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Combinatorial aspects of spatial mixing and new conditions for pressure representation Briceño Domínguez , Raimundo José 2016

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Combinatorial aspects of spatial mixing andnew conditions for pressure representationbyRaimundo Jose´ Bricen˜o Domı´nguezA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2016c© Raimundo Jose´ Bricen˜o Domı´nguez 2016AbstractOver the last few decades, there has been a growing interest in a measure-theore-tical property of Gibbs distributions known as strong spatial mixing (SSM). SSMhas connections with decay of correlations, uniqueness of equilibrium states, ap-proximation algorithms for counting problems, and has been particularly usefulfor proving special representation formulas and the existence of efficient approx-imation algorithms for (topological) pressure. We look into conditions for theexistence of Gibbs distributions satisfying SSM, with special emphasis in hardconstrained models, and apply this for pressure representation and approximationtechniques in Zd lattice models.Given a locally finite countable graph G and a finite graph H, we considerHom(G ,H) the set of graph homomorphisms from G to H, and we study Gibbsmeasures supported on Hom(G ,H). We develop some sufficient and other neces-sary conditions on Hom(G ,H) for the existence of Gibbs specifications satisfyingSSM (with exponential decay). In particular, we introduce a new combinatorialcondition on the support of Gibbs distributions called topological strong spatialmixing (TSSM). We establish many useful properties of TSSM for studying SSMon systems with hard constraints, and we prove that TSSM combined with SSMis sufficient for having an efficient approximation algorithm for pressure. We alsoshow that TSSM is, in fact, necessary for SSM to hold at high decay rate.Later, we prove a new pressure representation theorem for nearest-neighbourGibbs interactions on Zd shift spaces, and apply this to obtain efficient approxima-tion algorithms for pressure in the Z2 (ferromagnetic) Potts, (multi-type) Widom-Rowlinson, and hard-core lattice gas models. For Potts, the results apply to ev-ery inverse temperature except the critical. For Widom-Rowlinson and hard-corelattice gas, they apply to certain subsets of both the subcritical and supercriticalregions. The main novelty of this work is in the latter, where SSM cannot hold.iiPrefaceThe thesis is split in two main parts (Part I and Part II), based on the followingthree articles:1. Representation and poly-time approximation for pressure of Z2 lattice mod-els in the non-uniqueness region. Joint work with Stefan Adams, Brian Mar-cus, and Ronnie Pavlov. Journal of Statistical Physics, 162(4), 1031-1067.(See [1].)2. The topological strong spatial mixing property and new conditions for pres-sure approximation. Accepted for publication in Ergodic Theory and Dy-namical Systems. (See [15].)3. Strong spatial mixing in homomorphism spaces. Joint work with RonniePavlov. Submitted. (See [16].)Part I (Chapter 2, Chapter 3, and Chapter 4) is mainly based on paper 2 andpaper 3. Part II (Chapter 5, Chapter 6, and Chapter 7) is mainly based on paper 1and paper 2.Some of the writing of this thesis was done while the author was visiting theSimons Institute for the Theory of Computing at University of California, Berkeley.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Results overview . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 15I Combinatorial aspects of the strong spatial mixing property . 172 Basic definitions and notation . . . . . . . . . . . . . . . . . . . . . 182.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.1 Boards . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.2 The d-dimensional integer lattice . . . . . . . . . . . . . 202.1.3 Constraint graphs . . . . . . . . . . . . . . . . . . . . . 212.2 Configuration spaces . . . . . . . . . . . . . . . . . . . . . . . . 23ivTable of Contents2.2.1 Homomorphism spaces . . . . . . . . . . . . . . . . . . 242.2.2 Shift spaces in Zd . . . . . . . . . . . . . . . . . . . . . 242.3 Gibbs measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.1 Borel probability measures and Markov random fields . . 262.3.2 Shift-invariant measures . . . . . . . . . . . . . . . . . . 272.3.3 Nearest-neighbour interactions . . . . . . . . . . . . . . 282.3.4 Hamiltonian and partition function . . . . . . . . . . . . 292.3.5 Gibbs specifications . . . . . . . . . . . . . . . . . . . . 312.3.6 Nearest-neighbour Gibbs measures . . . . . . . . . . . . 313 Mixing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1 Spatial mixing properties . . . . . . . . . . . . . . . . . . . . . . 333.2 Combinatorial mixing properties . . . . . . . . . . . . . . . . . . 363.2.1 Global properties . . . . . . . . . . . . . . . . . . . . . . 363.2.2 Local properties . . . . . . . . . . . . . . . . . . . . . . 373.3 Topological strong spatial mixing . . . . . . . . . . . . . . . . . 383.4 TSSM in Zd shift spaces . . . . . . . . . . . . . . . . . . . . . . 413.4.1 Existence of periodic points . . . . . . . . . . . . . . . . 433.4.2 Algorithmic results . . . . . . . . . . . . . . . . . . . . 453.5 Examples: Zd n.n. SFTs . . . . . . . . . . . . . . . . . . . . . . 473.5.1 A strongly irreducible Z2 n.n. SFT that is not TSSM . . . 473.5.2 A TSSM Z2 n.n. SFT that is not SSF . . . . . . . . . . . 523.5.3 Arbitrarily large gap, arbitrarily high rate . . . . . . . . . 543.6 Relationship between spatial and combinatorial mixing properties 563.6.1 SSM criterion . . . . . . . . . . . . . . . . . . . . . . . 563.6.2 Uniform bounds of conditional probabilities . . . . . . . 573.6.3 SSM with high decay rate . . . . . . . . . . . . . . . . . 604 SSM in homomorphism spaces . . . . . . . . . . . . . . . . . . . . . 634.1 Dismantlable graphs and homomorphism spaces . . . . . . . . . 634.2 Dismantlable graphs and WSM . . . . . . . . . . . . . . . . . . 654.3 The unique maximal configuration property . . . . . . . . . . . . 714.4 SSM and the UMC property . . . . . . . . . . . . . . . . . . . . 72vTable of Contents4.5 UMC and chordal/tree decomposable graphs . . . . . . . . . . . 754.5.1 Chordal/tree decompositions . . . . . . . . . . . . . . . 754.5.2 A natural linear order . . . . . . . . . . . . . . . . . . . 774.6 The looped tree case . . . . . . . . . . . . . . . . . . . . . . . . 824.7 Examples: Homomorphism spaces . . . . . . . . . . . . . . . . . 87II Representation and poly-time approximation for pressure . . 945 Entropy and pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 955.1 Topological entropy . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Topological pressure . . . . . . . . . . . . . . . . . . . . . . . . 995.3 Pressure representation . . . . . . . . . . . . . . . . . . . . . . . 1025.3.1 A generalization of previous results . . . . . . . . . . . . 1035.3.2 The function pˆi and a new pressure representation theorem 1066 Classical lattice models and related properties . . . . . . . . . . . . 1126.1 Three Zd lattice models . . . . . . . . . . . . . . . . . . . . . . 1126.1.1 The (ferromagnetic) Potts model . . . . . . . . . . . . . 1126.1.2 The (multi-type) Widom-Rowlinson model . . . . . . . . 1136.1.3 The hard-core lattice gas model . . . . . . . . . . . . . . 1136.2 The bond random-cluster model . . . . . . . . . . . . . . . . . . 1156.3 The site random-cluster model . . . . . . . . . . . . . . . . . . . 1186.4 Additional properties . . . . . . . . . . . . . . . . . . . . . . . . 1226.4.1 Spatial mixing properties . . . . . . . . . . . . . . . . . 1226.4.2 Stochastic dominance . . . . . . . . . . . . . . . . . . . 1246.5 Exponential convergence of pin . . . . . . . . . . . . . . . . . . . 1276.5.1 Exponential convergence in the Potts model . . . . . . . 1276.5.2 Exponential convergence in the Widom-Rowlinson model 1316.5.3 Exponential convergence in the hard-core lattice gas model 1357 Algorithmic implications . . . . . . . . . . . . . . . . . . . . . . . . 1417.1 Previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.2 New results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142viTable of Contents8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151viiList of Figures2.1 A sample of the boards Z2 and T3. . . . . . . . . . . . . . . . . . 192.2 The graphs Kn and Kn , for n = 5. . . . . . . . . . . . . . . . . . 212.3 The graph Hϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 The graphs S6, So6, and S6 . . . . . . . . . . . . . . . . . . . . . . 223.1 The weak and strong spatial mixing properties. . . . . . . . . . . 343.2 A topologically mixing Z2 shift space. . . . . . . . . . . . . . . . 373.3 The topological strong spatial mixing property. . . . . . . . . . . 393.4 Construction of a periodic point using TSSM. . . . . . . . . . . . 443.5 Proof that Ω= Hom(Z2,K4) does not satisfy TSSM nor SSM. . . 513.6 Representation of the proof of Theorem 3.6.5. . . . . . . . . . . . 614.1 Decomposition in the proofs of Lemma 4.2.3 and Proposition 4.2.5. 694.2 A chordal/tree decomposition. . . . . . . . . . . . . . . . . . . . 764.3 The n-barbell graph Bn. . . . . . . . . . . . . . . . . . . . . . . . 824.4 A “channel” in Hom(Z2,T ) with two incompatible extremes. . . . 854.5 A dismantlable graph H such that Hom(Z2,H) is not TSSM. . . . 884.6 Two incompatible configurations α and β (both in red). . . . . . . 894.7 A scenario where SSM fails for any Gibbs (Z2,H,φ)-specification. 894.8 The graph Hq, for q = 5. . . . . . . . . . . . . . . . . . . . . . . 914.9 A Markov chain embedded in a Z2 Markov random field. . . . . . 926.1 A ?-connected set Θ (in black), ∆ = ∂ ?Θ∩Λ (in dark grey), andΛc (in light grey) for Λ= Qy,z. . . . . . . . . . . . . . . . . . . . 1216.2 A configuration θ ∈ EC? . On the left, the associated sets Σ~0(θ) andK(θ). On the right, the sets I(C?) and O(C?) for Γ(θ) = C?. . . . 138viiiList of Figures7.1 Decomposition in the proof of Proposition 7.2.1. . . . . . . . . . 143ixNomenclaturex,y, . . . Vertices, p. 18.∼, ∼G Vertex adjacency relation (in graph G), p. 18.Loop(G) Set of vertices with a loop, p. 18.A,B, . . . Subsets of vertices in a board, p. 19.N(x) Neighbourhood of vertex x, p. 19.∆(G) Maximum degree, p. 19.G Board graph (simple, connected, locally finite), p. 19.Zd d-dimensional integer lattice, p. 19.Td d-regular tree, p. 19.dist(x,y) Graph distance between x and y, p. 19.∂A Outer boundary of A, p. 20.∂A Inner boundary of A, p. 20.Nn(A) n-neighbourhood of A, p. 20.