (the edge set of Γ possibly infinite). Let f be theunique voltage relative to A∪ Z, being constant on A, zero on Z and such that its associatedcurrent flow θ = ∇f has unit flux on A. Then,R (A,Z) = infσ∈`2−(E;r)div θ=div σE (σ) ,that is, the effective resistance from A to Z is the minimal energy E (σ) amongst all flowsσ ∈ `2−(E; r) that have the same flux as the unit current flow θ (“Thompson’s principle”).Proof. In this case `2−(E; r) = F ⊕ ♦ by (3.1.3). Consider the orthogonal projection from`2−(E; r) onto F. Suppose σ ∈ `2−(E; r) satisfies div θ = div σ. By definition, σ− θ belongs tothe kernel of this projection, that is, belongs to F⊥ = ♦. Since θ is a current, θ ∈ ♦⊥ =F.Then, σ = θ + (σ − θ) is the orthogonal decomposition of σ in `2−(E; r). Thus (C.5.14)‖σ‖2`2−(E;r) = ‖θ‖2`2−(E;r)+ ‖σ − θ‖2`2−(E;r) ≥ ‖θ‖2`2−(E;r)= R (A,Z) .The result now follows.Proposition ( C.6.2 ) The effective conductance from A to Z is the minimal Dirichlet en-ergy amongst the functions assuming value one on A and zero on Z (“Dirichlet’s principle”).In symbolsC (A,Z) = infϕ=1 on Aϕ=0 on ZD (ϕ) .Furthermore, there exists an optimiser function ϕ, indeed, ϕ is the voltage function (relativeto A ∪ Z) assuming value one on A and zero on Z.Proof. Consider ϕ = f the voltage function assuming value one on A and zero on Z. Set λto be flux of θ over A. Then (C.5.7) and (C.5.13) (applied to g = f)C (A,Z) = λ = D (f) .184Appendix C. Electrical networks and the space of flowsThis shows that C (A,Z) is an upper bound of the infimum of the quantities D (ϕ) for ϕ asstated; there remains to show C (A,Z) ≤ D (ϕ) . Suppose now ϕ is any function satisfyingϕ = 1 on A and ϕ = 0 on Z. Then ϕ ∈ `2(V; c) (since ‖ϕ‖2`2(V;c) = c(A) +∑x/∈A∪Zc(x)ϕ(x)2,and the sum has finitely many terms). The Cauchy-Schwarz inequality shows(∇ϕ|∇f)`2−(E;r) ≤ D (ϕ)12 D (f)12 = D (ϕ)12 C (A,Z)12 .On the other hand, since div and ∇ are adjoint of each other, and using (C.5.13) withg = ϕ,(∇ϕ|∇f)`2−(E;r) = λ = C (A,Z) .Comparing one reaches C (A,Z) ≤ D (ϕ) .Observe that, in order for the proof of Thompson’s principle to go forward, we need toassume Γ had a finite vertex set, this contrast highly with the more benevolent hypothesis ofDirichlet’s principle (only A integrable was needed). The reason of this contrast lies in thefact that if A is integrable, any bounded function on A and zero on Z belongs to `2(V; c),while the decomposition of `2−(E; r) = ♦ ⊕F does not always occur. It is true, however,that `2−(E; r) = ♦⊕F⊕∇HD for any network satisfying the usual assumptions. This pointsto two possible generalisation of Thompson’s principle in which we consider projection onto♦⊥ or onto F (recall that ♦⊥ is the subspace of currents by (C.3.2)). We will return to thispoint later.Finally, we have that the effective resistance or conductance are symmetric under some-what restrictive hypothesis. We need that V should be an integrable set (c(V) <∞). Indeed,we have been handling only the case where A was integrable, since any voltage was assumedfrom A to Z (Z was thought as being grounded), as such, the voltages assumed value zero onZ always. But if now, we want voltages from Z to A, then we need to assume Z to be inte-grable as well. This leads to both A and Z to be integrable sets, and since the complementof their union is finite, the whole V is integrable. We finally mention that if a set A is notintegrable, the concept of what C (A,Z) should mean is awkward (we are in the situation inwhich a battery exercises infinite energy on A).Proposition ( C.6.3 ) If V is integrable (c(V) < ∞), then R (A,Z) = R (Z,A) andC (A,Z) = C (Z,A) .Proof. By Dirichlet’s principle, there exists an optimiser ϕ such that D (ϕ) = C (A,Z) .Then, 1−ϕ is optimal for C (Z,A) . Taking reciprocals (C.5.9) we obtain the case for effectiveconductances.185Appendix C. Electrical networks and the space of flows§ C.7. Rayleigh’s monotonicty law.Consider the network Γ = (G, c) which satisfies the usual assumptions and suppose c′ isanother assignment of conductances on G satisfying c(e) ≤ c′(e) for every edge e. Naturally,is easy to believe that CΓ (A,Z) ≤ CΓ′ (A,Z) . This is, however, not that obvious from aprobabilistic lens since changing a conductance of a single edge (say by doubling it) doesnot have a proportional effect in the probability of using that edge (it will not double theprobability) and worse, it is not at all easy to see how this affects the voltage function. Wewill use Dirichlet’s principle to show that an increase in conductance leads to an increase ineffective conductance. First notice that Γ′ will automatically satisfy the usual assumptions.Next, let A and Z be two disjoint subset of vertices such that (A ∪ Z){ is finite and A isintegrable for both assignments of conductances (c(A), c′(A) < ∞). Suppose ϕ (resp. ϕ′)is an optimal function for CΓ (A,Z) (resp. CΓ′ (A,Z)). Then, ϕ(x) = ϕ′(x) = 1 for x ∈ Aand ϕ(x) = ϕ′(x) = 0 for x ∈ Z. Also, Dirichlet’s principle shows the first of the followinginequalities and c ≤ c′, the secondCΓ (A,Z) = DΓ (ϕ) ≤ DΓ (ϕ′) ≤ DΓ′ (ϕ) = CΓ′ (A,Z)Thus, we have shown the following result.Proposition ( C.7.1 ) Let Γ = (G, c) be a network satisfying the usual assumptions andsuppose c′ is a second assignment of conductances on G such that c ≤ c′. Denote Γ′ = (G, c′),which clearly satisfy the usual assumptions. If A and Z are two disjoint subset of vertices,such that A has finite conductance for both assignments c and c′, and the complement ofA ∪ Z is finite, then CΓ (A,Z) ≤ CΓ′ (A,Z) (“Rayleigh’s monotonicity law”).§ C.8. Common network reductions.Here we are going to show a few ways to make a given network simpler and such that thevoltages are not affected by these changes. We are mostly interested in the intuition behindall these network reductions.Cutting of an edge.Intuitively, this means that a given edge e is no longer in the network. Mathematically, thiscan be translated to either making its conductance 0 or else constructing a new networkwith the edge deleted. We will show that both constructions leads to the same effectiveconductances. Suppose then that Γ = (G, c) is any network satisfying the usual assumptions186Appendix C. Electrical networks and the space of flowsand that e0 is an edge of Γ. Consider the following two networks Γ′ = (G′, c′) which is thenetwork induced by Γ on E\{±e0} (here E is the edge set of G); this corresponds to deletinge0 (and its opposite). Consider also Γ0 = (G, c0) in which we define c0(e) = c(e) for alle 6= ±e0 and c0(±e0) = 0; this corresponds to setting the conductance of ±e0 to zero. First,notice that Γ′ will satisfy the usual assumptions if and only if Γ0 will do the same. Second,the network random walks on Γ′ and Γ0 have the same transition probabilities. Assume thenthat Γ′ and Γ0 satisfy the usual assumptions. If A and Z are two disjoint set of vertices suchthat c(A) <∞ and the complement of their union is finite, then, for a function ϕ : V→ R tobe a voltage on Γ′ relative to A∪ Z (with ϕ|A = 1 and ϕ|Z = 0) it is necessary and sufficientthat ϕ should be a voltage on Γ0 relative to A ∪ Z. Furthermore,CΓ′ (A,Z) = CΓ0 (A,Z) .Also, by Rayleigh’s monotonicity law, cutting edges reduces conductance (assuming that theedge cut leaves a network satisfying the usual assumptions).Shorting of an edge.This correspond to putting a superconductor between two nodes, so that there is now zeroresistance between them. Mathematically this can be thought as either taking the limit ofthe conductance to infinity, or else, constructing a new network where the end points of theedge have been identified to a single vertex. Again, we shall show that both procedures reachthe same value for the effective conductance. Construct Γ′ from Γ by contracting edge e0 andalso set Γλ = (G, cλ), where cλ(e) = c(e) if e 6= e0 and cλ(e0) = λ. Obviously, both Γ′ andΓλ will satisfy the usual assumptions since Γ does. Suppose A and Z are as in the previousparagraph and assume further that e0 is not adjacent to both A and Z simultaneously. Denoteby ϕ′ the voltage function on Γ′ relative to A ∪ Z being one on A and zero on Z, and by ϕλthe corresponding voltage of Γλ having the analogous properties. Observe that ϕ′ inducescanonically a function f on Γλ by setting f(x) = ϕ′(x) for x /∈ {e−0 , e+0 } and f(e−0 ) = f(e+0 )is the value of ϕ′ at the vertex which is the end points of the contracted edge e0. ClearlyDΓλ (f) = DΓ′ (ϕ′) and Dirichlet’s principle also givesCΓλ (A,Z) = DΓλ (ϕλ) ≤ DΓλ (f) = DΓ′ (ϕ′) = CΓ′ (A,Z) .We are going to show that, in fact, limλ→∞CΓλ (A,Z) = CΓ′ (A,Z) . Rayleigh’s monotonicitylaw shows that CΓλ (A,Z) is increasing in λ and since it is bounded, it converges to somenumber, which we will show is CΓ′ (A,Z) via some subsequence. The maximum principle(3.1.3) gives that 0 ≤ ϕλ ≤ 1, thus (ϕλ(x))x∈V ∈ [0, 1]V, and the latter space is a metrisable187Appendix C. Electrical networks and the space of flowscompact space since V is denumerable. Then, there exists a subsequence, also denoted byϕλ, such that for every vertex x, ϕλ(x)→ ϕ∞(x) ∈ [0, 1]. Furthermore,λÄϕλ(e−0 )− ϕλ(e+0 )ä2 ≤ DΓλ (ϕλ) ≤ DΓ′ (ϕ′) <∞;thus we may assume more, that λÄϕλ(e−0 )− ϕλ(e+0 )ä2converges to some non negative numberζ ≤ DΓ′ (ϕ′) (by perhaps passing yet through another subsequence). A fortiori, ϕ∞(e+0 ) =ϕ∞(e−0 ) and thus, ϕ∞ can be thought as a function on the network Γ′. Consider now ψλ :E→ R given byψλ(e) =c(e)2(ϕλ(e−)− ϕλ(e+))2 if e 6= ±e0,λ2(ϕλ(e−)− ϕλ(e+))2 if e = ±e0,so thatDΓλ (ϕλ) =∑e∈Eψλ(e),and ψλ converges simply toψ(e) =c(e)2(ϕ∞(e−)− ϕ∞(e+))2 if e 6= ±e0,ζ2if e = ±e0.We may now apply Fatou’s lemma and Dirichlet’s principle, to obtainζ +DΓ′ (ϕ∞) = ζ +∑e6=e0ψ(e) ≤ lim · infλ→∞∑e∈Eψλ(e)= lim · infλ→∞DΓλ (ϕλ) ≤ DΓ′ (ϕ′) ≤ DΓ′ (ϕ∞) .This chain of inequalities show that ζ = 0 and then thatDΓ′ (ϕ′) = DΓ′ (ϕ∞) ≤ lim · infλ→∞DΓλ (ϕλ) ≤ lim · supλ→∞DΓλ (ϕλ) ≤ DΓ′ (ϕ′) .Thus,DΓ′ (ϕ′) = DΓ′ (ϕ∞) = limλ→∞DΓλ (ϕλ) .The claims have been substantiated. Furthermore, by Rayleigh’s monotonicity principle,shorting an edge increases effective conductance.188Appendix C. Electrical networks and the space of flowsConductors in parallel.Assume that between two nodes x and y there is a family of edges connecting them, say(ej) and all e−j = x, e+j = y (the possibility of an infinite denumerable family not excluded).Construct a new network Γ′ = (G, c′) by setting c′(e) = c(e) for all e 6= ej and creating a newedge e′ such that (e′)− = x, (e′)+ = y, and making its conductances to be c′(e′) =∑jc(ej) ≤c(x) < ∞. For a function f to be a voltage relative to Γ it is necessary and sufficient thatit should be a voltage relative to Γ′. This is obvious from the definition of the transitionprobability of the network random walk. Furthermore, since voltages do not change, theeffective conductances and resistances from a set to another can be calculated using any ofthe two networks.Remark ( C.8.1 ) It is by virtue of this reduction, that when studying uniform spanningtrees or forests often one may impose the restriction of every network being a graph, that is,to ban multiple edges. In other words, to assume that multiple edges do not occur imposesno restrictions to any result governing uniform spanning forests.We can go a further in a particular case. Suppose Γ is a network satisfying the usualassumptions with a finite vertex set. Suppose that x and y are neighbours and suppose thatwe split all edges connecting them into two disjoint families (ej) and (kl). Construct a newnetwork Γ′ by replacing all the edges ej with a new edge e′ with conductance c′(e′) =∑jc(ej),as before; this time, however, we leave the edges kl intact. If ϕ is any function such thatϕ(x) = 1 and ϕ(y) = 0, thenDΓ (ϕ) =∑lc(kl)(ϕ(x)− ϕ(y))2 +DΓ′ (ϕ) =∑lc(kl) +DΓ′ (ϕ) .Taking the infimum over ϕ such that ϕ(x) = 1 and ϕ(y) = 0 leads to(C.8.2) CΓ (x, y) =∑lc(kl) + CΓ′ (x, y) .Resistors in series.As usual, consider a pair of disjoint sets A and Z such that the complement of their unionis finite and c(A) < ∞. Assume now that there is a path γ = (vj)0≤j≤m such that forj = 1, . . . ,m − 1, the only neighbours of vj in Γ are vj−1 and vj+1, assume further that novertex in the path uses any vertex in A∪ Z. Suppose that ej are the unique edges satisfyinge−j = vj−1 and e+j = vj for j = 1, . . . ,m. Construct a new network Γ′ withV′ = V \ {v1, . . . , vm−1}E′ = (E \ {±e1, . . . ,±em}) ∪ {±e′},189Appendix C. Electrical networks and the space of flowswhere e′ is a new edge satisfying (e′)− = v0 and (e′)+ = vm. Set the resistances r′(e) fore ∈ E′ to ber′(e) =r(e) if e 6= ±e′m∑j=1r(ej) if e = ±e′.If f is a voltage function on Γ relative to A ∪ Z, assuming value zero on Z, then f ′ = f |V′ isa voltage on Γ′ relative to A∪Z satisfying the same boundary conditions as f. By induction,suffices to consider the case m = 2. It is then clear that f ′ has the appropriate boundaryconditions on A and on Z, so there remains to check it is harmonic outside A∪Z. Let θ′ = ∇f ′(which is a flow in Γ′) and θ = ∇f (a flow in Γ). It is easy to check that ζ = θ(e1) = θ(e2)with the given assumptions, and then the choice of resistance r′ impliesθ′(e) =θ(e) for all e 6= ±e′ζ for e = ±e′.That f ′ is harmonic outside A ∪ Z signifies that θ is orthogonal to every star s′x of Γ′ for allx /∈ A ∪ Z. Clearly,(θ′|s′x)`2−(E′,r′) = (θ|sx)`2−(E;r) = 0if x 6= v0 and x 6= v2; if, x = v0, then (C.1.4.2)Äθ′∣∣∣s′v0ä`2−(E′,r′) = ∑e−=v0e6=e′c(e) (θ|χe)`2−(E;r) + c′(e′)Äθ′∣∣∣χe′ä`2−(E′;r′)=∑e−=v0e6=e′θ(e) + c′(e′)r′(e′)θ(e′)=∑e−=v0e6=e′θ(e) + θ(e1) =∑e−=v0θ(e) = (θ|sv0)`2−(E;r) = 0;the case of v2 is similar. Thus, by uniqueness, f′ is the claimed harmonic function of Γ′.Furthermore, since voltages do not change, the effective conductance between sets does notchange either.Pruning.This corresponds to deleting a portion of the graph that does not affect whatsoever theeffective conductance from a set to another. Obviously, there must be some geometricalsetting for this to be doable. We will say that a set of vertices B is a bush if there exists avertex b ∈ B obeying the following: for every vertex o /∈ B and any path γ with γ− = o and190Appendix C. Electrical networks and the space of flowsγ+ ∈ B, b is a vertex of γ. When B is a bush, B˚ = B \ {b}; this follows from the fact that ifx ∈ B and y ∼ x is not in B, then x = b (the only points in B allowed to have neighboursoutside B is the special vertex b).In other words, a bush is a set B such that the only way to connect one of its vertices withits complement is by passing through a special vertex b ∈ B. What pruning states is that wemay delete bushes from the network if these bushes do not contain vertices connected neitherto the battery (in A) nor grounded (in Z). Formally, suppose A and Z are two disjoint subsetof vertices, such that the complement of their union is finite and c(A) <∞. Let B be a bushcontaining no vertex of Aor Z. (Notice B is finite.) Suppose b ∈ B is the vertex satisfyingthe definition of bush. Define G′ = (V′,E′), where V′ = (V \ B) ∪ {b} and E′ = E \ E|B,where E|B are the edges of B (both end points in B). (Observe that the edges adjacent tob and a vertex outside of B remain intact.) Set c′ = c|E′ and Γ′ = (G′, c′). First we showΓ′ satisfies the usual assumptions. The only non obvious assumption is connectedness. Infact, Γ′ is connected, for if o and x are vertices of Γ′ that cannot be connected via a pathin Γ′, then there exists a minimum length path γ in Γ with γ− = o and γ+ = x. Clearly γmust use edges of B, thus b is a vertex of γ, say b is the jth vertex of γ, and the (j + 1)thvertex, say z, must be in B. By minimality, b 6= z, thus, z ∈ B˚. By induction, it is easy tosee that that all subsequent (after z) vertices in γ also belong to B˚, so x = γ+ ∈ B˚, which isabsurd. Thus, Γ′ is connected. Now, let u : A→ R be a bounded function. If f is the uniquevoltage function on Γ relative to A ∪ Z, such that f(x) = u(x) for x ∈ A and f = 0 on Z,then f ′ = f |V′ is the unique voltage function on Γ′ satisfying the same properties. First, weclaim that f |B is a constant function with value f(b). This follows easily from the maximumprinciple applied to f and −f on the set B˚. Indeed, B is finite, so f attains a maximum anda minimum on B = B˚ which must be at points in ∂B˚ = {b}. Thus, the maximum and theminimum of f coincide and f is constant as claimed. Next, we check that f ′ is harmonicrelative to Γ′ at every vertex in V′ \ (A∪ Z). It is obvious that this will be the case at everyx ∈ V′ \ (A∪Z) with x 6= b since we have not change any neighbour nor conductance aroundx. There remains to show f ′ is harmonic at b. Notice that c′(b) =∑e−=be+ /∈Bc(e). Harmonicity off at b givesf(b) =∑e−=bc(e)c(b)f(e) =∑e−=be+ /∈Bc(e)c(b)f(e) + f(b)∑e−=be+∈Bc(e)c(b)Some algebra givesf ′(b) = f(b) =∑e−=be+ /∈Bc(e)c′(b)f(e) =∑e−=be+ /∈Bc′(e)c′(b)f ′(e),191Appendix C. Electrical networks and the space of flowsshowing f ′ is harmonic at b (relative to Γ′). Furthermore, the effective conductance betweenA and Z is the same in the two networks.Collapse of zero current edges.Suppose e0 is an edge were there is no