∂n(A) n-boundary of A, p. 20.4 Lexicographic order in Zd , p. 20.P Lexicographic past of Zd , p. 20.Bn n-block, p. 20.xNomenclatureQy,z Broken [y,z]-block, p. 20.Rn n-rhomboid, p. 21.? Superindex for concepts of Zd with diagonal edges, p. 21.H Constraint graph (finite, possibly with loops), p. 21.HConstraint graph with loops everywhere, p. 21.Kn Complete graph with n vertices, p. 22.Hϕ Constraint graph of hard-core model, p. 22.Sn n-star graph, p. 22.A Alphabet, p. 23.Ω Configuration space (shift space, homomorphism space, etc.), p. 23.α,β , . . . Configurations, p. 23.ω,υ , . . . Points in Ω, p. 23.L (Ω) Language of Ω, p. 23.[α]Ω Cylinder set in Ω for a configuration α , p. 23.unionsq, ⊔ Disjoint union of sets, p. 23.Hom(G ,H) Homomorphism space, p. 24.σ ,σx Shift action in Zd , p. 24.F Family of finite configurations/constraints, p. 25.Ei Set of n.n. constraints in the ith direction, p. 25.ΩF Shift space induced by F, p. 25.Ωdϕ Support of the Zd hard-core lattice gas model, p. 25.O(ω) Orbit of ω ∈A Zd , p. 26.xiNomenclatureFA σ -algebra generated by cylinder sets with support A, p. 26.µ,ν Borel probability measures on A V , p. 26.supp(µ) Support of µ , p. 26.M1(Ω) Set of Borel probability measures with support contained in Ω, p. 26.A,B, . . . Events, p. 27.M1,σ (Ω) Subset of shift-invariant measures inM1,σ (Ω), p. 27.δω δ -measure supported on ω , p. 27.νω Shift-invariant measure induced by a periodic point ω , p. 27.Φ Interaction for configurations on vertices and edges, p. 28.φ Function on finitely many configurations inducing Φ, p. 28.ξ Generic inverse temperature, p. 29.H ΦA Hamiltonian function in A, p. 30.ZΦA Partition function of A, p. 30.pi( f )A Free-boundary probability measure on AA, p. 30.H ΦA,ω Hamiltonian in A with ω-boundary, p. 30.ZΦA,ω Partition function of A with ω-boundary, p. 30.piωA ω-boundary probability measure on AA, p. 30.pi Nearest-neighbour Gibbs (Ω,Φ)-specification, p. 31.G(pi) Set of Gibbs measures for pi , p. 31.ΣB(α1,α2) Set of B-disagreement between α1 and α2, p. 34.dTV Total variation distance, p. 35.P,Q Couplings, p. 35.xiiNomenclatureDω,x Digraph of descending paths, p. 55.Q(pi) SSM criterion, p. 56.pc(Zd) Threshold of Zd Bernoulli site percolation, p. 56.cpi(ω) Infimum of piωA (ω|{x}) over all finite A and x ∈ A, p. 58.cpi Infimum of cpi(ω) over all ω ∈Ω, p. 58.Φmax Greatest absolute value of Φ over all n.n. configurations, p. 58.φλ Constrained energy function for λ , p. 66.,≺ Linear order on V or A and coordinate-wise extension, p. 71.ωα Unique maximal configuration for α , p. 71.φλ Constrained energy function for λ and , p. 72.Bn n-barbell graph, p. 82.h(Ω) Topological entropy of Ω, p. 96.h(µ) Measure-theoretic entropy of µ , p. 99.C (Ω) Set of continuous functions from Ω to R, p. 99.PΩ( f ) Topological pressure of f ∈ C (Ω), p. 99.PΩ(Φ) Pressure of interaction Φ, p. 100.ZˆΦBn Partition function of locally admissible configurations in Bn, p. 100.AΦ Continuous function induced by Φ, p. 101.pµ Limit of pµ,P∩Bn as n→ ∞, p. 103.Iµ Information function for µ , p. 103.cµ Infimum of pµ,S(ω) over all S and ω , p. 104.c−µ (ν) Infimum of pµ,S(ω) over all SbP and ω ∈ supp(ν), p. 104.xiiiNomenclaturepiy,z(ω) Abbreviation of piQy,z(ω), p.ω) Abbreviation of pi~1n,~1n(ω), p. 106.pˆi(ω) Limit of pin(ω) as n→ ∞, p. 106.Ipi Modified information function for pi , p. 106.cpi(ν) Infimum of cpi over all ω ∈ supp(ν), p. 107.Λ,∆, . . . Subsets of vertices in Zd , p. 112.piFPβ Potts model with inverse temperature β , p. 113.βc(q) Critical inverse temperature Z2 Potts model with q types, p. 113.piWRζ Widom-Rowlinson model with inverse temperature ζ , p. 113.piHCλ Hard-core lattice gas model with activity λ , p. 114.ωq Fixed point qZd, p. 114.ω(e)/ω(o) Zd hard-core lattice gas (even/odd) checkerboard points, p. 114.ϕ(i)p,q,Λ Free/wired (i = 0/1) bond random-cluster distribution, p. 116.ϕ(i)p,q Bond random-cluster limiting distributions, p. 116.pc(q) Critical value Z2 bond random-cluster model, p. 116.P(1)p,q,Λ Edwards-Sokal coupling, p. 117.p∗ Dual value of the bond-random cluster parameter p, p. 118.ψ(i)p,q,Λ Free/wired (i = 0/1) site random-cluster limiting distribution, p. 118.∂↓Qy,z Past boundary of Qy,z, p. 123.∂↑Qy,z Future boundary of Qy,z, p. 123.≤D Stochastic dominance relation, p. 124.ϕp,q(x↔ y) Two-point connectivity function, p. 125.xivAcknowledgementsFirst and foremost, I would like to express my deep gratitude to my advisor, BrianMarcus. I could not have imagined having a better mentor, boss, and collabora-tor for my PhD studies. I thank him for his continuous support, patience, energy,shared knowledge, and the always friendly and stimulating environment that hemanages to create around him. I have learnt from him even more than I initially ex-pected, ranging from mathematics (symbolic dynamics, ergodic theory, and more)to baseball, Lincoln, and life. I am sure that I still have many things to learn fromhim; I will miss a lot our weekly caffeinated meetings, full of math and joy.I am also very grateful to Ronnie Pavlov for his collaboration and encourage-ment during this whole period. I have learnt a great deal working with him, and itwill take me some time to stop associating the happy beginning of summers withhis always fruitful academic visits. I also thank him and Nic Ormes for organizingthe 4th Pingree Park Dynamics Workshop and letting me show part of my workthere.I am very thankful to Anthony Quas for hosting me at University of Victoriaand giving me the opportunity to talk in the Dynamics Seminar. I also thank himfor agreeing to be my external examiner and doing such a thorough review of mythesis, and for pointing out meaningful connections between the topological strongspatial mixing property and Kirszbraun Theorem.I would like to acknowledge Asaf Nachmias for the very excellent courses thathe gave here at UBC which helped me to understand much better relationshipsbetween symbolic dynamics and statistical mechanics, and also for his support andencouragement. I am deeply grateful to him and Ron Peled for giving me theopportunity to continue my research in the next years with their group at Tel AvivUniversity, which I am sure will be a great experience.My sincere thanks also goes to Stefan Adams for showing me part of the fasci-xvAcknowledgementsnating world of statistical mechanics, for his collaboration, and his warm welcomeat University of Warwick. I also thank the ergodic theory group and the statisticalmechanics group over there for letting me speak in their seminars and being sofriendly to me.I would like to thank the organizers of the program in Counting Complexityand Phase Transitions at the Simons Institute for the Theory of Computing. Inspecial, I thank Alistair Sinclair for giving me the opportunity to participate inthe program and I thank the many amazing people I met over there. In particular,Andrei Bulatov, Vı´ctor Dalmau, David Gamarnik, Leslie Ann Goldberg, PavolHell, Mark Jerrum, Martin Loebl, and Prasad Tetali, who shared some of their timeand listened to my questions and problems.I am also indebted to Andrew Rechnitzer for sharing his knowledge regardinglattice combinatorics and statistical physics models, and for his collaboration in thislast period. I would also like to thank Peter Winkler for his career advices, RicardoGo´mez for his nice reception in Guanajuato and in the Mathematical Congress ofthe Americas, and Arman Raina and Adrien She for their nice work and experi-ments related to the hard-core model.Many thanks to my academic brothers, Nishant Chandgotia and Felipe Garcı´a-Ramos, for their support and company during most of my time at UBC. Certainly,you both added a lot of spice to our trips and my life here in Vancouver, and I hope(and I know) that the future will bring us back together.I would also like to acknowledge some other members of the symbolic dynam-ics and ergodic theory community, in particular Mike Boyle, Douglas Lind, andKarl Petersen for always being a guiding light and an example to follow, MichaelHochman and Tom Meyerovitch for one of the main results that inspired this the-sis, and Michael Schraudner, who introduced me to this world, encouraged me inearlier stages of my career, and also organized and invited me to two great con-ferences (DySyCo 2012 and SyDyGr 2014) in my motherland, Chile. I also thankIva´n Rapaport and Alejandro Maass for trusting in my capacities, and supportingme in my career and the decision to pursue a PhD, and the whole Department ofMathematical Engineering (DIM) at Universidad de Chile for the solid training andeducation that I received there.Thanks to all the UBC Mathematics Department, in particular people who con-xviAcknowledgementstributed to my mathematical culture such as Omer Angel, David Brydges, JoelFriedman, among many others, and all the staff for helping me to deal with “thesystem” and always having the mood to share a good laugh.I am very grateful to all the great (new and old) friends that were part of mylife during these years, all the Auditorium Annex and its joyful miseries, all themembers of The Random Walkers and AA-Friday-FC, the Chilean community inVancouver and Berkeley, and all the coffee shops, transitory people, and cups ofcoffee that were passive (and many times active) witnesses of my endeavours.Finally, I would like to thank and honour all my family, especially my brotherSebastia´n, for understanding, supporting, and believing in me through all theseyears.I gratefully acknowledge Ian Blake, Joel Friedman, Nicholas Harvey, BrianMarcus, Asaf Nachmias, Anthony Quas, Andrew Rechnitzer, and James Zidek forbeing part of my examining and supervisory committee, and the financial supportfrom the University of British Columbia during all my PhD studies.xviiTo the memory of my dog Ron,pure and beautiful stubbornness.xviiiChapter 1Introduction1.1 PreliminariesThe focus of this thesis is the study of spin systems through the view of two comple-mentary areas: statistical mechanics and symbolic dynamics. Both share a commonground with different emphasis, namely• the study of measures on graphs (typically, a lattice such as the d-dimensio-nal integer lattice Zd) where vertices take values on a set of symbols withhard constraints, i.e. measures are not fully supported because some config-urations of symbols are forbidden due to local restrictions; and• the computation of thermodynamic quantities. In particular, we are inter-ested in the development of techniques useful for representing and approxi-mating topological entropy and its generalization, (topological) pressure.Consequently, the main goal of this work is twofold. First, we aim to intro-duce combinatorial and measure-theoretic conditions, establish connections amongthem, and see how combinatorial aspects of hard constrained systems interact withmeasure-theoretic mixing properties. Secondly, we use these insights to improveresults on representation and efficient approximation algorithms for pressure.A