current flowing, that means that the voltage at its endpoints is the same. Thus, we are in a similar situation of that of shorting, except that thistime we know in advance that the voltage is the same at the end points of the conductor. Weare going to show that if we contract the edge, then delete it from the network, no effectiveconductance or resistance gets affected by this. Suppose Γ is a network satisfying the usualassumptions and let A and Z as always: disjoint, the complement of their union is finiteand A is integrable. Let f be any voltage function relative to A ∪ Z, assuming value zeroon Z and having values bounded on A. Suppose e0 is any edge such that f(e−0 ) = f(e+0 ).Construct the network Γ′ by first contracting Γ through e0 and then deleting e0. Denote byz the vertex of Γ′ that replaced the end points of e0. Induce a function f ′ on the vertices ofΓ′ canonically using f, that is to say, f ′(x) = f(x) for x 6= z and f ′(z) = f(e−0 ) = f(e+0 ).Then, f ′ is the unique voltage function on Γ′ satisfying the same boundary conditions asf ; furthermore, DΓ′ (f ′) = DΓ (f) . First, the conclusion regarding Dirichlet’s energies isobvious since ∇f(e0) = 0. Next, we show f ′ is harmonic relative to Γ′ outside A ∪ Z.This is obvious at every vertex of Γ′ except z. If c′ denotes the conductances of Γ′, thenc′(z) = c(e−0 ) + c(e−0 )− 2c(e0). Hence,c′(z)f ′(z) =Äc(e−0 )− c(e0)äf(e−0 ) +Äc(e+0 )− c(e0)äf(e+0 )=∑e−=e−0e 6=e0c(e)f(e) +∑e−=e+0e6=−e0c(e)f(e)=∑(e′)−=zc′(e′)f(e′),where the last sum refers to the edges of Γ′. This shows that f ′ is harmonic at z, hence theresult.Remark ( C.8.3 ) Every self-loop has zero current. It is because of this reason hat oftenself-loops are banned in electrical networks; they contribute nothing to effective conductancesor resistances. They also can never appear in trees, and so they are also often banned whenstudying uniform spanning trees or forest.192Appendix C. Electrical networks and the space of flowsDeletion of zero-conductance conductors.If a network Γ, satisfying the usual assumptions, has an edge e0 which has zero conductance,then we may delete it from the network and its deletion will have no effect whatsoever neitherin the usual assumptions of Γ nor on effective resistances nor on effective conductancesbetween sets (taken as usual). This is obvious from Dirichlet’s principle.Remark ( C.8.4 ) It is because of this reason that often zero-conductance conductors arebanned to appear in the definition of a network. However, we saw earlier that cutting anedge can be thought as simply setting the conductance of an edge to zero, so banning themand then letting them to have conductance zero is rather awkward.A final comment on network reductions.Because of remarks (C.8.1), (C.8.3) and (C.8.4) it is often assumed the a given network sat-isfies that its assignment of conductances never assumes value zero, and that the underlyinggraph of the network is simple (multiple edges and self-loops banned)§ C.9. Effective resistance and conductance to infinity:Doyle and Snell’s criterion.Consider a vertex, which is assumed fixed throughout this section, o of the network Γ (whichis assumed to satisfy the usual assumptions). Because of the usual assumptions on Γ, thenetwork random walk defines an irreducible Markov chain (every state is accessible fromany other state). Then, a necessary and sufficient condition for the network random walk(Sn)n∈Z+ to be transient is that Po (τ+o =∞) > 0. Consider then an exhaustion (Gn) ofG and let Vn the vertex set of Gn. We assume Vn to be finite and defining an increasingsequence. By considering only large values of n, we may assume o ∈ V1. Set τn = τZn to bethe hitting time of the complement Zn of Vn in V; then τn →∞ as n→∞ a.s. On the eventS0 = o, {τ+o = ∞} =⋂n∈N{τn < τ+o } and the family of events ({τn < τ+o })n∈N is decreasingsince τn+1 ≥ τn. Therefore,PoÄτ+o =∞ä= limn→∞PoÄτn < τ+oä= limn→∞C (o,Zn)c(o);furthermore, the left-hand side is independent of the choosing of exhaustion, so the right-hand side limit exists whatever the exhaustion may be. We define therefore the effective193Appendix C. Electrical networks and the space of flowsconductance from o to infinity to be(C.9.1) C (o,∞) = CΓ (o,∞) = limn→∞C (o,Zn) = c(o)PoÄτ+o =∞ä,where Zn is as above. The effective resistance from o to infinity is defined as(C.9.2) R (o,∞) = RΓ (o,∞) = C (o,∞)−1 ,with the convention that ∞−1 = 0. The foregoing have established the following well-knowncriterion.Proposition ( C.9.3 ) (Doyle and Snell, 1984) Let Γ be a network satisfying the usualassumptions. In order for the network random walk of Γ to be transient, it is necessary andsufficient that for some vertex o, its effective conductance to infinity should satisfy C (o,∞) >0; when this is the case, C (x,∞) > 0 for all vertices x (“Doyle and Snell’s criterion”).A consequence of Rayleigh’s monotonicity law and Doyle and Snell’s criterion is thefollowing.Proposition ( C.9.4 ) Suppose Γ = (G, c) is a network satisfying the usual assumptionsand suppose c′ is a second assignment of conductances on the edges of Γ such that c ≤ c′.Denote Γ′ = (G, c′). Then Γ′ also satisfies the usual assumptions. Further, for any vertex o,CΓ (o,∞) ≤ CΓ′ (o,∞) . In particular, if the network random walk of Γ is transient, so is thenetwork random walk on Γ′.§ C.10. Flows to infinity: Terry Lyons’ criterion.Start with a network Γ satisfying the usual assumptions and we may assume that all zero-conductance edges have been deleted from Γ (C.8.4). In other words, we may assume thatc(e) > 0 for every edge e of Γ. As in the previous section, consider a vertex o of G. If Z is a setnot containing o and whose complement is finite, then x 7→ c(x)−1GZ{(o, x) = c(o)−1GZ{(x, o)is the unique voltage function with value R (o,Z) at o and zero on Z, see (C.5.6). Bydefinition, GZ{(x, o) is the expected number of times a random walk started at x will visito before hitting Z. Let (Vn) define an increasing sequence of finite subset of vertices suchthat o ∈ V1 and set Zn = V{n. Assume further that the network induced on Vn by that ofΓ satisfies the usual assumptions. Consider then Sx a network random walk started at x.DefineIxn(o) =∑m∈Z+1{Sxm=o,τZn (Sx)>m},194Appendix C. Electrical networks and the space of flowswhich is the number of visits to o by the random walk Sx before hitting Zn. By definition,GVn(x, o) = E (Ixn(o)) .As n → ∞, the stopping times τZn(Sx) → ∞, and an application of the monotone conver-gence theorem shows that E (Ixn(o)) → E (Ix(o)) , where Ix(o) is the total number of visitsto o by Sx. In other words, what we showed waslimn→∞GVn(x, o) = G(x, o).If fn(x) = c(x)−1GVn(o, x) = c(o)−1GVn(x, o) for x ∈ V, then fn is the unique voltagefunction with value zero on Zn and value R (o,Zn) at o (C.5.6). We just showed thatfn(x) → f(x) = c(x)−1G(x, o) and by definition f(o) = R (o,∞) . Also, the superpositionprinciple shows thatfn(x)fn(o)is the unique harmonic function such that assumes value one at oand zero on Zn, and thusfn(x)fn(o)= Px (τo < τZn) . Henceforth,f(x)f(o)= Px (τo <∞) , a fortiorif is harmonic at every x 6= o.For every n, construct Γn from Γ as follows:(a) contract Γ by the edges of Zn;(b) delete self-loops based at Zn in the contracted network.Write En for the edge set of Γn and cn for its conductance. Observe Γn has a finite vertex set.Then, Γn satisfies the usual assumptions. The only non obvious of these is that cn(Zn) <∞,but since we deleted all self-loops at Zn,cn(Zn) =∑e−∈Zne+ /∈Znc(e) ≤ ∑x∈Vnc(x) <∞since the last sum only has finitely many terms. Furthermore, the quantitiesPx (τo < τZn)do not depend on the choice between the network random walk of Γ or that of Γn for x /∈ Zn,by (C.5.4). Let θn = ∇fn be the gradient of fn relative to Γn and let θ = ∇f be the gradientof f relative to Γ. Construct yet another auxiliary network Γn,aux as the network induced onVn by Γ with all edges connecting two points in ∂Vn ⊂ Zn deleted; equivalently, Γn,aux isthe network induced on Vn by that of Γ with all vertices in ∂Vn and the edges connectingvertices between Vn and ∂Vn added. Then, Γn,aux satisfies the usual assumptions since we195Appendix C. Electrical networks and the space of flowshad assumed c(e) > 0 for every edge e of Γ. Let σn = θ|En , which can be thought as thegradient relative to Γn,aux of the restriction of f to Vn. If cn,aux is the conductance of Γn,auxthen cn,auxÄVnä<∞. Indeed,cn,auxÄVnä= c(Vn) + cn,aux(∂Vn) = c(Vn) +∑e−∈∂Vne+∈Vnc(e) = c(Vn) + cn(Zn) <∞.We can also think of σn as a flow in Γn. We claim that div θn(x) = div σn(x) for all verticesx of Γn. We know that div θn(o) = 1 by (C.5.6). Extend θn to all of E by sending zerocurrent on the extension. Then, θn converges simply to θ in Γ. Observe that (C.5.14) showsEΓ (θn) = R (o,Zn) → R (o,∞) . If R (o,∞) < ∞, then EΓ (θn) is bounded by a constantM > 0 independent of n, thus (C.2.9) shows that θn → θ weakly. Then, 1 = (θn|so)`2−(E;r) →(θ|so)`2−(E;r) and so div θ(o) = 1. Observe that as soon as a vertex x belongs to Vn, all edgesadjacent to x will belong to En, thus div σn(o) = 1. If x is a vertex of Γn that is neithero nor Zn, then div θn(x) = 0 by Kirchhoff’s node law and similarly div σn(x) = 0. Byconservation of flux (C.5.12) div θn(Zn) = −1 and div σn(Zn) =∑x∈Zndiv σn(x) = −1, wherediv σn(Zn) refers to the divergence of σn relative to Γn and div σn(x), relative to Γn,aux.Thus, Thompson’s principle shows thatEΓ (θn) = EΓn (θn) ≤ EΓn (σn) = EΓ (σn) ≤ EΓ (θ) ≤ lim · infn→∞ EΓ (θn) = R (o,∞) .We have established the following: if R (o,∞) < ∞ then there exists a flow θ such thatdiv θ = 1o, and E (θ) = R (o,∞) <∞.Reciprocally, suppose that there exists a flow θ with div θ = 1o and E (θ) <∞. ConstructΓn and Γn,aux as before and let θn the unique current flow arising from a voltage having unitflux at o and set σn = θ|En . As before, div θn = div σn and we may extend θn to all of Γ bysending zero current through the extension. Thus, Thompson’s principle and (C.5.14) givesR (o,Zn) = EΓ (θn) = EΓn (θn) ≤ EΓn (σn) ≤ EΓ (θ) .We have shown that: if there exists a flow θ in Γ with div θ = 1o and E (θ) < ∞, thenR (o,∞) <∞.We have established the following celebrated criterion for transience of the network ran-dom walk on a network bearing in mind Doyle and Snell’s criterion.Proposition ( C.10.1 ) [Lyo88] Let Γ be a network satisfying the usual assumptions. Inorder for the network random walk of Γ to be transient, it is necessary and sufficient that forsome vertex o there should exists a flow θ such that div θ = 1o and E (θ) <∞ (“T. Lyons’196Appendix C. Electrical networks and the space of flowscriterion”). Furthermore, when T. Lyons’ criterion holds, there exists a function f : V→ Rsuch that, with θ = ∇f,f(x)f(o)= Px (τo <∞) , x ∈ Vf(o) = E (θ) = R (o,∞) .§ C.11. Variational definitions of R (o,∞) and C (o,∞) .Analogously to Dirichlet’s and Thompson’s principles, we can state a “variational defini-tions” for the effective conductance or resistance from a vertex o to ∞. The work done inthe previous section to prove T. Lyons’ criterion open the path to start with Thompson’sprinciple.Proposition ( C.11.1 ) Let Γ be a network satisfying the usual assumptions. Then,R (o,∞) = infθ∈`2−(E;r)div θ=1oE (θ) ;(“Thompson’s principle”). Furthermore, there exists one and only flow attaining this infi-mum.Proof. As stated earlier, the work is already done in the previous section, there it was shownthat (with notation of that section) if θ ∈ `2−(E; r) satisfies div θ = 1o, thenR (o,Zn) ≤ E (θ) ,thus R (o,∞) ≤ E (θ) and the minimum is attained by the flow θ of (C.10.1). There onlyremains to show that the minimiser is unique. However, in any Hilbert space there holdsthe parallelogram law‖x‖2 + ‖y‖22=∥∥∥∥x+ y2∥∥∥∥2 + ∥∥∥∥x− y2∥∥∥∥2 .Thus, if x and y lie inside a convex subset C of a Hilbert space and they both simultaneouslysatisfy ‖x‖ = ‖y‖ = infz∈C‖z‖ then x = y, as is seen immediately from the parallelogram lawsincex+ y2∈ C by convexity. Applying this result to the convex set C = {σ ∈ `2−(E; r) |div σ = 1o}, the uniqueness of the flow with minimal energy follows.The unique flow θ that attains the minimum in Thompson’s principle for R (o,∞) willbe called unit current flow from o to ∞.To motivate a variational definition of Dirichlet’s principle, consider a sequence of sets(Zn) as in the definition of effective conductance, see (C.9.1). Then, applying Dirichlet’s197Appendix C. Electrical networks and the space of flowsprinciple (C.6.2),C (o,∞) = limn→∞C (o,Zn) = limn→∞ infϕ(o)=1ϕ=0 on ZnD (ϕ) .Since the complement of Zn is finite by hypothesis, the natural guess is that(C.11.2) C (o,∞) = infϕ(o)=1ϕ has finite supportD (ϕ) .The proof of this equality follows immediately upon realising that the union, as n runsthrough N, of the sets of functions ϕ : V → R such that ϕ(o) = 1 and ϕ = 0 on Zn is theset of functions ϕ : V→ R with finite support such that ϕ(o) = 1.§ C.12. Free and wired effective resistance and conduc-tance between points.The concept of C (A,Z) have been defined only for special cases: A was assumed to havefinite conductance (c(A) <∞), the two sets were disjoint and the voltage function definingC (A,Z) was assumed to have value zero on Z, rendering Z essentially null. With theseassumptions, it was seen that we may think of C (A,Z) as the effective conductance in afinite network where the edges of both A and Z had been contracted and the self-loops atZ had been deleted from the network. Thus, we have always dealt only with the conceptof C (a, z) , between two points in a finite network. In this section, we want to generalisethe concept C (a, z) to “truly infinite networks.” The natural way to do this is to do somelimiting process. We will help ourselves with Thompson’s principle, as opposed to Dirichlet’s.We are going to motivate the definition of R (a, z) in infinite networks. Recall from(C.4.1) that for any network Γ satisfying the usual assumptions,`2−(E; r) = ♦⊕F⊕∇HD.Also, Thompson’s principle gives, for a network with finitely many vertices (C.6.1)R (a, z) = minσ∈`2−(E;r)div σ=div θE (σ) ,where θ is the unit current flow from a to z.What is important here is how the proof proceeded.Since θ is a current, it belongs to ♦⊥ = F (C.3.2) and it was seen that θ = pr♦⊥σ = prFσ,for any σ as stated (here prU is the orthogonal projection onto the subspace U). Finally,198Appendix C. Electrical networks and the space of flowsfrom (C.1.4.2) and assuming c(e) > 0,(χe|sx)`2−(E;r) = r(e)sx(e) =1 if e− = x,−1 if e+ = x,0 otherwiseBy linearity, if γ is a path of edges with γ− = a and γ+ = z, then div θγ = div θ (C.1.6).Thus, we have: if Γ has a finite vertex set and γ is a path of edges with initial point aand terminal point z, then pr♦⊥θγ = prFθγ is the unique minimiser in Thompson’s principle.Thus, in what follows we will concentrate in extending the definition of R (e−, e+) for a givenedge e.Suppose now that Γ is a network satisfying the usual assumptions and let Vn be anincreasing sequence of finite sets of vertices such that their union is the vertex set of Γ.Assume further that the networks induced on Vn by that of Γ satisfy the usual assumptions.Let e be any edge of E such that c(e) > 0 and assume that the end points of e already belongto V1. As usual, set Zn = V{n. Construct the following two networks:(a) Γn is the network induced on Vn.(b) ΓWn is the network obtained from Γ after having contracting it through the edges ofthe complement of Vn and then deleting self-loops at the contracted point, which wewill call Zn. It easily follows that ΓWn satisfies the usual assumptions.Suppose En is the edge set of Γn. Since Vn is finite, `2−(En; r) = ♦n ⊕Fn. We know thatpr♦⊥nχe = χe − pr♦nχe is the unique minimiser in Thompson’s principle. Furthermore, everycycle in Γn is also a cycle in Γ, thus ♦n can be canonically regarded as a subspace of`2−(E; r). Since Vn is increasing, the subspaces ♦n also define an increasing sequence whoseunion contains all cycles of Γ, thus⋃♦n = ♦ with the closure taken in `2−(E; r). Then,∥∥∥pr♦⊥nχe − pr♦⊥χe∥∥∥`2−(E;r) =∥∥∥pr♦nχe − pr♦χe∥∥∥`2−(E;r) → 0.(One way to see this is true is by considering Hilbert bases Hn of ♦n such that Hn ⊂ Hn+1.In this setting, a Hilbert basis of ♦ is H =⋃nHn, and the projections satisfy:pr♦nχe =∑γ∈Hn(γ|χe)`2−(E;r) γ,pr♦χe =∑γ∈H(γ|χe)`2−(E;r) γ,199Appendix C. Electrical networks and the space of flowswith the family ((γ|χe)`2−(E;r) γ)γ∈H summable, see [Ash72, Sect. 3.2].). Thus, we showedthat pr♦⊥nχe → pr♦⊥χe in `2−(E; r). In particular, for any edge f and recalling evaluation iscontinuous (C.1.3.1)pr♦⊥χe(f) = limn→∞ pr♦⊥nχe(f),Äpr♦⊥χe∣∣∣sxä`2−(E;r)= limn→∞Äpr♦⊥nχe∣∣∣sxä`2−(E;r)= 1e−(x)− 1e+(x)EÄpr♦⊥χeä= limn→∞EÄpr♦⊥nχeä= limn→∞RΓnÄe−, e+ä.Define the following flowsθFe = pr♦⊥χeθFa,z = pr♦⊥θγ(C.12.1)where γ is any path from a to z in Γ. (This obviously gives a definition of θFa,z that does notdepend on the choice of path for if γ and γ′ were paths from a to z, then γ ∨ (−γ′) would acycle, hence it would belong to ♦.) The previous arguments justify to call this flow the unitcurrent flow along e or from a to z with free boundary conditions or more simply the unitfree current along e or from a to z. We define the free effective resistance along e or betweena and z, to beRFΓÄe−, e+ä= EÄθFeä= limn→∞RΓnÄe−, e+ä.RFΓ (a, z) = EÄθFa,zä= limn→∞RΓn (a, z) .