classical exampleA good example of a hard constrained system is theZd hard-core lattice gas model.In this model, every site x ∈ Zd is identified with a random variable ωˆ(x) (a spin)that takes values in the set of symbols A = {0,1}. In order to obtain a joint dis-tribution ωˆ = (ωˆ(x))x∈Zd representing an interplay between spins, we consider aninteraction or, more specifically, a nearest-neighbour (n.n.) interaction Φ. A n.n.11.1. Preliminariesinteraction Φ is a function that associates some weight to configurations of sym-bols supported on finite subsets of Zd , by considering local contributions in sitesand bonds. In the case of the hard-core lattice gas model with inverse tempera-ture ξ > 0, given a finite subset A ⊆ Zd and a configuration ω ∈ A Zd , the jointdistribution piωA on A with exterior ω|Ac is given bypiωA ( ωˆ|A = α) = 1{α∈[ω|Ac ]} ·exp(−ξ#1(α))ZΦA,ω, (1.1)where α ∈ A A and ω|Ac denotes the restriction of ω to Ac. Here the n.n. inter-action Φ implicitly determines #1(α) (the number of 1’s in the configuration α),the set [ω|Ac ] (the set of “admissible” configurations), and the partition functionZΦA,ω (a very relevant normalizing factor). By “admissible” configurations we re-fer to α’s such that the new configuration α ω|Ac obtained from ω after replacingω|A by α does not violate any constraint. In the hard-core lattice gas model case,where pairs of adjacent 1’s are forbidden, these configurations can be understood asindependent sets, a familiar concept in combinatorics and graph theory involvinglocal constraints. Notice that ω (in particular, the boundary configuration ω|∂A)has some influence over the distribution inside the volume A. A good part of ourresearch has to do with the study of the influence that boundaries have in the dis-tributions piωA for general systems.The extension of these distributions to the infinite volume Zd is via (nearest-neighbour) Gibbs measures, which are a particular kind of Borel probability mea-sure µ such that for every finite A ⊆ Zd and ω ∈ A Zd , Eµ (·|FAc)(ω) coincideswith piωA µ-a.s. Here, Eµ (·|FAc)(ω) denotes the conditional expectation with re-spect the product σ -algebra onA Ac . The collection pi = {piωA }A,ω is usually knownas the Gibbs specification for Φ. Sometimes there is more than a single measureµ for a given Gibbs specification pi . In that case we talk about the existence of aphase transition (see [35]), one of the central issues in statistical physics.It is common to consider an inverse temperature parameter ξ > 0 to modu-late the strength of interactions. Depending on ξ , the interaction ξΦ can giverise to multiple Gibbs measures (the supercritical regime) or just a single one (thesubcritical regime). The latter case, when there is no phase transition, is related21.1. Preliminarieswith systems where the influence of boundaries decays with the distance. Severalclassical lattice models, like the hard-core lattice gas model, exhibit these two be-haviours for different regimes of ξ . Other instances that exhibit this phenomenonare the (ferromagnetic) Potts model and the (multitype) Widom-Rowlinson model(see [1]).Pressure and topological entropyThe pressure of an interaction is a crucial quantity studied in statistical mechanicsand dynamical systems. In the former, it coincides (up to a sign) with the specificGibbs free energy of a statistical mechanical system (e.g. [35, Part III] and [70,Chapter 3-4]). In the latter, it is a generalization of topological entropy and hasmany applications in a wide variety of classes of dynamical systems, ranging fromsymbolic to smooth systems (e.g. [13, 47, 76]).The pressure of a Zd lattice model with translation-invariant n.n. interactionΦis defined as the asymptotic exponential growth rate of the partition function ZΦBn :P(Φ) := limn→∞logZΦBn|Bn| , (1.2)where ZΦBn denotes the partition function with free boundary (i.e. there is no inter-action with the exterior) and Bn = [−n,n]d ∩Zd is an increasing sequence of boxesthat exhausts Zd as n→ ∞Roughly speaking, the pressure tries to capture the complexity of a given sys-tem by associating to it a nonnegative real number. This value can be representedin several ways: sometimes as a closed formula, other times as a limit and, in thecases of our interest, as the integral of a conditional probability distribution or asthe output of an algorithm.Often, it is a difficult task to compute it. When d = 1, there is a closed-form ex-pression for P(Φ) in terms of the largest eigenvalue of an adjacency matrix formedfrom Φ (see [52, p. 99]). In contrast, when d ≥ 2 there are very few n.n. inter-actions Φ for which P(Φ) is known exactly (e.g. Ising model [66], dimer model[46], square ice-type model [54]). In fact, there are computability constraints forapproximating pressure that in general cannot be overcome. In [44] it was proven31.1. Preliminariesthat, when d ≥ 2, the set of numbers that can arise as topological entropies for Zdshifts of finite type (Zd SFT) coincide with the set of right-recursive enumerablenumbers. When d = 1, the closed-form expression mentioned above always givesan efficient approximation algorithm.A Zd SFT Ω can be understood as the support of a lattice model or also asa particular case of a hard constrained Zd lattice model where the interaction Φis uniform, and it is the main object of study in the area known as multidimen-sional symbolic dynamics. In this case, the topological entropy coincides with thepressure and it is given by the formulah(Ω) = limn→∞log |LBn(Ω)||Bn| , (1.3)whereLBn(Ω) is the set of all configurations with shape Bn that appear in Ω.Despite the general computability constraints, one can still expect to be ableto approximate efficiently the topological entropy and pressure of some systems(formally, to prove the existence of a polynomial time approximation algorithm),and also hope to delineate the general characteristics of systems where this is pos-sible. Part of of this work has to do with developing approximation techniques andfinding conditions for computing these quantities.The process of calculating the pressure P(Φ) (or more particularly, the topo-logical entropy h(Ω)) can be broken up into two steps: representation and approx-imation. Since P(Φ) is usually defined in terms of integrals and/or limits that arevery hard or too slow to compute directly, it is useful to develop alternative for-mulas to represent it in such a way that the new representation can be efficientlyapproximated. In this regard, of particular interest is a correlation decay propertyof Gibbs measures known as strong spatial mixing (SSM), which is a strengtheningof another property known as weak spatial mixing (WSM).Spatial mixing propertiesLet f :N→R be a function such that f (n)↘ 0 as n→ ∞. We say that a Zd latticemodel satisfies WSM with decay function f if for any finite A⊆Zd , B⊆A, β ∈A B,41.1. Preliminariesand any pair of admissible configurations ω1,ω2 ∈A Zd ,∣∣piω1A ( ωˆ|B = β )−piω2A ( ωˆ|B = β )∣∣≤ |B| f (dist(B,∂A)) , (1.4)i.e. given a set B, the influence on the distribution ωˆ|B of the boundary configura-tions ω1|∂A and ω2|∂A decays with the graph distance from B to ∂A, according tof . In other words, WSM implies asymptotic independence between the configura-tion of a finite volume and the boundary configuration outside a large ball aroundthis volume.On the other hand, a Zd lattice model satisfies SSM with decay function f iffor any finite A ⊆ Zd , B ⊆ A, β ∈ A B, and any pair of admissible configurationsω1,ω2 ∈A Zd ,∣∣piω1A ( ωˆ|B = β )−piω2A ( ωˆ|B = β )∣∣≤ |B| f (dist(B,Σ∂A(ω1,ω2))) , (1.5)where Σ∂A(ω1,ω2) = {x ∈ ∂A : ω1(x) 6= ω2(x)}.The main difference between SSM and WSM is that SSM considers the dis-tance of B to only the sites in the boundary Σ∂A(ω1,ω2) where ω1 and ω2 differ.In this case, we can still have a decay of correlation if B is close to ∂A but thedisagreements between ω1 and ω2 are far apart.WSM has direct connections with the nonexistence of phase transitions (see[78]). SSM is a stronger version of WSM and is related with the absence of bound-ary phase transitions [61].There has been a growing interest (see [61, 79, 39, 33]) in SSM, due to its con-nections with fully polynomial-time approximation schemes (FPTAS) for countingproblems which are #P-hard (e.g. approximation algorithms for counting indepen-dent sets [79]; see also [6, 31]), and mixing time of the Glauber dynamics in someparticular systems (see [45, 28]). Of particular interest for us, is that it has alsoproven to be useful for pressure representation and approximation (see [32, 58,15]).Examples of systems that satisfy these properties in some regime include theIsing and Potts models (see [61, 40]), and even some cases where hard constraintsare considered, such as the hard-core lattice gas model (see [79]) and q-colourings51.1. Preliminaries(see [33]).Combinatorial mixing propertiesWhen dealing with hard constraints, we face the problem that sometimes two ormore configurations are not compatible. This is an extra difficulty when prov-ing spatial mixing properties or a useful pressure representation, since many tools(probability couplings, equivalences, etc.) have been developed only for the non-constrained cases. By combinatorial mixing properties we refer to conditions onthe support Ω of a hard constrained system that allow us to “glue” together con-figurations in an admissible way. Two relevant properties are strong irreducibilityand the existence of a safe symbol.A Zd SFT Ω is strongly irreducible with gap g ∈ N if for any A,B ⊆ Zd suchthat dist(A,B)≥ g, and for every α ∈LA(Ω) and β ∈LB(Ω),α ∈LA(Ω),β ∈LB(Ω) =⇒ αβ ∈LA∪B(Ω), (1.6)where αβ denotes the configuration obtained after combining the configurations αand β into a single one. In simple terms, a Zd SFT Ω is strongly irreducible if anytwo admissible (partial) configurations can be glued together in a (full) configura-tion ω ∈Ω, provided their supports are sufficiently far apart.Given a Zd SFT Ω⊆A Zd , a safe symbol s ∈A for Ω is a symbol that can beadjacent to any other symbol in A (think of 0 in the hard-core lattice gas model).This property allows us to replace any symbol by s and preserve admissibility ofconfigurations while we modify them. The existence of a safe symbol has provento be useful to develop pressure representation and approximation theorems (see[32, Assumption 1]). In addition, the existence of a safe symbol implies strongirreducibility.Strong irreducibility and the existence of a safe symbol are qualitatively dif-ferent properties. The former is a global condition that involves the lattice as awhole. On the other hand, the existence of a safe symbol is a local condition.An interesting family of examples is the case of proper colourings of the lattice(q-colourings), where