(C.12.2)We remark once and for all that RFΓ (a, z) = RFΓ (z, a) for any pair of vertices a, z, sinceθFa,z = −θFz,a.In the network ΓWn we proceed similarly. Suppose EWn is is edge set; by definition, EWn ⊂ E.Its vertex set is VWn = Vn∪{Zn} which is finite. Thus, we may write `2−ÄEWn ; rä= ♦Wn ⊕FWn ,where ♦Wn is the closure of the span of the cycles in `2−ÄEWn ; rä, and FWn that of the starsin `2−ÄEWn ; rä. Here we observe that as soon as a vertex x is in Vn, all the neighbours of xare vertices of ΓWn . Thus, if x is a vertex of Vn, then the star at x relative to Γ is also thestar at x relative to ΓWn . Thus, FWn is identified canonically with a subspace of F. And asbefore,⋃nFWn = F, the closure in `2−(E; r). The arguments for the unit free current nowshow θWe = prFχe is the limit in `2−(E; r) of prFWn χe. It satisfies, with f any edge of Γ,prFχe(f) = limn→∞ prFWn χe(f),ÄprFχe∣∣∣sxä`2−(E;r)= limn→∞ÄprFWn χe∣∣∣sxä`2−(E;r)= 1e−(x)− 1e+(x)EÄprFχeä= limn→∞EÄprFWn χeä= limn→∞RΓWnÄe−, e+ä.200Appendix C. Electrical networks and the space of flowsDefineθWe = prFχeθWa,z = prFχe,(C.12.3)where γ is any path in Γ from a to z. (Again, F ⊂ ♦⊥, so this definition does not dependon γ.) It is because of the arguments given that we may call these flows the unit currentflow (along e or between a and z) with wired boundary conditions or simply the unit wiredcurrent (along e or between a and z). We define the wired effective resistance along e orbetween a and z to beRWΓÄe−, e+ä= EÄprFχeä= limn→∞RΓWnÄe−, e+ä.RWΓ (a, z) = EÄpr♦⊥θγä= limn→∞RΓWn (a, z) .(C.12.4)We remark once and for all that RWΓ (a, z) = RWΓ (z, a) for any pair of vertices a, z becauseθWa,z = −θWz,a. Furthermore, θWa,z is the unique flow satisfying(C.12.5) EΓÄθWa,zä= minσ∈`2−(E;r)div σ=1a−1zEΓ (σ) ,(“Thompson’s principle for wired effective resistance”). The proof of this is easy, notice thatdiv θWa,z = 1a − 1z, this implies that if σ is as stated, then σ − θWa,z ∈ ker div = F⊥, andsince θWa,z ∈ F, it follows that σ = θWa,z +Äσ − θWa,zäis the orthogonal decomposition of σ in`2−(E; r) =F⊕F⊥, showing that E (σ) > EÄθWa,zäunless σ = θWa,z.We finally define the free effective conductance between a and z to be CW (a, z) =RF (a, z)−1 and the wired effective conductance, CW (a, z) = RW (a, z)−1 .Proposition ( C.12.6 ) Suppose Γ = (G, c) is a network satisfying the usual assumptions.Let a and z be any two of its vertices. Let c′ be a second assignments of conductances suchthat c ≤ c′ and denote Γ′ = (G, c′). ThenCWΓ (a, z) ≤ CWΓ′ (a, z) ,C FΓ (a, z) ≤ C FΓ′ (a, z) ,(“Rayleigh’s monotonicity law”).Proof. By definition and with obvious notations,CWΓ (a, z) = limn→∞CΓWn (a, z) ≤ limn→∞C(Γ′n)W (a, z) = CWΓ′ (a, z) .A verbatim argument shows the case for free effective resistance.201Appendix C. Electrical networks and the space of flowsProposition ( C.12.7 ) Let a and z be two vertices of Γ, which is a network assumed tosatisfy the usual hypotheses. ThenRW (a, z) ≤ R (a,∞) +R (z,∞) .Proof. We first will show the triangle inequality for effective resistances in a network withfinitely many vertices. So, assume first Γ has finitely many vertices and let x, y, z be anythree of its vertices. Suppose θx,y and θy,z are the unit current flows from x to y and from y toz, respectively. Then, θx,y +θy,z is a flow satisfying div θ(v) = 1x(v)−1z(v). By Thompson’sprinciple,R (x, z) ≤ E (θx,y + θy,z) = (θx,y + θy,z|θx,y + θy,z)`2−(E;r)= E (θx,y) + E (θy,z) + 2 (θx,y|θy,z)`2−(E;r)= R (x, y) +R (y, z) + 2 (θx,y|θy,z)`2−(E;r) .To finish showing the triangle inequality for effective resistances in the finite-vertex setting,we will show that(θx,y|θy,z)`2−(E;r) ≤ 0.To see this last inequality recall that θx,y = ∇ϕx,y with ϕx,y the respective voltage functionssatisfying adequate boundary conditions. Then, we can apply the adjointness between ∇and div , and Kirchhoff’s node law,(θx,y|θy,z)`2−(E;r) = (ϕx,y|div θy,z)`2(V;c) = ϕ(y)− ϕ(z) ≤ 0,where the last inequality is a consequence of the maximum principle. This established thetriangle inequality for effective resistances in the finite-vertex setting.To prove the proposition, consider as usual an increasing sequence (Vn) of sets of verticessuch that Vn ↑ V and denote by ΓWn the network obtained from Γ by contracting all edgesof the complement Zn of Vn and then deleting self-loops based at Zn. Then, ΓWn have finitelymany vertices and by the triangle inequality and (C.6.3)RΓWn (a, z) ≤ RΓWn (a,Zn) +RΓWn (z,Zn) .Letting n→∞ shows the result bearing in mind the definition of effective resistance from anode to infinity.Proposition ( C.12.8 ) Suppose Γ is a network satisfying the usual assumptions. Supposethat x and y are neighbours and suppose that we split all edges connecting them into two202Appendix C. Electrical networks and the space of flowsdisjoint families (ej) and (kl). Construct Γ′ by replacing all edges (ej) with a new edge e′having conductance c′(e′) =∑jc(ej) <∞. ThenCWΓ (x, y) =∑lc(kl) + CWΓ′ (x, y)C FΓ (x, y) =∑lc(kl) + CFΓ′ (x, y)Proof. This follows from (C.8.2) bearing in mind the definitions of wired and free effectiveconductance as limits over networks with finitely many vertices.203Appendix DThe spaces D and HDWe study further the spaces D and HD (C.2.3) in order to provide conditions under which∇HD = {0}. Since we are going to be investigating properties of harmonic functions, recallthe definition of the transition operator (3.1.1) and its main properties (3.1.2).Recall from Annex C what does it mean for a network Γ = (G, c), G = (V,E), to satisfythe usual assumptions. Fix such a network throughout this Annex; no other networks willbe considered.Most of the work in this chapter is based on [Bar17] and [Soa94] with the necessaryadaptations to our case; in particular, we do not assume locally finiteness of the underlyinggraph.§ D.1. The topological spaces D and HD.Consider a vertex o of Γ. We introduce the following inner product on D which depend on o,(f |g)o = f(o)g(o) + (∇f |∇g)`2−(E;r) ,‖f‖2o = (f |f)o = f(o)2 +D (f) .(D.1.1)It is easy to check that (f, g) 7→ (f |g)o defines an inner product, except perhaps for the detailin proving that (f |f)o = 0 is sufficient for f = 0. To see this observe that (f |f)o = 0 impliesf(o) = 0 and ∇f = 0, the second of these relations implies f(e+) = f(e−) for every edge ewith c(e) > 0. Since the random walk is irreducible, for every vertex v, there exists a pathγ of positive-conductance edges with γ− = o and γ+ = v, thus f(o) = f(v). Incidentally, weshowed204Appendix D. The spaces D and HDProposition ( D.1.2 ) For a function f : V → R to be constant it is necessary andsufficient that D (f) = 0. Furthermore, if f is constant and ϕ : V → R is any function,D (f + ϕ) = D (ϕ) .The previous inner product depends on the choosing of the vertex o, however, the topologyit defines is independent of the choice of o. To see this, consider any other vertex o′. We willshow that there exist constants c, C > 0 such that for every function f ∈ D, we havec ‖f‖o ≤ ‖f‖o′ ≤ C ‖f‖o .This will prove that the two normed spaces (D, ‖·‖o) and (D, ‖·‖o′) possess the same topology.By symmetry, suffices to show the inequality ‖f‖o′ ≤ C ‖f‖o . To show it, consider a minimallength path (ej)1≤j≤m from o to o′ whose edges all have positive conductances (this is possibledue to irreducibility of the random walk). By minimality, the edges are not repeated. TheCauchy-Schwarz inequality applied to the coefficients 1,»r(e1), . . . ,»r(em), givesf(o′)2 =f(o) + m∑j=1»r(ej)(»c(ej)Äf(e+j )− f(e−j )ä)2≤1 + m∑j=1r(ej)f(o)2 + m∑j=1c(ej)Äf(e+j )− f(e−j )ä2≤1 + m∑j=1r(ej) îf(o)2 +D (f)ó = c ‖f‖2o .This inequality is important in its own.Proposition ( D.1.3 ) For each vertex v ∈ V there exists a constant c = cv > 0 suchthat |f(v)| ≤ c ‖f‖o , that is to say, the function f 7→ f(v) from D into R is linear andcontinuous. Thus, evaluations on D are continuous linear forms.We are also interested in HD. From (C.2.8), HD is the set of function f such thatdiv ∇f = f − Pf = 0.Since ∇ can be thought as a function D→ `2−(E; r) and div : `2−(E; r)→ `2(V; c), we reachthat div ◦ ∇ : D → `2(V; c) and HD is the kernel of this linear function. Furthermore, by(C.2.6), we know‖div ∇f‖`2(V;c) ≤√2 ‖∇f‖`2−(E;r) =√2D (f) .We have established the following proposition.205Appendix D. The spaces