given q colours the constraints impose that two adjacent ver-tices cannot have the same colour. It is easy to check that the underlying Zd SFT61.1. Preliminariesdoes not contain any safe symbol. However, if we have enough colours the systemdoes satisfy a weaker combinatorial condition introduced in [58] called single site-fillability (SSF). SSF is a local condition that also implies strong irreducibility andhas shown to be useful for having have pressure representation and approximationtheorems (see [58]).Part of the work presented here involves pushing this even further and lookingfor weaker combinatorial properties still useful for these purposes. Of particularinterest is the study of characteristics that hard constraints should satisfy in orderto be (to some extent) compatible with the SSM property.Pressure representationIn [32, 58] it was proven that SSM (with decay function f (n) = Ce−γn) togetherwith some extra combinatorial properties are enough for having a pressure repre-sentation and approximation algorithms for P(Φ). The basic idea was motivatedby the Variational Principle (see [47, Section 4.4]), given by the formulaP(Φ) = supµ∈M1,σ (Ω)(h(µ)+∫AΦdµ), (1.7)where Ω is the support of the Zd lattice model, M1,σ (Ω) denotes the set of allshift-invariant (or stationary) Borel probability measures µ supported on Ω, andAΦ is an auxiliary function defined as AΦ(ω) =−Φ(ω|{~0})−∑di=1Φ(ω|{~0,~ei}),where~0 denotes the origin and {~ei}di=1 denotes the canonical basis of Zd . Here, thequantity h(µ) is the measure-theoretic entropy of µ . Any measure that achieves thesupremum is known as an equilibrium state, which (under very mild conditions)coincide with a Gibbs measures for Φ.Given an equilibrium state µ , we have that P(Φ) = h(µ)+∫AΦdµ . It is knownthat the measure-theoretic entropy h(µ) can be expressed as the integral with re-spect to µ of the information function Iµ . The information function is defined µ Iµ(ω)=− logµ(ωˆ(~0) = ω(~0)∣∣∣ω|P), whereP denotes the lexicographic pastof the origin~0. In other words, given ω ∈A Zd , Iµ(ω) is the negative logarithm ofthe conditional probability of having the value ω(~0) at the origin, given that everysite in the lexicographic past coincides with ω . Then we can write h(µ) =∫Iµdµ71.1. Preliminariesand we haveP(Φ) =∫(Iµ +AΦ)dµ, (1.8)for any equilibrium state µ . The idea in [32, 58] was to represent P(Φ) as theintegral of the same integrand, but with respect to a simpler measure ν , i.e.P(Φ) =∫(Iµ +AΦ)dν . (1.9)This can be regarded as a pressure representation. A pressure representationbecomes especially useful for approximating P(Φ) in the case ν is a periodic pointmeasure, i.e. a measure which assigns equal weight to each distinct translation ofa given periodic configuration ω ∈ A Zd . Then ∫ (Iµ +AΦ)dν becomes a finitesum. The terms in this sum corresponding to AΦ are easy to compute. In this way,the problem of approximating P(Φ) reduces to approximate Iµ on a single peri-odic configuration ω and its translates. We remark that to approximate Iµ(ω) isequivalent to just approximate µ(ωˆ(~0) = ω(~0)∣∣∣ω|P), the conditional probabilityof a single site taking some particular value. Then this new way to express P(Φ)by-passes the need of computing the integral with respect to µ . A pressure rep-resentation requires some assumptions on the measures µ and ν , and also on thesupport Ω. In general, it may fail. For example, when considering 3-colourings inZ2 (see [58]), the representation formula in Equation (1.9) is not always true.Pressure approximationGiven a pressure representation as in Equation (1.9), the problem of approximatingP(Φ) reduces to know how to approximate Iµ efficiently. In [32] an approach wasgiven to approximate µ(ωˆ(~0) = ω(~0)∣∣∣ω|P), and therefore Iµ(ω), in polynomialtime in some region of the subcritical regime for the Zd hard-core lattice gas model.This approach was based on the computational tree method developed by Weitz in[79], which is a powerful technique in the binary case (i.e. |A | = 2). In [58], analternative method was develop for the Z2 case to approximate Iµ efficiently, basedon the transfer matrix method. Both approaches rely in the assumption that themeasures involved satisfy the SSM property (with exponential decay) plus somecombinatorial property on the support. We review this approach and also deal with81.2. Results overviewthe (a priori, radically different) case when the SSM property fails.1.2 Results overviewThe results are divided in two main parts:I. Development of a robust framework for working with hard constrained sys-tems in general graphs, and study measure-theoretic correlation decay prop-erties on them.II. Study of techniques for representing and efficiently computing pressure inthe uniqueness and non-uniqueness regions in Zd lattice models.Part I: Combinatorial aspects of the strong spatial mixing propertyThere is a good amount of literature devoted to study the regimes where SSM holdsfor particular systems. In particular, the Ising and Potts models [53, 80], the hard-core lattice gas model [79], and q-colourings [31] have received special attention.Given a locally finite countable graph G (the board), a finite set of symbols A ,some hard constraints, and an interaction Φ, we can define very general systems(Ω,Φ), where Ω is the configuration space, the subset of configurations not violat-ing any constraint.We develop a new framework for studying SSM in general hard constrainedspin systems. This is done in part through the introduction of a combinatorialmixing property named topological strong spatial mixing (TSSM). A configurationspaceΩ is TSSM with gap g∈N if for any subsets A, B, and S such that dist(A,B)≥g (with respect to the graph distance in G ), and for every α ∈LA(Ω), β ∈LB(Ω),and σ ∈LS(Ω),ασ ∈LA∪S(Ω),σβ ∈LS∪B(Ω) =⇒ ασβ ∈LA∪S∪B(Ω). (1.10)To develop the TSSM property we combine notions from combinatorics inlattice models (like the existence of a safe symbol and SSF) and from symbolicdynamics, in particular strong irreducibility. The result is a hybrid concept strictlyweaker than all the combinatorial properties used in [32] and [58], and strictly91.2. Results overviewstronger than strong irreducibility (notice that if we take S = /0 in Equation (1.10)we recover the definition of strong irreducibility).We prove that, under certain hypotheses, if a system satisfies SSM, then itssupport satisfies TSSM. As two of many applications, we show that the space of4-colourings on the Z2 lattice does not satisfy SSM for any interaction Φ (seeProposition 3.5.6; it has been suggested in the literature [74] that a uniform Gibbsmeasure on this system should satisfy WSM), and that TSSM implies the existenceof periodic points in Zd shift spaces, for every d ∈N (see Proposition 3.4.3). Later,in Part II, we use TSSM to generalize work in [32] and [58], giving conditions forsimple representation and efficient approximation of pressure.One of the main ideas is that in order to have a measure-theoretic correlationdecay property like SSM, there may be some necessary combinatorial conditionson the configuration space. We also explore a complementary question: given aconfiguration space satisfying a particular combinatorial property, can we alwaysfind a Gibbs specification satisfying SSM supported on it? In general, if a con-figuration space just satisfies strong irreducibility, the answer is no. For example,the space of 4-colourings on Z2 satisfies strong irreducibility, but no measure sup-ported on it can satisfy SSM.In [17], Brightwell and Winkler did a complete study of the family of disman-tlable graphs, including several interesting alternative characterizations. Amongthe equivalences discussed in that work, many involved a board G , a finite graph H(the constraint graph, assumed to be dismantlable), and the set of all graph homo-morphisms from G to H, which we denote here by Hom(G ,H). We call such a setof graph homomorphisms a homomorphism space, and it can be understood as aconfiguration space Ω. In this context, we should interpret the set of vertices V(H)of H as the set of symbols A in some spin system living on vertices of G . The ad-jacencies given by the set of edges E(H) of H indicate the pairs of symbols that areallowed to be next to each other in G , and the edges that are missing can be seenas hard constraints in our system (i.e. pair of spin values that cannot be adjacent inG ). Examples of such systems are very common. If we consider G = Z2 and Hϕ ,with V(Hϕ) = {0,1} and E(Hϕ) containing every edge but the loop connecting 1with itself, then Hom(Z2,Hϕ) represents the support of the hard-core lattice gasmodel in Z2, i.e. the set of independent sets in the square lattice.101.2. Results overviewWe are interested in combinatorial mixing properties that are satisfied by ahomomorphism space Hom(G ,H), i.e. properties that allow us to “glue” togetherpartial configurations in G . For example, the homomorphism space Hom(Z2,H),where V(H) = {0,1} and H has a unique edge connecting 0 with 1, has only twoelements, both checkerboard patterns of 0s and 1s. This homomorphism spacelacks good combinatorial mixing properties since, for example, it is not possibleto “glue” two 0s together which are separated by an odd distance horizontally orvertically. Note that this is not the case for Hom(Z2,Hϕ), where the only differenceis that Hϕ has in addition an edge connecting 0 with itself. A gluing propertyof particular interest is strong irreducibility. In [17], dismantlable graphs werecharacterized as the only graphs H such that Hom(G ,H) is strongly irreducible forevery board G .In addition, we can consider a n.n interaction Φ that associates some “energy”to every vertex and edge of a constraint graph H. From this we can constructa Gibbs (G ,H,Φ)-specification pi , which is an ensemble of probability measuressupported in finite portions of G . Specifications are a common framework forworking with spin systems and defining Gibbs measures µ . From this point it ispossible to start studying spatial mixing properties, which combine the geometryof G , the structure of H, and the distributions induced by Φ. In [17], dismantlablegraphs were also characterized as the only graphs H for which for every board Gof bounded degree there exists a n.n. interaction Φ such that the Gibbs (G ,H,Φ)-specification pi has no phase transition (i.e. there is a unique Gibbs measure forpi).We consider the problem of existence of strong spatial mixing measures sup-ported on homomorphism spaces. First, we extend the results of Brightwell andWinkler on uniqueness, by characterizing dismantlable graphs as the only graphsH for which for every board G of bounded degree there exists a n.n. interactionΦ such that the Gibbs (G ,H,Φ)-specification pi satisfies WSM (with exponentialdecay function; see Section 4.2). Then we give sufficient conditions (see Sec-tion 4.3) on H and Hom(G ,H) for the existence of Gibbs (G ,H,Φ)-specificationssatisfying SSM (with exponential decay function). Since SSM implies WSM, anecessary condition for SSM to hold in every board G is that H is dismantlable.We exhibit examples showing that SSM is a strictly stronger property, in terms of111.2. Results overviewcombinatorial properties of H and Hom(G ,H), than WSM. In particular, there existdismantlable graphs H where SSM fails in Hom(G ,H), for some boards G .To our knowledge, this is the first attempt to characterize the constraint graphsH that are suitable for SSM to hold for general boards G . Related work was done in[27], where the family of dismantlable graphs was related with rapid mixing of theGlauber dynamics in finite boards with free boundary conditions. In contrast, wedemonstrate that the role of boundary conditions cannot be ignored when we arelooking for Gibbs specifications satisfying SSM in models with hard constraints.Other combinatorial and measure-theoretic mixing properties have been con-sidered in the literature of lattice models. We also explore the relationships be-tween some of them. In Part II, we see how they are useful in some cases forrepresentation and approximation. In particular, in the context of stationary ZdGibbs measures µ satisfying SSM, we establish many useful properties that showhow TSSM on the support of µ is sufficient for a special pressure representationand efficient approximation algorithms.Part II: Representation and poly-time approximation for pressureThe main focus of Part II is to find simple representations of pressure and use thisto develop efficient (in principle) algorithms to approximate it, profiting from themeasure-theoretical and combinatorial mixing properties studied in Part l.There is much work in the literature on numerical approximations for pressure(see [7, 29]). We take a theoretical computer science point of view (see [50]): analgorithm for computing a real number r is said to be poly-time if for every N ∈N,the algorithm outputs an approximation rN to r, which is guaranteed to be accuratewithin 1N and takes time at most polynomial in N to compute. In that case, wesay that r is poly-time computable. One of our goals is to prove the existence ofpoly-time algorithms for P(Φ) under certain assumptions on Φ and the support Ω.Following the works of Gamarnik and Katz [32], and Marcus and Pavlov [58],we provide extended versions of representation theorems of pressure in terms ofconditional probabilities and also conditions for more general approximation al-gorithms. In [32], for obtaining such representation and approximation theorems,they assumed the existence of a safe symbol, which is a very strong combinato-121.2. Results overviewrial condition, together with SSM (and an exponential assumption on the decayfunction of SSM for algorithmic purposes). Later, in [58], these assumptions werereplaced by more general and technical conditions in the case of representation,and the SSF property, which generalized the safe symbol case both in the represen-tation and in the algorithmic results. Here, making use of the theoretical machinerydeveloped in [58], we have relaxed those conditions even more by using the moregeneral property of TSSM.Next, we continue the development of representing the pressure with a sim-plified expression. We prove a new pressure representation theorem and differ-ent techniques to tackle the problem of pressure approximation in supercriticalregimes, where long range correlations do not decay and therefore SSM cannothold. We considered three classical lattice models: the (ferromagnetic) Potts, (mul-titype) Widom-Rowlinson, and hard-core lattice gas models. For Widom-Rowlin-son and hard-core lattice gas, the techniques apply to certain subsets of both thesubcritical and supercritical regions, where the main novelty is in the latter. Inthe case of the Potts model with q colours, we obtain a pressure representationfor any β > 0 and a poly-time approximation algorithm for any β 6= βc(q), whereβc = log(1+√q) denotes the critical inverse temperature.To do this the main tools are coupling techniques in Markov random fields (see[11]) and the relation of these models (in particular, Potts and Widom-Rowlinson)with random-cluster models (see [42]). Two of the most exploited couplings werethe van den Berg and Maes coupling for Markov random fields inducing paths ofdisagreement (see [11]), and the Edwards-Sokal coupling, relating the Potts modelwith the bond random-cluster model (see [42]).The pressure representation theorems developed in [32] and [58] were givenin terms of the information function Iµ , that depends on a stationary Gibbs mea-sure µ for an interaction Φ. In order to have analogous representation formulas forsupercritical regimes, we had to modify the representation formula by introduc-ing a closely related function Ipi , which depends only on the Gibbs specification{piωA }A,ω . This turned out to be necessary but at the same time natural, since pres-sure depends only on the interaction and not on any particular Gibbs measure µ .The results hold in every dimension d. Among other conditions, these resultsrequire conditions onΩ and a convergence condition for certain sequences of finite-131.2. Results overviewvolume half-plane measures (different convergence conditions in the different re-sults). In the case d = 2, if the convergence holds at exponential rate, then oneobtains a poly-time algorithm for approximating P(Φ). For d > 2, if the exponen-tial convergence holds, one can deduce an algorithm for approximating P(Φ) withsub-exponential but not polynomial rate. We remark that the finite-volume half-plane measures mentioned above typically are constant on their bottom boundariesand thus are related to wetting models (see [68, 73]).In [32], the convergence condition is SSM of a Gibbs measure µ for the n.n.interaction Φ. This condition is known to imply that there is a unique Gibbs mea-sure for Φ and thus can be applied only in the uniqueness (subcritical) region ofa given lattice model. The convergence conditions in [58] are weaker but also ap-ply primarily to this region. However, since our convergence condition dependsonly on the interaction, one might expect that the pressure representation and ap-proximation results can apply in the non-uniqueness region as well. Indeed, theydo. As illustrations, we apply these results to explicit subcritical and supercriticalsub-regions of three main classical lattice models in Z2. In particular, for the pres-sure approximation results for these models, we establish the required exponentialconvergence conditions (see Section 6.5). However, we believe that our results areapplicable to a much broader class of models, in particular satisfying weaker con-ditions on Ω. We remark that the SSM condition of [32] is a much stronger versionof our condition, and so in this sense our results generalize some results of thatpaper (in particular, for the Z2 hard-core lattice gas model).In the case of the 2-dimensional Potts model, we obtain a pressure representa-tion and efficient pressure approximation for all β 6= βc(q), where βc(q) = log(1+√q) is the critical value which separates the uniqueness and non-uniqueness re-gions. Our proof in the non-uniqueness region generalizes a result from [22] forq = 2 (i.e. the Ising model) and we closely follow their proof, which relies heav-ily on a coupling with the bond random-cluster model and planar duality. For theuniqueness region, our result follows from [4].For the Widom-Rowlinson and hard-core lattice gas models, our results are notas complete as in the Potts case, since the subcritical and supercritical regions forthese two models haven’t been completely determined. We also expect our resultscan be improved, because they only apply to proper subsets of the currently known141.3. Thesis structureuniqueness/non-uniqueness regions.For the Widom-Rowlinson model, in the supercritical region, we use a varia-tion of the disagreement percolation technique introduced in [11], combined withthe connection between the Widom-Rowlinson model and the site random-clustermodel. In the subcritical region, we apply directly the results in [11].For the hard-core lattice gas model, in the supercritical region, we combinethe coupling in [11] and a Peierls argument used by Dobrushin (see [26]). In thesubcritical region, we use a recent result on SSM for the hard-core lattice gas modelin Z2.For the Potts model, we also extend the pressure representation, by a continuityargument, to give an expression for the pressure at criticality (see Corollary 13). Itis of interest that there is an exact, explicit, but non-rigorous, formula for the pres-sure at criticality due to Baxter (see [8]). So, our rigorously obtained expressionshould agree with that formula, though we do not know how to prove this state-ment. It seems that Baxter’s explicit expression gives a poly-time approximationalgorithm, but we cannot justify that our expression is poly-time computable.1.3 Thesis structureThe thesis is split in mainly two parts.Part I consists of Chapter 2, Chapter 3, and Chapter 4.In Chapter 2, we give the basic notions of graph theory (Section 2.1), configu-ration spaces (Section 2.2), and Gibbs measures (Section 2.3).In Chapter 3, we define spatial mixing (Section 3.1) and combinatorial (Section3.2) properties, we introduce the topological strong spatial mixing property (Sec-tion 3.3), establish properties and characterizations of it, and relationships to somemeasure-theoretic quantities (Section 3.4). Next, we provide connections betweenmeasure-theoretic and combinatorial mixing properties (Section 3.6); in particular,Theorem 3.6.5 provides evidence that TSSM is closely related with SSM. In Sec-tion 3.5, we give several examples illustrating different types of mixing propertiesin the context of Zd SFTs.In Chapter 4, we introduce the necessary background for studying homomor-phism spaces and dismantlable graphs (Section 4.1). After that, we explore the151.3. Thesis structureconnections between dismantlability and WSM (Section 4.2). Then we introducethe unique maximal configuration (UMC) property (Section 4.3) and show that thisproperty is sufficient for having a Gibbs specification satisfying SSM with arbi-trarily high exponential decay of correlations (Section 4.4). We introduce a fairlygeneral family of graphs H, strictly contained in the family of dismantlable graphs,such that Hom(G ,H) satisfies the UMC property for every board G (Section 4.5).Theorem 4.6.3 provides a summary of relationships and implications among theproperties