D and HDProposition ( D.1.4 ) Denote by ID the identity function in D. Then ID − P = div ◦ ∇ :D→ `2(V; c) is a continuous linear function satisfying that, for every function f ∈ D,‖f − Pf‖2`2(V;c) ≤ 2D (f) ≤ 2 ‖f‖o .Furthermore, HD is the kernel of ID − P.In particular, ID − P is a continuous linear function D → `2(V; c) whose operator normis bounded from above by√2, regardless of the choice of norm in D.In a similar manner, the definitions of the norms in D and `2−(E; r) has the followingcorollary. The gradient ∇ : D→ `2−(E; r) satisfies, for every f ∈ D,(D.1.5) ‖∇f‖`2−(E;r) ≤ ‖f‖o .In other words, ∇ is a continuous linear function with norm ≤ 1 when viewed D→ `2−(E; r),regardless of the choice of norm in D.Proposition ( D.1.6 ) The inner product (D.1.1) defines a structure of Hilbert space onD; the topology of this structure is independent of the choice of o. For a sequence (fn) inD to converge to f ∈ D it is necessary and sufficient that ∇fn → ∇f in `2−(E; r) andfn(o)→ f(o) in R for some vertex o; equivalently, for (fn) to converge to f in the topologyof D it is necessary and sufficient that D (f − fn) → 0 and fn(o) → f(o) (convergences inR) for some vertex o. Furthermore, the subspace HD is closed in D.Proof. We already proved that (D.1.1) defines an inner product and the topology its normdefines does not depend on o. We now prove D is complete with respect to (any of) thenorm(s) ‖f‖o . Since‖f‖2o = f(o)2 +D (f) = f(o)2 + E (∇f)2 ,it follows readily that fn → f in D is equivalent to fn(o) → f(o) and ∇fn → ∇f, which inturn is the same as saying fn(o)→ f(o) and D (f − fn)→ 0.We now prove D is a complete space. Consider a sequence (fn) that is fundamental inD with respect to ‖·‖o ; that is to say, the sequenceÇsupm≥n‖fn − fm‖oån∈Ntends to zero inR. The definition of the norm ‖·‖o shows at once that (fn(o)) is a fundamental sequencein R, hence convergent to some λ, and (∇fn) is fundamental in `2−(E; r), hence convergingto some θ since `2−(E; r) is also complete. By (C.3.2), it follows that ∇fn ∈ ♦⊥ and this aclosed subspace, hence the limit of ∇fn belongs to ♦⊥. Therefore, θ = ∇g for some g ∈ D.Let f = g + λ− g(o), where λ and g(o) denote constant functions. Then fn(o)→ f(o) andD (fn − f) = D (fn − g)→ 0 by (D.1.2). Therefore, fn → f in D.Finally, HD is closed since it is the kernel of a continuous linear function (D.1.4).206Appendix D. The spaces D and HD§ D.2. Convergence in D.In the previous proposition, the topology of D was characterised as two convergences in R,one of them being that of the Dirichlet energy. We proceed to develop a few tools in regardsto this energy.Proposition ( D.2.1 ) Suppose I is an interval in R, λ > 0 and that ϕ : I → R is a λ-Lipschitz function, that is to say, |ϕ(s)− ϕ(t)| ≤ λ|s− t| for all s, t ∈ I. For every Dirichletfunction h such that h(V) ⊂ I, the compound function ϕ ◦h also belongs to D and it satisfiesD (ϕ ◦ h) ≤ λ2D (h) .Proof. Expanding the definition of D (ϕ ◦ h) we use the Lipschitz condition and the resultis immediate.Proposition ( D.2.2 ) Let f, g be two functions defined on V. Set ϕ = min(f, g) andψ = max(f, g). Then, for all pair of vertices x, y,|f(x)− ϕ(x)− f(y) + ϕ(y)| ≤ |f(x)− g(x)− f(y) + g(y)|,and|f(x)− ψ(x)− f(y) + ψ(y)| ≤ |f(x)− g(x)− f(y) + g(y)|.In particular,D (f − ϕ) ≤ D (f − g) and D (f − ψ) ≤ D (f − g) .Proof. The case of ψ is proven with obvious modification to the case of ϕ, and we will showonly the latter. By symmetry between the pairs of x and y, and f and g, there are three casesto consider. In the first case, ϕ(x) = f(x) and ϕ(y) = f(y), and in the second, ϕ(x) < f(x)and ϕ(y) < f(y), but in any of these, the result is obvious. In the third case, ϕ(x) = f(x)and ϕ(y) < f(y), but then f(x) ≤ g(x) and ϕ(y) = g(y), implying|f(x)− ϕ(x)− f(y) + ϕ(y)| = f(y)− g(y)≤ f(y)− g(y) + g(x)− f(x)= |f(y)− g(y)− f(x) + g(x)|.The last part follows upon noticing thatD (f − ϕ) = ∑e∈E12c(e)Äf(e−)− ϕ(e−)− f(e+) + ϕ(e+)ä2≤∑e∈E12c(e)Äf(e−)− g(e−)− f(e+) + g(e+)ä2 = D (f − g) ,which is the desired inequality.207Appendix D. The spaces D and HDWe introduce the common notation t+ = max(t, 0) for t ∈ R; we set f+ to be the functionx 7→ f(x)+ and t− = (−t)+ and f− = (−f)+. The functions f+ and f− are often referred toas the positive and negative part of f, respectively. Despite the name, f− ≥ 0.Proposition ( D.2.3 ) For any pair of functions f, g : V→ R, we have the inequalitiesDÄf+ − g+ä+D Äf− − g−ä ≤ D (f − g) ≤ 2D Äf+ − g+ä+ 2D Äf− − g−ä .Proof. Observe first that(t− s)2 = (t+ − s+)2 + (t− − s−)2 − 2(t+ − s+)(t− − s−) ≥ (t+ − s+)2 + (t− − s−)2since (t+ − s+)(t− − s−) ≥ 0; we also have the inequality(t− s)2 ≤ 2(t+ − s+)2 + 2(t− − s−)2,which follows from (a+ b)2 ≤ 2a2 + 2b2.Proposition ( D.2.4 ) For a function f to belong to D it is necessary and sufficient that itspositive and negative parts should belong to D. Moreover, for a sequence (fn) in D to convergeto some f ∈ D it is necessary and sufficient that the sequences (f+n ) and (f−n ) converge tof+ and f− in D, respectively.Proof. By (D.2.3), f ∈ D ⇐⇒ f± ∈ D. Next, it is clear that f±n (o) → f±(o) and again(D.2.3) applied to g = fn shows that D (f − fn) → 0 is the same as saying that bothD (f± − f±n ) → 0. The conclusion follows from the characterisation of convergence in Dgiven by (D.1.6).§ D.3. The topological space D0.Let D0 be the closure in D of the set of Dirichlet functions with finite support, in otherwords,KR (V) = {f : V→ R | f has finite support},D0 = KR (V).(D.3.1)Clearly, KR (V) is a vector subspace of D and it is a well-known fact from linear algebra thatthe closure, relative to a norm, of a vector subspace is also a vector subspace. Furthermore,D0 being closed in the complete space D, is complete in and of itself. Thus, we reach thefollowing result.208Appendix D. The spaces D and HDProposition ( D.3.2 ) D0 is a Hilbert space for any choice of norm ‖·‖o .Proposition ( D.3.3 ) We have `2(V; c) ⊂ D0; furthermore, if a sequence fn ∈ `2(V; c)converges to f in `2(V; c), then fn converges to f in D0.Proof. The inequality|fn(x)− f(x)| ≤ c(x)−1 ‖fn − f‖`2(V;c)shows that evaluation at x, f 7→ f(x), is a continuous linear form on `2(V; c). Also, theinequality (C.2.4) shows that if ‖fn − f‖`2(V;c) → 0, then D (fn − f) → 0. Therefore, iffn → f in `2(V; c) then fn → f in D. Thus, suffices to show KR (V) is dense in `2(V; c)relative to the topology of `2(V; c). Consider (Vn) an increasing sequence of finite sets suchthat Vn ↑ V. If f ∈ `2(V; c), then f1Vn → f and |f1Vn| ≤ |f |, so the Lebesgue’s dominatedconvergence theorem shows that every function in `2(V; c) is the limit of functions with finitesupport, in other words KR (V) is dense in `2(V; c) in this space.Proposition ( D.3.4 ) Let f ∈ D and ϕ ∈ `2(V; c). Then f − Pf is square integrable(D.1.4) and(∇f |∇ϕ)`2−(E;r) = (f − Pf |ϕ)`2−(E;r) ,(“Gauss-Green formula”). The formula may fail if f and ϕ are only assumed in the spaceD0.Proof. We may apply (C.2.7) to ∇f ∈ `2−(E; r) and ϕ ∈ `2(V; c). Bearing in mind div ◦∇ =ID − P, the result follows.We now provide an example where the Gauss-Green formula fails if f, ϕ ∈ D0. We usethe example in [Bar17]. We consider the canonical network on N and we choose ϕ = 1and f(n) =n∑k=1(−1)kk. Define ϕj = 1{1,...,j} +j∑k=1j − kj1{k} and fj = f1{1,...,j}, is is easyto see that ϕj → ϕ and fj → f in D and since ϕj, fj have finite support, both f and ϕbelong to D0. Now, (f −Pf)(k) = (−1)k+1[k + 12k(k + 1)]and so, the inner-product (f − Pf |ϕ)converges conditionally (compare with(−1)kk). Since∇ϕ = 0, the left hand side is absolutelyconvergent to zero, thus Gauss-Green formula fails.Observe that D0 is the closure of the span of the functions 1x, where x is a vertex. Now,a direct calculation shows,(D.3.5) ∇1x =∑e−=xc(e)χe,209Appendix D. The spaces D and HDwhich amounts to saying that the gradient of 1x is the star at x. Therefore, since ∇ : D →`2−(E; r) is continuous,(D.3.6) ∇(D0) =F.Proposition ( D.3.7 ) For every vertex x, the function Gx =∞∑n=0P n1x (3.1.2) belongs toD0. Thus, for every finitely supported function ϕ, we have∞∑n=0P nϕ ∈ D0.Proof. The second claim follows from the first for we have for such ϕ,∞∑n=0P nϕ =∑x∈Sϕ(x)∞∑n=0P n1x,where S is the support of ϕ.It remains to show that∞∑n=0P n1x ∈ D0 for every x ∈ V. By (3.1.2) and Lebesgue-Tonelli’sthereom, with (Sn) the network random walk,Gx(y) =∞∑n=0Ey (1x(Sn)) = Ey( ∞∑n=01x(Sn)).Consider τk = τV\Vk the exit time of Vk (k ≥ 1) by the random walk, and setGx,k(y) =∞∑n=0EyÄ1x(Sn)1{τk>n}ä= Ey( ∞∑n=01x(Sn)1{τk>n}).Then Gx,k → Gx, simply, by virtue of the