studied (we encourage the reader to take a quick look to it before a moredetailed reading). Later, we focus in the particular case where H is a (looped)tree T and conclude that the properties on T yielding WSM for some measure onHom(G ,T ) coincide with those yielding SSM (Section 4.6). At the end of thischapter, we provide examples illustrating the qualitative difference between thecombinatorial properties necessary for WSM and SSM to hold in homomorphismspaces (Section 4.7).Part II consists of Chapter 5, Chapter 6, and Chapter 7.In Chapter 5, we introduce topological entropy (Section 5.1) and topologicalpressure (Section 5.2). We discuss some variational principles and then we exhibitsome pressure representation theorems (Section 5.3).In Chapter 6, we review the three specific Zd lattice models to which we applyour main results (Section 6.1), and we review the bond random-cluster (Section 6.2)model and site random-cluster model (Section 6.3), where Lemma 6.3.2 is a novelresult concerning the latter. Next, we review some spatial mixing and stochasticdominance concepts (Section 6.4) and use this to help establish exponential con-vergence results (Section 6.5).In Chapter 7, we combine the pressure representation theorems and exponen-tial convergence results in order to obtain efficient approximation algorithms forpressure (Section 7.1).Finally, in Chapter 8, we discuss possible future directions of research.16Part ICombinatorial aspects of thestrong spatial mixing property17Chapter 2Basic definitions and notation2.1 GraphsA graph is an ordered pair G = (V,E) (or G = (V (G),E(G)) if we want to em-phasize the graph G), where V is a finite set or a countably infinite set of elementscalled vertices, and E is contained in the set of unordered pairs {{x,y} : x,y ∈V},whose elements we call edges. We denote x ∼ y (or x ∼G y if G is not clear fromthe context) whenever {x,y} ∈ E, and we say that x and y are adjacent. A vertexx is said to have a loop if {x,x} ∈ E. The set of looped vertices of a graph Gwill be denoted Loop(G) := {x ∈V : {x,x} ∈ E} and G will be called simple ifLoop(G) = /0. A graph G is finite if |G|< ∞, where |G| denotes the cardinality ofV (G), and infinite otherwise.Fix n∈N. A path (of length n) in a graph G will be a finite sequence of distinctedges {x0,x1},{x1,x2}, . . . ,{xn−1,xn}. A single vertex x will be considered to be apath of length 0. A cycle (of length n) will be a path such that x0 = xn. Notice that aloop is a cycle. A vertex y will be said to be reachable from another vertex x if thereexists a path (of some length n≥ 0) such that x0 = x and xn = y. For A1,A2 ⊆V , apath from A1 to A2 is any path whose first vertex is in A1 and whose last vertex isin A2. A graph will be said to be connected if every vertex is reachable from anyother vertex, and a tree if it is connected and has no cycles. A graph which is a treeplus possibly some loops, will be called a looped tree.For a vertex x, we define its neighbourhood N(x) as the set {y ∈V : y∼ x}. Agraph G will be called locally finite if |N(x)|<∞, for every x ∈V . A locally finitegraph will have bounded degree if ∆(G) := supx∈V |N(x)|<∞. In this case we saythat G is of maximum degree ∆(G). Given d ∈ N, a graph of maximum degree dis d-regular if |N(x)|= d, for all x ∈V .We say that a graph G′ = (V ′,E ′) is a subgraph of G = (V,E) if V ′ ⊆ V and182.1. GraphsE ′ ⊆ E. For a subset of vertices A⊆V , we define the subgraph of G induced by Aas G[A] := (A,E[A]), where E[A] := {{x,y} ∈ E : x,y ∈ A}.2.1.1 BoardsA board G = (V ,E ) will be any simple, connected, locally finite graph with atleast two vertices, where V is the set of vertices (or sites) and E is the set of edges(or bonds). We will use the letters x, y, etc. for denoting the vertices in a board.Boards and configurations on them (see Section 2.2) will be the main playgroundin this work.Example 2.1.1. Given d ∈ N, we have the two following families of boards:• The d-dimensional integer lattice Zd (see Subsection 2.1.2).• The d-regular tree Td (also known as Bethe lattice).Figure 2.1: A sample of the boards Z2 and T3.Given a board G = (V ,E ), we can define a natural distance function betweenvertices x,y ∈ V , namelydist(x,y) := min{n : ∃ a path of length n s.t. x = x0 and xn = y}, (2.1)which can be extended to sets A1,A2 ⊆ V as dist(A1,A2) = minx∈A1,y∈A2 dist(x,y).We denote B b A whenever a finite set B ⊆ V is contained in a set A ⊆ V .When denoting subsets of V that are singletons, brackets will usually be omitted,e.g. dist(x,A) will be regarded to be the same as dist({x},A). Notice that if x ∈192.1. GraphsA, then dist(x,A) = 0. Given A ⊆ V , we define its (outer) boundary as ∂A :={x ∈ V : dist(x,A) = 1} and its closure as A := A∪ ∂A. The inner boundary ofA will be the set ∂A := ∂Ac, i.e. the set of sites x in A which are adjacent tosome other site in Ac. Given n ∈ N, we call Nn(A) := {x ∈ V : dist(x,A)≤ n}the n-neighbourhood of A, and ∂nA :=Nn(A)\A, the n-boundary of A. Noticethat N0(A) = A, N1(x) = N(x)∪{x}, and ∂1(A) = ∂A. For A b V , we define itsdiameter as diam(A) := maxx,y∈A dist(x,y).2.1.2 The d-dimensional integer latticeThe board of main interest will be the d-dimensional integer lattice Zd . Givend ∈ N, we define Zd = (V (Zd),E (Zd)) to be the 2d-regular (countably infinite)graph such thatV (Zd) = Zd , and E (Zd) ={{x,y} : x,y ∈ Zd ,‖x− y‖= 1}, (2.2)with ‖x‖ = ∑di=1 |xi| the 1-norm, for x = (x1, . . . ,xd) ∈ Zd . In a slight abuse ofnotation, we will use Zd to denote both the set of vertices and the board itself.A natural order on Zd is the so-called lexicographic order, where y 4 x (orequivalently, x< y) iff y = x or yi < xi, for the smallest 1≤ i≤ d for which yi 6= xi.Considering this order, we define the (lexicographic) past of Zd asP :={x ∈ Zd \{~0} : x4~0}, (2.3)where~0 denotes the origin. Given y,z ∈ Zd such that y,z ≥~0 (here ≥ denotes thecoordinate-wise comparison of vectors), we also define the [y,z]-block as the setBy,z :={x ∈ Zd :−y≤ x≤ z}, (2.4)and the broken [y,z]-block asQy,z := By,z \P ={x<~0 :−y≤ x≤ z}. (2.5)In addition, given n ∈N, we define the n-block as Bn := B~1n,~1n and the brokenn-block as Qn = Bn \P , where~1 denotes the vector (1, . . . ,1) ∈ Zd . Notice that202.1. GraphsBn = [−n,n]d ∩Zd . We also define the n-rhomboid as Rn :=Nn({~0}).In Zd we can also define an alternative notion of adjacency and therefore, an al-ternative notion of boundary, inner boundary, closure, path, connectedness, etc., byreplacing the 1-norm ‖·‖with the ∞-norm ‖·‖∞, defined as ‖x‖∞ =maxi=1,...,d |xi|,for x ∈ Zd . When referring to these notions with respect to the ∞-norm, we willalways add a ? superscript and talk about ?-adjacency x ?∼ y , ?-boundary ∂ ?A,inner ?-boundary ∂ ?A, ?-closure A?, ?-path, ?-connectedness, etc. Notice that twovertices x and y are ?-adjacent if they are adjacent in a version of the d-dimensionalinteger lattice Zd including in addition diagonal edges. We will denote this versionof the lattice by Zd,?.2.1.3 Constraint graphsA constraint graph H = (V,E) is a finite graph, where loops are allowed. Themain role of constraint graphs will be to prescribe adjacency rules for values ofvertices in a given board (see Subsection 2.2.1). A difference between boards andconstraint graphs is that the latter must be finite. Another one is that constraintgraphs are allowed to have loops. We will denote by H= (V,E) the constraintgraph obtained by adding loops to every vertex, i.e. E= E∪{{u,u} : u ∈ V}and Loop(H) = V. We will use the letters u, v, etc. for denoting vertices in aconstraint graph.Figure 2.2: The graphs Kn and Kn , for n = 5.A constraint graph will be called complete if u ∼ v, for every u,v ∈ V suchthat u 6= v. The complete graph with n vertices will be denoted Kn (notice thatLoop(Kn) = /0). A constraint graph will be called loop-complete if u ∼ v, for212.1. Graphsevery u,v ∈ V. Notice that the loop-complete graph with n vertices is K n . Thegraphs Kn and Kn are very important examples (see Example 2.3.1) of constraintgraphs, which relate to proper colourings of boards and unconstrained models,respectively. Other relevant examples are the following.Example 2.1.2. The constraint graph given byHϕ := ({0,1},{{0,0},{0,1}}) , (2.6)shown in Figure 2.1.2, is related to the hard-core model (see Example 2.3.1).Figure 2.3: The graph Hϕ .Given n ∈ N, the n-star graph is defined asSn = ({0,1, . . . ,n},{{0,1}, . . . ,{0,n}}) . (2.7)In addition, it will be useful to consider the constraint graphsSon = (V(Sn),E(Sn)∪{{0,0}}) , (2.8)and Sn (see Figure 2.4). Notice that Hϕ = So1.Figure 2.4: The graphs S6, So6, and S6 .222.2. Configuration spaces2.2 Configuration spacesFix a board G = (V ,E ) and let A be a finite set of symbols called alphabet.We endow A with the discrete topology and A V with the corresponding producttopology. We will call any closed subset Ω ⊆ A V a configuration space. Theelements in Ω will be called points and will be denoted with the Greek letters ω ,υ , etc.Two families of configuration spaces considered here are homomorphism spa-ces (see Section 2.2.1) and Zd shift spaces (see Section 2.2.2).Given A ⊆ V , a configuration will be any map α : A→ A (or equivalently,α ∈ A A), denoted with lowercase Greek letters α , β , etc. The set A is called theshape of α , and a configuration will be said to be finite if its shape is finite. Noticethat, in particular, a point is a configuration with shape V . For any configurationα with shape A and B⊆ A, α|B denotes the sub-configuration of α occupying B,i.e. the map from B to A obtained by restricting the domain of α to B. Given asymbol a ∈A , aA will denote the configuration of all a’s on A.Given a configuration space Ω, a configuration α ∈A A is said to be globallyadmissible if there exists ω ∈Ω such that ω|A = α . The languageL (Ω) of Ω isthe set of all finite globally admissible configurations, i.e.L (Ω) :=⋃AbVLA(Ω), (2.9)where LA(Ω) := {ω|A : ω ∈Ω}, for A ⊆ V . Given A ⊆ V and a configurationα ∈A A, we define the cylinder set[α]Ω := {ω ∈Ω : ω|A = α} . (2.10)We will say that [α]Ω has support A. When omitting the superscriptΩ, we willconsider [α] to be the cylinder set for Ω=A V .For A1 and A2 disjoint sets, α1 ∈A A1 , and α2 ∈A A2 , α1α2 will be the config-uration on A1unionsqA2 (here unionsq denotes the disjoint union) defined by (α1α2)|A1 = α1and (α1α2)|A2 = α2. In particular, [α1α2]Ω = [α1]Ω∩ [α2]Ω.Notice that α ∈A A is globally admissible iff α ∈LA(Ω) iff [α]Ω 6= /0.232.2. Configuration spaces2.2.1 Homomorphism spacesA natural way to define a rich family of configuration spaces is by relating boardsand constraint graphs via graph homomorphisms.Definition 2.2.1. A graph homomorphism α : G1→G2 from a graph G1 = (V1,E1)to a graph G2 = (V2,E2) is a mapping α : V1→V2 such