monotone convergence theorem. If x /∈ Vk ory /∈ Vk, then Gx,k(y) = 0 and each Gx,k ∈ KR (V) ⊂ `2(V; c) ⊂ D0. By the Gauss-Greenformula (D.3.4), for N ∈ N and m ∈ Z+,(∇Gx,N |∇Gx,N+m)`2−(E;r) = (Gx,N |Gx,N+m − PGx,N+m)`2(V;c) .Since Gx,N has support contained in VN , the only values that matter are Gx,N(y) andGx,N+m(y)− PGx,N+m(y) for y ∈ VN . By first-step analysis (for y ∈ VN)Gx,N+m(y)− PGx,N+m(y) = 1x(y).Therefore,(∇Gx,N |∇Gx,N+m)`2−(E;r) =∑y∈VNc(y)Gx,N(y)1x(y) = c(x)Gx,N(x).210Appendix D. The spaces D and HDWith this in mind,D (Gx,N −Gx,N+m) = D (Gx,N) +D (Gx,N+m)− 2 (∇Gx,N |∇Gx,N+m)`2−(E;r)= c(x)Gx,N(x) + c(x)Gx,N+m(x)− 2c(x)Gx,N(x)= c(x)ÄGx,N+m(x)−Gx,N(x)ä.Since Gx,k(y)→ Gx(y) for every y, (Gx,k)k∈N is a fundamental sequence in D0, and thereforeconverges to some limit H ∈ D0, but evaluation is continuous by (D.1.3) and so H = Gx, asclaimed.Proposition ( D.3.8 ) In order that f ∈ D0, it is necessary and sufficient that f+ andf− should belong to D0. Furthermore, for f to belong to D0 it is necessary and sufficientthat there should exist two sequences (sn) and (tn) of functions with finite support such that0 ≤ sn ≤ f+, 0 ≤ tn ≤ f− and sn → f+, tn → f− in the topology of D.Proof. We know that f ∈ D0 if and only if fn → f in the topology of D for some sequenceof functions with finite support. For such a sequence, it is clear that f±n have finite supportand by (D.2.4), we have that f±n → f±, showing that both f± ∈ D0; the converse is obvioussince f = f+ − f−. We prove now the second claim, the sufficiency is obvious for thenf+ and f− belong to D0. We prove the necessity. Since f ∈ D0, there is a sequence (fn)of functions with finite support such that fn → f in D. Then f±n → f± in D by (D.2.4).Consider sn = min(f+n , f+) and tn = min(f−n , f−). Clearly 0 ≤ sn ≤ f+, 0 ≤ tn ≤ f− andsn(o) → f+(o), tn(o) → f−(o). By (D.2.2), D (sn − f+) ≤ D (f+n − f+) and D (tn − f−) ≤D (f−n − f−) , showing the claim by (D.1.6).Using D0, we may now provide another version of Dirichlet’s principle.Proposition ( D.3.9 ) Let o be a vertex of a network Γ satisfying the usual assumptions.ThenC (o,∞) = infϕ∈D0ϕ(o)=1D (ϕ) ,(“Dirichlet’s principle”).Proof. By (C.11.2), we haveC (o,∞) = inf D (ϕ) ,where ϕ(o) = 1 and ϕ has finite support; such ϕ is a Dirichlet function. Consider Z0(respectively Z) to be the set of ϕ with finite support (respectively that belongs to D0) suchthat ϕ(o) = 1 and denote by Z0 its closure in D. Then Z = e−1o ({1}), where eo : D0 → R is211Appendix D. The spaces D and HDthe evaluation ϕ 7→ ϕ(o); since eo is continuous (D.1.3), Z is closed. Clearly Z0 ⊂ Z, henceZ0 ⊂ Z. Now, if ϕ ∈ Z there exists a sequence ψn of functions with finite support such thatψn → ϕ in D, define ϕn = ψn + (1− ψn(o))1o. Clearly ϕn(o) = 1, so ϕn ∈ Z0, and‖ϕ− ϕn‖o ≤ ‖ϕ− ψn‖o + |1− ψn(o)| ‖1o‖o → 0,showing that ϕ ∈ Z0. Hence Z = Z0. So, the claim thatinfϕ∈Z0D (ϕ) = infϕ∈ZD (ϕ)is now obvious since D (·) is continuous D→ R.§ D.4. The Hilbert space D/R.Consider now the quotient set D/R, that is, two Dirichlet functions are identified if itsdifference is a constant function. Define on the set D/R× D/R the following function(D.4.1)Äf˜ , g˜ä 7→ Äf˜ ∣∣∣g˜äD/R= (∇f |∇g)`2−(E;r) ,where f ∈ f˜ and g ∈ g˜, this is well-defined by (D.1.2). This defines on D/R a structure ofHilbert space. This easily follows from the the characterisation of the topology of D (D.1.6).IfÄf˜näis a fundamental sequence in D/R, then we can choose representatives fn ∈ f˜n forwhich fn(o) = 0 and so (fn) is fundamental in D with respect to ‖·‖o and it thereforeconverges to a function f ∈ D, which satisfies D (fn − f)→ 0. Then f˜ is the limit ofÄf˜nä.We transport now the definition of ∇ from being a function D → `2−(E; r) to a functionD/R→ `2−(E; r) by setting ∇f˜ = ∇f for any f ∈ f˜ .Proposition ( D.4.2 ) The function ∇ : D/R→ `2−(E; r) has image ♦⊥ and ∇ : D/R→ ♦⊥is an isomorphism of Hilbert spaces (it is a bijection that preserves inner product).Proof. By (C.3.2), ∇f˜ ∈ ♦⊥ and reciprocally, a flow in ♦⊥ is of the form ∇f = ∇f˜ , wheref ∈ D and f˜ is the class of f. Hence, all reduces to show that ∇ : D/R → ♦⊥ is anisomorphism of Hilbert spaces. It follows immediately from (D.4.1) that ∇ preserves innerproduct and therefore, it also preserves the norm, that is,∥∥∥∇f˜∥∥∥D/R= ‖∇f‖`2−(E;r) and so,∇ : D/R→ ♦⊥ is bijective and this completes the proof.Denote now by D˜0 the set of classes f˜ as f runs through D0. Since the operator ∇ :D/R→ ♦⊥ is an isomorphism of Hilbert spaces (D.4.2), and ♦⊥ =F⊕∇HD is an orthogonaldecomposition of ♦⊥ by (C.4.1), it follows that D/R = ∇−1(F) ⊕ HD/R is an orthogonal212Appendix D. The spaces D and HDdecomposition. Upon applying the equivalence relation, the equality ∇(D0) = F (D.3.6) isthe same as saying D˜0 = ∇−1(F). We have therefore an orthogonal decomposition of theHilbert space D/R(D.4.3) D/R = D˜0 ⊕ HD/R.An immediate corollary of this decomposition is that D0 ∩ HD is contained in the set ofconstant functions. This intersection reduces to {0} in many cases as the next propositionshows.Proposition ( D.4.4 ) The following three conditions are equivalent on network Γ satisfyingthe usual assumptions.(a) D0 contains the constant functions;(b) D0 = D;(c) the network random walk in Γ is recurrent.Proof. We know that D0 ⊂ D always and it is obvious that condition (b) implies condition(a). We now prove that 1 ∈ D0 is sufficient for D0 = D, so assume 1 ∈ D0. Since 1 ∈ D0,there exists a sequence fn of functions with finite support such that fn → 1 in the topologyof D. Since 1 is already its positive part, we may assume fn to be positive as well (D.2.4).By passing through a subsequence (also denoted by (fn)), we may assume ‖fn − 1‖o ≤ n−2.Introduce gn = nfn, which are also non-negative and finitely supported functions. ThenD (gn) = n2D (fn) ≤ n2n−4 = n−2,and gn(v)→∞ for every vertex v since fn(v)→ 1 and evaluation is continuous (D.1.3).Given u ∈ D, we have u = u+ − u− with both u± ∈ D (D.2.4) and so, suffices to showevery non-negative Dirichlet function belongs to D0. For such a function u, set the sequence(un) by un = min(u, gn). Then, un is a sequence of non negative and finitely supportedfunctions such that for every vertex x, there is an index nx such that un(x) = u(x) for alln ≥ nx. Consider En to be the set of edges e for which ∇u(e) 6= ∇un(e). Since the networkis denumerable, we may order the vertex set V = (vj)j∈N. For every positive integer J thereexists nJ ∈ N such that for all n ≥ nJ , no edge joining vertices among v1, . . . , vJ will be inEn. Then,∑e∈En12r(e)∇u(e)2 can be made arbitrarily small if n is large enough, that is to say,∑e∈En12r(e)∇u(e)2 = o (1) as n→∞. By definition of En,‖u− un‖2o =∑e∈En12r(e)Ä∇u(e)−∇un(e)ä2.213Appendix D. The spaces D and HDBy (D.2.2),|u(e+)− u(e−)− un(e+) + un(e−)|2 ≤ 2|u(e+)− u(e−)|2 + 2|gn(e+)− gn(e−)|2,and this bound implies Ä∇u(e)−∇un(e)ä2 ≤ 2∇u(e)2 + 2∇gn(e)2.Therefore‖u− un‖2o ≤∑e∈Enr(e)∇u(e)2 + ∑e∈Enr(e)∇gn(e)2 ≤ o (1) + 2n−2 = o (1)as n→∞. Thus, showing D = D0 as desired.It remains to show that (c) is equivalent to (b). For this, observe that by (D.3.9)‖1− D0‖2D = inf{‖1− f‖2o : f ∈ D0}≤ inf{‖1− f‖2o : f ∈ D0, f(o) = 1} = C (o,∞) ,with equality if the constant functions belong to D0, Therefore, we have established‖1− D0‖2D ≤ C (o,∞) ,with equality if 1 ∈ D0. We may now apply Doyle and Snell’s criterion (C.9.3). Then, 1 ∈ D0if the network is recurrent since then C (o,∞) = 0, and if 1 /∈ D0 then C (o,∞) > 0 and thenetwork is transient. This concludes the proof.