that{x,y} ∈ E1 =⇒ {α(x),α(y)} ∈ E2. (2.11)Given two graphs G1 and G2, we will denote by Hom(G1,G2) the set of allgraph homomorphisms α : G1→ G2 from G1 to G2.Fix a board G and a constraint graph H. We will call homomorphism spacethe set Hom(G ,H). In this context, the graph homomorphisms that belong to Ω=Hom(G ,H) can be understood as points in a configuration space contained inA V ,where A = V(H). Notice that a point ω ∈Ω is a “colouring” of V with elementsfrom V(H) such that x∼G y =⇒ ω(x)∼H ω(y). In other words, ω is a colouringof G that respects the constraints imposed by H with respect to adjacency.Example 2.2.1. For d ∈ N, two examples of homomorphism spaces are• Hom(Zd ,Hϕ), the set of elements in {0,1}Zd with no adjacent 1s, and• Hom(Td ,Kq), the set of proper q-colourings of the d-regular tree.2.2.2 Shift spaces in ZdFix the board to be Zd , for some d ∈N. For an alphabetA , we can define the shiftaction σ : Zd ×A Zd → A Zd given by (x,ω) 7→ σx(ω), where x ∈ Zd , ω ∈A Zd ,and (σx(ω))(y) =ω(x+y), for y ∈ Zd . In this case, we callA Zd the full shift andwe can consider the metric m(ω,υ) := 2− inf{‖x‖:ω(x)6=υ(x)}, for ω,υ ∈A Zd , whichinduces the product topology in A Zd, i.e. (A Zd,m) is a compact metric space.We can also extend the shift action σ to configurations with arbitrary shapes,i.e. given α ∈A A and x ∈ Zd , we define σx(α) ∈A A−x as the configuration suchthat (σx(α))(y) = α(x+ y), for y ∈ A− x = {y− x : y ∈ A}.242.2. Configuration spacesWe are interested in configuration spaces Ω ⊆ A Zd that are shift-invariant,i.e. σx(Ω) = {σx(ω) : ω ∈Ω}=Ω, for all x ∈ Zd . In order to obtain such spaces,given a family of finite configurations F⊆⋃AbZdA A, we defineΩF :={ω ∈A Zd : σx(ω)|A /∈ F, for all Ab Zd , for all x ∈ Zd}. (2.12)Here,Ω=ΩF⊆A Zd is called a Zd shift space and it is the set of all points thatdo not contain a translation of an element from F as a sub-configuration. Noticethat a Zd shift spaceΩ is always a shift-invariant set, i.e. σx(Ω) =Ω, for all x∈Zd .In fact, a subset Ω ⊆ A Zd is a Zd shift space iff it is shift-invariant and closed inthe product topology (see [56, Theorem 6.1.21] for a proof of the case d = 1. Thegeneral case is analogous). More than one family F can define the same Zd shiftspace Ω and, when Ω can be defined by a finite family F, it is said to be a Zd shiftof finite type (Zd SFT). A Zd SFT is a Zd nearest-neighbour (n.n.) SFT if Fcan be partitioned in sets {Ei}di=1 such that each Ei contains configurations onlyon shapes on edges of the form {~0,~0+~ei}, where ~e1, . . . ,~ed denote the canonicalbasis. We will mostly restrict our attention to Zd n.n. SFTs. In such case, we callF =⊔di=1Ei a set of n.n. constraints and, given a set A ⊆ Zd and a configurationα ∈A A, we say that α is feasible for F if for every x∈ A and for every i= 1, . . . ,dsuch that {x,x+~ei} ⊆ A, we have that σ−x(α|{x,x+~ei})/∈ Ei. Notice that the Zdn.n. SFT induced by F is the set ΩF ={ω ∈A Zd : ω is feasible for F}.Notice that homomorphism spaces Hom(Zd ,H) are a particular case of Zdn.n. SFTs, where additional symmetries are preserved besides translations (inparticular, any automorphism of Zd). For example, the homomorphism spaceΩdϕ = Hom(Zd ,Hϕ) can be also regarded as a Zd n.n. SFT. When d = 2, Ω2ϕ itis known as the hard square shift.In some contexts, a feasible configuration α ∈A A is said to be locally admis-sible and it is called globally admissible if in addition it extends to a point of Ω.A globally admissible configuration is always locally admissible, but the converseis false (see Section 3.5.7 for an example). In addition, notice that if a configura-tion α ∈A A is globally (resp. locally) admissible, then α|B is also globally (resp.locally) admissible, for any B⊆ A.A point ω ∈A Zd is periodic (of period k) in the ith direction if σk~ei(ω) =ω ,252.3. Gibbs measuresfor k ∈N, and periodic if it is periodic in the ith direction, for every i= 1, . . . ,d. Aperiodic point ω is a fixed point if σx(ω) = ω , for all x ∈ Zd . We define the orbitof a point ω as the set O(ω) :={σx(ω) : x ∈ Zd}. Notice that a point ω is periodiciff |O(ω)|< ∞, and ω is a fixed point iff |O(ω)|= 1.2.3 Gibbs measuresIn this section we develop the Gibbs formalism, focusing on the case where theconfiguration space is a homomorphism space or/and a Zd n.n. SFT. We will re-fer to the conjunction of these two classes as invariant nearest-neighbour (n.n.)configuration spaces.2.3.1 Borel probability measures and Markov random fieldsGiven a set A ⊆ V , we denote by FA the σ -algebra generated by all the cylindersets [α] with support A, and we equip A V with the σ -algebraF =FV . A Borelprobability measure µ onA V is a measure such that µ(A V ) = 1, determined byits values on cylinder sets of finite support.For notational convenience, when measuring cylinder sets we just use the con-figuration α instead of [α]. For instance, µ (α1α2|β ) represents the conditionalmeasure µ ([α1]∩ [α2]|[β ]). Given B⊆ A⊆ V and a measure µ onFA, we denoteby µ|B the restriction (or projection or marginalization) of µ toFB.The support supp(µ) of a Borel probability measure µ is defined as the closedsetsupp(µ) :={ω ∈A V : µ(ω|A)> 0, for all Ab V}. (2.13)Given a configuration space Ω, we will denote by M1(Ω) the set of Borelprobability measures whose support supp(µ) is contained in Ω.A measure µ ∈M1(Ω) is a Markov random field on G (G -MRF) if, for anyA b V , α ∈ A A, and B b V such that ∂A ⊆ B ⊆ V \A, and any β ∈ A B withµ(β )> 0, it is the case thatµ (α|β ) = µ (α|β |∂A) . (2.14)262.3. Gibbs measuresIn other words, an MRF is a measure where every finite configuration con-ditioned on a boundary configuration is independent of the configuration on the“exterior”.2.3.2 Shift-invariant measuresWhen G =Zd , a measure µ onA Zd such that µ(σx(A))= µ(A), for all measurablesets A ∈F and x ∈ Zd , is called shift-invariant (or stationary). In this context,the support supp(µ) turns out to be always a Zd shift space (a closed and shift-invariant subset of A Zd). Given a Zd shift space Ω, M1,σ (Ω) denotes the set ofshift-invariant measures contained inM1(Ω). A measure µ ∈M1,σ (Ω) is ergodicif for all shift-invariant A ∈F (i.e σx(A) = A, for all x ∈ Zd), it is the case thatµ(A) ∈ {0,1}, and measure-theoretic strong mixing, if for every nonempty setsA,Bb Zd and any α ∈A A, β ∈A B,lim‖x‖→∞µ ([α]∩σ−x([β ])) = µ (α)µ (β ) , (2.15)i.e. for every ε > 0, there exists n ∈ N such that‖x‖ ≥ n =⇒ |µ ([α]∩σ−x([β ]))−µ (α)µ (β )|< ε. (2.16)If µ ∈M1,σ (Ω) is measure-theoretic strong mixing, then µ is ergodic (see [76]for these and related notions). Given any point ω ∈A Zd , we define the δ -measuresupported on ω as the measureδω(A) =1 if ω ∈ A,0 otherwise, (2.17)for any A ∈F .If ω is a periodic point with orbit O(ω) = {ω1, . . . ,ωk}, we define νω to be theshift-invariant Borel probability measure supported on O(ω) given byνω :=1k(δω1 + · · ·+δωk) . (2.18)272.3. Gibbs measuresGibbs measures will be the main class of Borel probability measures studied inthis work. Before defining them, we need to introduce nearest-neighbour interac-tions and Gibbs specifications.2.3.3 Nearest-neighbour interactionsLetΩ be a configuration space for a board G = (V ,E ). We define the set of config-urations on vertices as L v(Ω) :=⋃x∈V L{x}(Ω), and the set of configurationson edges asL e(Ω) :=⋃{x,y}∈E L{x,y}(Ω). We will abbreviateL v,e(Ω) :=L v(Ω)∪L e(Ω). (2.19)A nearest-neighbour (n.n.) interaction onΩ will be any real-valued functionΦ :L v,e(Ω)→ R (2.20)that evaluates configurations inL v,e(Ω). We will always assume thatΦmax := supα∈L v,e(Ω)|Φ(α)|< ∞. (2.21)Constrained energy functionsGiven a constraint graph H = (V,E), a constrained energy function will be anypair (H,φ) such that φ is a real-valued function φ : V∪E→ R from vertices andedges in H. Given a homomorphism space Ω= Hom(G ,H), a constrained energyfunction φ induces naturally a n.n. interaction Φ in Ω by takingΦ(α) =φ (α(x)) if α ∈L{x}(Ω), x ∈ V ,φ ({α(x),α(y)}) if α ∈L{x,y}(Ω), {x,y} ∈ E . (2.22)Example 2.3.1. Let q∈N and ξ > 0. Many constrained energy functions representwell-known classical models.• Ferromagnetic Potts (K q ,ξφFP): φFP∣∣V ≡ 0, φFP({u,v}) =−1{u=v}.282.3. Gibbs measures• Anti-ferromagnetic Potts (K q ,ξφAP): φAP∣∣V ≡ 0, φAP({u,v}) =−1{u6=v}.• Proper q-colourings (Kq,φPC): φPC∣∣V∪E ≡ 0.• Hard-core (Hϕ ,ξφHC): φHC(0) = 0, φHC(1) =−1, φHC∣∣E ≡ 0.• Multi-type Widom-Rowlinson (S q ,ξφWR): φWR(v) =−1{v 6=0}, φWR∣∣E ≡ 0.Usually, the parameter ξ is referred to as the inverse temperature.All the previous models are isotropic, i.e. when G = Zd they have the sameinteraction in every coordinate direction {~0,~ei}, for i = 1, . . . ,d.Zd lattice energy functionsGiven d ∈ N and the board G = Zd , we can define a n.n. interaction that eval-uates configurations on edges according to their orientation. Given a set of n.n.constraints F =⊔di=1Ei, any real-valued function φ from A {~0} and A {~0,~ei} \Ei(i = 1, . . . ,d) will be a Zd lattice energy function for F. A Zd lattice energy func-tion φ for F induces naturally a n.n. interaction Φ on the Zd n.n. SFT ΩF by takingΦ(α) =φ(σ−x(α)|{~0})if α ∈L{x}(Ω),φ(σ−x(α)|{~0,~ei})if α ∈L{x,x+~ei}(Ω),(2.23)for x ∈ Zd and i = 1, . . . ,d. Notice that in this case Φ is shift-invariant (i.e. thevalue of a configuration on an edge is the same for any translation of it), which isnot a requirement in general. However, the shift-invariance of Φ fits naturally withthe shift-invariance of ΩF, so we will usually assume this in the context of Zd shiftspaces. Clearly, a shift-invariant n.n. interaction is defined by only finitely manynumbers (namely, the values defining φ ).2.3.4 Hamiltonian and partition functionLet Ω⊆A V be an invariant n.n. configuration space (i.e. a homomorphism spaceor a Zd n.n. SFT), and let Φ be a n.n. interaction. Given A b V , we define the292.3. Gibbs measuresHamiltonian in A asH ΦA :LA(Ω)→ R, whereH ΦA (α) := ∑x∈AΦ(α|{x})+ ∑{x,y}∈E [A]Φ(α|{x,y}), (2.24)for α ∈LA(Ω). We define the partition function of A asZΦA := ∑α∈LA(Ω)exp(−H ΦA (α)) , (2.25)and the free-boundary probability measure on A A given bypi( f )A (α) :=exp(−H ΦA (α))ZΦAif α ∈LA(Ω),0 otherwise.