§ D.5. Riesz’ and Royden’s decompositions.By the previous proposition, when the network random walk in G is transient, D0∩HD = {0}and we have the following decomposition as a direct sumD = D0 + HD.Therefore, in the transient case, every function f ∈ D is uniquely expressible as(D.5.1) f = fD0 + fHD,where fD0 ∈ D0 and fHD ∈ HD (“Royden’s decomposition”). Observe that in general fD0and fHD are not orthogonal vectors in D, even when ∇fD0 is orthogonal to ∇fHD, for(fD0|fHD)o = fD0(o)fHD(o).We recall the definition of superharmonic function as a function f : V → R such thatPf ≤ f. If (Sn)n∈Z+ is the network random walk with S0 constant, then (Pf(Sn))n∈Z+defines a supermartingale, as it is easily seen from definitions (proof is identical to (3.1.4)).214Appendix D. The spaces D and HDProposition ( D.5.2 ) Consider a network satisfying the usual assumptions Γ. The super-harmonic functions in D0 are necessarily non-negative.Proof. Suppose f ∈ D0 is superharmonic. Observe that D (|f |) ≤ D (f) (D.2.1) and that|f | = f + 2f−, this last equality impliesD (|f |) = D (f) + 4D Äf−ä+ 4 Ä∇f ∣∣∣∇f−ä`2−(E;r).Consider fn having finite support and fn → f in D, then f−n → f− in D by (D.2.4). Hence,by the Gauss-Green formula (D.3.4),Ä∇f ∣∣∣∇f−ä`2−(E;r)= limn→∞Ä∇f ∣∣∣∇f−n ä`2−(E;r) = limn→∞ Äf − Pf ∣∣∣f−n ä`2(V;c) ≥ 0since f − Pf ≥ 0. Whence, D (f) ≤ D (|f |) and therefore, D (f) = D (|f |) and D (f−) = 0.This shows that f− is a constant function. Since f− ∈ D0, it follows from (D.4.4) thatf− = 0. Thus, f = f+ ≥ 0.Proposition ( D.5.3 ) Suppose Γ is a network satisfying the usual assumptions. Everynon-negative superharmonic function f is uniquely expressible as f =∞∑n=0P nu + h, where uis non-negative and h is harmonic (“Riesz’ decomposition”); in this decomposition, h ≥ 0.Proof. The superharmonicity of f implies that (P nf) is non-increasing and therefore, sincef ≥ 0, the sequence of functions P nf converges simply to a function h. We will show h isharmonic. Clearly f −P nf ↑ f −h, thus the monotone convergence theorem shows that, forevery x ∈ V,limn→∞P (f − Pnf)(x) = P (f − h)(x).Since 0 ≤ Pf(x) ≤ f(x), we have limn→∞Pn+1f(x) → Ph(x). Thus h(x) = Ph(x), that isto say, h is harmonic at x. Consider now u = f − Pf ≥ 0. Then,N∑n=0P nu = f − PNfand taking the limit as N → ∞, the right hand side converges simply to f − h, and bydefinition, the left hand side converges and its limit is∞∑n=0P nu. The decomposition has beenestablished. We now prove this decomposition is unique. If∞∑n=0P nu1 + h1 =∞∑n=0P nu2 + h2are two decompositions of f, we may apply Lebesgue’s convergence theorem to show, forevery x ∈ V,P( ∞∑n=0P n(u2 − u1))(x) =∞∑n=0P n+1(u2 − u1)(x).215Appendix D. The spaces D and HDHowever,∞∑n=0P n(u2 − u1) = h1 − h2 = P (h1 − h2) =∞∑n=0P n+1(u2 − u1).Thus, u2 − u1 = 0 and then h1 − h2 = 0 as well.Proposition ( D.5.4 ) The Riesz decomposition for a superharmonic function f in D0 isof the form f =∞∑n=0P nu where u ≥ 0.Proof. The Riesz decomposition of f exists (D.5.3) and we may write f =∞∑n=0P nu + h,where u ≥ 0 and h ≥ 0 is harmonic. Consider now an exhaustion (Gk) of G and let Vkbe the vertex set of Gk. The exhaustion is chosen so that each Vk is finite. Set uk = u1Vkand fk =∞∑n=0P nuk, notice that uk belongs to D0 having finite support and so, fk ∈ D0 aswell (D.3.7). Consider now a sequence of functions 0 ≤ fk,n ≤ fk having finite support andfk,n → fk in D (D.3.8). Then, ∇fk,n → ∇fk in `2−(E; r); therefore, using the Gauss-Greenformula (D.3.4)D (fk) = limn→∞ (∇fk|∇fk,n)`2−(E;r) = limn→∞ (fk − Pfk|fk,n)`2(V;c) .Now, fk − Pfk = uk ≥ 0, and clearly uk ≤ u = f − Pf ∈ `2(V; c) by (D.1.4), hence(fk − Pfk|fk,n)`2(V;c) ≤ (f − Pf |fk,n)`2(V;c) .Using again the Gauss-Green formula, it follows that(f − Pf |fk,n)`2(V;c) = (∇f |∇fk,n)`2−(E;r) ≤ D (f)12 D (fk,n)12 .Whence, we proved thatD (fk) ≤ limn→∞D (f)12 D (fk,n)12 = D (f)12 D (fk)12 ,which in turn implies D (fk) ≤ D (f) (since D (fk) <∞).Thus far, we have constructed the functions uk = u1Vk and fk =∞∑n=0P nuk ∈ D0and showed that D (fk) ≤ D (f) . Observe that the monotone convergence theorem showslimk→∞fk(x) =∞∑n=0P nu(x) for every vertex x ∈ V. By Fatou’s lemma,D( ∞∑n=0P nu)≤ lim · infk→∞D( ∞∑n=0P nuk)= lim · infk→∞D (fk) ≤ D (f) <∞.216Appendix D. The spaces D and HDHence, in the Riesz decomposition of f =∞∑n=0P nu+ h, we proved that∞∑n=0P nu ∈ D and so,h ∈ HD. Then, if we can show that∞∑n=0P nu ∈ D0, the Riesz decomposition of f will also bethe Royden’s decomposition by uniqueness. Having this and the fact f already belongs toD0, will show together that h = 0, which is the desired conclusion. All reduces to show that∞∑n=0P nu ∈ D0.We now prove that∞∑n=0P nu ∈ D0 and this will terminate the proposition. To begin with,‖fk‖o ≤ ‖f‖o and therefore, fk lies in BD0 (0; ‖f‖o) the ball of centre zero and radius ‖f‖oin the Hilbert space D0. Remark from (D.3.6) that ∇(D0) = F and that ∇ : D0 → Fis continuous by (D.1.5). Now, ∇ : D0 → F is injective for if ∇f = 0 for some f ∈ D0then f is a constant function on V, but the network random walk is transient by hypothesis,showing that there are no constant functions in D0 other than the function zero (D.4.4), thusf = 0 and ∇ is injective as claimed. Whence, ∇ : D0 → F is a continuous linear bijectionbetween Hilbert spaces. Taking advantage of Banach-Schauder theorem, ∇ : D0 → Fhas a continuous inverse. Since F is separable (C.1.4), D0 is also separable. By invokingBanach-Alaoglu theorem, we see that BD0 (0; ‖f‖o) is a metrisable compact space in the weaktopology of D0. Thus, (fk) has limit points in BD0 (0; ‖f‖o) relative to the weak topology.But we know that evaluation is a continuous linear form D0 → R (D.1.3), and we alreadyproved that fk(x) →∞∑n=0P nu(x), this entails that∞∑n=0P nu is a limit point of (fk) (in fact,the unique limit point). Henceforth,∞∑n=0P nu ∈ D0 as desired.§ D.6. The space HD and the Dirichlet Liouville prop-erty.Recall from Chapter 3 that the Dirichlet Liouville property (DLP) states that any Dirichletharmonic function is constant, while the Liouville propery (LP), any bounded harmonicfunction is constant. We are going to show (LP) =⇒ (DLP). As such, we need a few moretools regarding HD.Proposition ( D.6.1 ) Suppose the network random walk on a network satisfying the usualassumptions is transient. Denote by HD+ the set of harmonic Dirichlet functions that arenon negative. Then HD = HD+ − HD+.217Appendix D. The spaces D and HDProof. Consider f ∈ HD and decompose f = f+ − f−. We know that f± ∈ D and by virtueof Royden’s decomposition, f± is uniquely expressible as f± = v± + u± with v± ∈ D0 andu± ∈ HD. It is easy to check that Pf± ≥ f± and since Pu± = u± for i = 1, 2, it followsthat Pv± ≥ v±. Hence, −v± are superharmonic functions in D0, and therefore −v± ≥ 0 by(D.5.3). Since f± = v± + u± ≥ 0, we get u± ≥ −v± ≥ 0 and thus u± are non-negativeharmonic functions. Finally, f = f+− f− = (v+− v−) + (u+− u−) and since f has a uniqueRoyden’s decomposition, if follows that v+ − v− = 0 and so f = u+ − u− is the difference oftwo non-negative harmonic Dirichlet functions as desired.Proposition ( D.6.2 ) The bounded harmonic Dirichlet functions are dense in HD; in fact,the set of bounded functions in HD+ is a dense subset on HD+.Proof. The proposition will be completely proven if we show the second claim. Indeed, anyf ∈ HD can be written as f = f1− f2 with both f1 and f2 positive harmonic Dirichlet func-tions, as proven in the previous proposition. Given f ∈ HD+, consider fn = min(f, n), whichis readily seen to be superharmonic. Royden’s decomposition allows expressing uniquelyfn = un + hn,where un ∈ D0 and hn ∈ HD. Since fn is superharmonic and hn harmonic, un is superhar-monic as well, and as un ∈ D0, it follows that un is a non negative (D.5.2) and un =∞∑m=0Pmvnfor some vn ≥ 0 (D.5.4). Then fn = un + hn is also the Riesz decomposition of f and sohn ≥ 0 (D.5.3). Now,D (f − fn) = D (un) +D (f − hn)− 2 (∇un|∇(f − hn))`2−(E;r) = D (un) +D (f − hn) ,since ∇un ∈ F and ∇(f − hn) ∈ ∇HD. By Lebesgue’s dominated convergence theorem,D (f − fn)→ 0, and therefore D (un)→ 0 and D (f − hn)→ 0. Since 0 ≤ un ≤ fn ≤ f andusing a subsequence should the need arise, (un(o)) can be assumed to converge in R to someconstant λ. Therefore, un(o) → λ and D (un) → 0, which together imply that un convergesin D0 to the constant function λ, this function belongs to D0 since this is a closed subspace.Since the network random walk is assumed transient, the only constant function in D0 is thezero function (D.4.4), therefore λ = 0. Since fn(o) = f(o) for all large values of n, we deducethat hn(o)→ f(o) and therefore, hn → f in D.This proposition is cornerstone for the following.Proposition ( D.6.3 ) If a network satisfying the usual assumption possesses the Liovilleproperty the it also possesses the Dirichlet Liouville property; in other words, (LP) =⇒(DLP).218Appendix D. The spaces D and HDProof. By (D.6.2), every harmonic function h ∈ HD can be approximated in D by constantfunctions. By (D.1.3), h is constant.That a network satisfies the Dirichlet Liouville property signifies that ∇HD = {0}. Thus,and keeping in mind (C.4.2), a sufficient condition for a network satisfying the usual as-sumptions, whose underlying graph does not possess multiple edges, to satisfy WSF = FSF isthat the network should possess the Liouville property, this is just a restatement of Theorem(3.4.1).219
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Uniform spanning forest on the integer lattice with drift in one coordinate.
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