(2.26)In addition, given ω ∈ Ω, we define the ω-boundary Hamiltonian in A asH ΦA,ω : {α ∈LA(Ω) : α ω|Ac ∈Ω}→ R, whereH ΦA,ω(α) = ∑x∈AΦ(α|{x})+ ∑{x,y}∈E [A∪∂A]:{x,y}∩A 6= /0Φ((αω|Ac)|{x,y}), (2.27)and the ω-boundary probability measure on A A given bypiωA (α) :=exp(−H ΦA,ω (α))ZΦA,ωif α ω|Ac ∈Ω,0 otherwise,(2.28)where ZΦA,ω := ∑α:α ω|Ac∈Ω exp(−H ΦA,ω(α)).Both, for pi( f )A and piωA , given B⊆ A and β ∈A B, we marginalize as follows:pi∗A(β ) = ∑α∈A A:α|B=βpi∗A(α), (2.29)where ∗ = ( f ) or ω . Notice that pi∗A is a G [A]-MRF (here we use the assumptionthat Ω is an invariant n.n. configuration space and Φ is a n.n. interaction).302.3. Gibbs measures2.3.5 Gibbs specificationsThe collection pi = {piωA : Ab V ,ω ∈Ω} will be called nearest-neighbour (n.n.)Gibbs (Ω,Φ)-specification (or just Gibbs specification if Ω and Φ are under-stood). If Φ≡ 0, we call the n.n. Gibbs (Ω,0)-specification pi the uniform Gibbsspecification on Ω (see the case of proper q-colourings in Example 2.3.1).Notice that a n.n. Gibbs specification pi for a shift-invariant n.n. interaction Φon a Zd n.n. SFT Ω is always stationary, in the sense that piσx(ω)A−x (σx(α)) = piωA (α),for every α ∈A A and x ∈ Zd .2.3.6 Nearest-neighbour Gibbs measuresA n.n. Gibbs (Ω,Φ)-specification pi is regarded as a meaningful representationof an ideal physical situation where every finite volume A in the space is in ther-modynamical equilibrium with the exterior. The extension of this idea to infinitevolumes is via a particular class of Borel probability measures onΩ called nearest-neighbour Gibbs measures, which are MRFs specified by n.n. interactions.Definition 2.3.1. Given a n.n. Gibbs (Ω,Φ)-specification pi , a nearest-neighbour(n.n.) Gibbs measure for pi is any measure µ ∈M1(Ω) such that for any Ab Vand ω ∈Ω with µ(ω|∂A)> 0, we have ZΦA,ω > 0 andEµ(1[α]Ω |FAc)(ω) = piωA (α) µ-a.s., (2.30)for every α ∈LA(Ω).Equation (2.30) is equivalent to what is known as the Dobrushin-Lanford-Ruelle (DLR) equation. It is stated only for cylinder events [α]Ω in A, but this isequivalent to the usual definition with general events A ∈F instead. Given a n.n.Gibbs (Ω,Φ)-specification pi , we will denote G(pi) the set of n.n. Gibbs measuresfor pi . If Ω 6= /0, then G(pi) 6= /0 (special case of a result in [26], see also [18] and[70, Theorem 3.7 and Theorem 4.2]). Notice that every n.n. Gibbs measure µ is aG -MRF because the formula for piωA only depends on ω|∂A.Often there are multiple n.n. Gibbs measures for a single pi . This phenomenonis usually called a phase transition. There are several conditions that guarantee312.3. Gibbs measuresuniqueness of n.n. Gibbs measures (i.e. |G(pi)| = 1). Some of them fall into thecategory of spatial mixing properties, introduced in the next chapter.In the caseZd n.n. SFTs, a Gibbs measure for a stationary Gibbs specification pimay or may not be shift-invariant, but G(pi) must contain at least one shift-invariantmeasure (see [35, Corollary 5.16]).32Chapter 3Mixing propertiesIn this chapter we introduce some mixing properties of measure-theoretic and com-binatorial type for a Gibbs (Ω,Φ)-specification pi . In general terms, a mixing prop-erty says that, either a measure or the support of it, does not have strong long-rangecorrelations. This last aspect will be relevant for obtaining succinct representationsof pressure, and when developing efficient algorithms for approximating it.3.1 Spatial mixing propertiesIn the following, let f : N→ R≥0 be a function, referred as decay function, suchthat f (n)↘ 0 as n→ ∞. We will loosely use the term “spatial mixing property” torefer to any measure-theoretical property satisfied by pi defined via a decay of cor-relation between events (or configurations) with respect to the distance separatingthe shapes they are supported on.The first property introduced here, weak spatial mixing (WSM), has direct con-nections with the nonexistence of phase transitions and has been studied in severalworks, explicitly and implicitly (see [17, 78]). The next one, strong spatial mix-ing (SSM), is a strengthening of WSM that also has connections with meaningfulphysical idealizations (see [61, 45, 28]) and has proven to be useful for developingapproximation algorithms (see [79, 39, 33, 6, 31, 80]).Definition 3.1.1. A Gibbs (Ω,Φ)-specification pi satisfies weak spatial mixing(WSM) with decay f if for any Ab V , B⊆ A, β ∈A B, and ω1,ω2 ∈Ω,∣∣piω1A (β )−piω2A (β )∣∣≤ |B| f (dist(B,∂A)). (3.1)If a Gibbs (Ω,Φ)-specification pi satisfies WSM, then there is a unique n.n.Gibbs measure µ for pi (see [78, Proposition 2.2]).333.1. Spatial mixing propertiesWe use the convention that dist(B, /0) =∞. Given two configurations α1 ∈A A1 ,α2 ∈A A2 , and B⊆ A1∩A2, we define their set of B-disagreement asΣB(α1,α2) := {x ∈ B : α1(x) 6= α2(x)} . (3.2)Considering this, we have the following definition, a priori stronger than WSM.Definition 3.1.2. A Gibbs (Ω,Φ)-specification pi satisfies strong spatial mixing(SSM) with decay f if for any Ab V , B⊆ A, β ∈A B, and ω1,ω2 ∈Ω,∣∣piω1A (β )−piω2A (β )∣∣≤ |B| f (dist(B,Σ∂A(ω1,ω2))) . (3.3)Notice that dist(B,Σ∂A(ω1,ω2))≥ dist(B,∂A), so SSM implies WSM.Figure 3.1: The weak and strong spatial mixing properties.We will say that a Gibbs specification pi satisfies SSM (resp. WSM) if it sat-isfies SSM (resp. WSM) with decay f , for some decay function f . For γ > 0, aGibbs specification pi satisfies exponential SSM (resp. exponential WSM) withdecay rate γ if it satisfies SSM (resp. WSM) with decay function f (n) =Ce−γn,for some C > 0. We say that a Gibbs specification pi satisfies SSM for a class ofsets C if Definition 3.1.2 holds restricted to sets A ∈ C , for C a (possibly infinite)family of finite sets.Definition 3.1.3 (Total variation distance). Let K be a finite set and let X1 andX2 be two K-valued random variables with probability distributions ρ1 and ρ2,respectively. The total variation distance between X1 and X2 (or equivalently,343.1. Spatial mixing propertiesbetween ρ1 and ρ2) is defined asdTV(ρ1,ρ2) :=12 ∑x∈X|ρ1(x)−ρ2(x)| . (3.4)In the literature, it is also common to find the definition of WSM and SSM withthe expression∣∣piω1A (β )−piω2A (β )∣∣ replaced by the total variation distance of piω1A ∣∣Band piω2A∣∣B. The definitions here are, a priori, slightly weaker (so the results whereSSM is an assumption are also valid for this alternative definition), but sufficientfor our purposes.Definition 3.1.4. A coupling of two probability measures ρ1 on a finite set K1 andρ2 on a finite set K1, is a probability measure P on the set K1×K1 such that, forany A⊆ K1 and B⊆ K1, we have thatP(A×K1) = ρ1(A), and P(K1×B) = ρ2(B). (3.5)Given a finite set K and two K-valued random variables X1 and X2 with prob-ability distributions ρ1 and ρ2, respectively, it is well-known that dTV(ρ1,ρ2) is alower bound on P(X1 6= X2) over all couplings P of ρ1 and ρ2 and that there is acoupling, called the optimal coupling, that achieves this lower bound.Lemma 3.1.1 ([60, Lemma 2.3]). Let pi be a Gibbs (Ω,Φ)-specification and f adecay function such that for any Ab V , x ∈ A, β ∈A {x}, and ω1,ω2 ∈Ω,∣∣piω1A (β )−piω2A (β )∣∣≤ f (dist(x,Σ∂A(ω1,ω2))) . (3.6)Then, pi satisfies SSM with decay f .Remark 1. The proof of Lemma 3.1.1 given in [60] is for MRFs µ satisfying ex-ponential SSM on G = Zd , but its generalization to Gibbs specifications, arbitrarydecay functions, and more general boards is direct. We do not know if there is ananalogous lemma for WSM.Example 3.1.1. Recall the constrained energy functions introduced in Example2.3.1. The following Gibbs specifications satisfy exponential SSM (and therefore,WSM).353.2. Combinatorial mixing properties• (K q ,ξφFP) on G = Z2, for any q ∈ N and small enough ξ (see [1, 11]).• (K q ,ξφAP) on G = Z2, for q≥ 6 and any ξ > 0 (see [40]).• (Kq,φPC) on G = Td , for q ≥ 1+ δ ∗d, where δ ∗ = 1.763 . . . is the uniquesolution to xe−1/x = 1 (see [34, 39]).• (Hϕ ,ξφHC) on G with ∆(G )≤ d, for any ξ such that eξ < λc(d) := (d−1)(d−1)(d−2)d(see [79]).• (S q ,ξφWR) on G = Zd , for any q ∈ N and small enough ξ (see [1, 11]).There are more general sufficient conditions for having exponential SSM (forinstance, see Subsection 3.6.1 or the discussion in [60]).3.2 Combinatorial mixing propertiesIn this section we consider combinatorial properties for a configuration space Ω.According to the way they are defined, we classify them as global or local.3.2.1 Global propertiesBy a global property we understand any property of Ω that allow us to “glue”together globally admissible configurations in a single point ω ∈Ω. The two prop-erties introduced here have in common that they allow us to put together configu-rations provided they are separated enough.Definition 3.2.1. A Zd shift space Ω is topologically mixing if for any A,B b Zdthere exists a separation constant gA,B ∈ N such that for every α ∈A A, β ∈A B,and any x ∈ Zd with dist(A,x+B)≥ gA,B,[α]Ω, [β ]Ω 6= /0 =⇒ [α]Ω∩σ−x([β ]Ω) 6= /0. (3.7)Definition 3.2.2. A configuration space Ω is strongly irreducible with gap g ∈Nif for any A,Bb V with dist(A,B)≥ g, and for every α ∈A A, β ∈A B,[α]Ω, [β ]Ω 6= /0 =⇒ [αβ ]Ω 6= /0. (3.8)363.2. Combinatorial mixing propertiesIn other words, in the context of Zd shift spaces, strong irreducibility meansthat the separation constant gA,B from Definition 3.2.1 can be chosen to be uniformin A and B.Figure 3.2: A topologically mixing Z2 shift space.Remark 2. Since a configuration space is a compact space, it does not make adifference if the shapes of A and B are allowed to be infinite in the definition ofstrong irreducibility.3.2.2 Local propertiesNow we introduce two local properties concerning constraint graphs and n.n. con-straints, which will later be shown to have implications on the global properties(like, for example, strong irreducibility) of invariant n.n. configuration spaces Ω(homomorphism spaces and Zd n.n. SFTs, respectively).The first property is the existence of a special symbol which can be adjacent toevery other symbol (including itself).Definition 3.2.3. Given a constraint graph H, we say that s ∈ V is a safe symbolif {s,v} ∈ E, for every v ∈ V. Analogously, given an alphabet A , a set of n.n.constraints F, and the corresponding Zd n.n. SFT Ω=ΩF, we say that s ∈A is asafe symbol for Ω if s{~0}δ is locally admissible for every configuration δ ∈A ∂{~0}.Example 3.2.1 (A Zd n.n. SFT with a safe symbol). The constraint graph Hϕ hasa safe symbol (see Figure 2.1.2). In the support of the Zd hard-core lattice gasmodel (the Zd n.n. SFT Ωdϕ ), 0 is a safe symbol for every d (see [32]).373.3. Topological strong spatial mixingThe following property was introduced in [58] in the context of Zd shift spaces,and here we proceed to adapt it to the case of homomorphism spaces.Definition 3.2.4. A homomorphism space Hom(G ,H) is single-site fillable (SSF)(or we say that it satisfies single-site fillability) if for every site x ∈ V and B ⊆∂{x}, any graph homomorphism β : G [B]→ H can be extended to a graph homo-morphism α : G [B∪{x}]→ H (i.e. α is such that α|B = β ). Analogously, a Zdn.n. SFT Ω is SSF if for some set of n.n. constraints F such that Ω=ΩF, for everyδ ∈A ∂{~0}, there exists a ∈A such that a{~0}δ is locally admissible.Note 1. An invariant n.n. configuration space Ω is SSF iff every locally admissibleconfiguration is globally admissible (see [58] for the Zd n.n. SFT case).In the definition of SSF above, the symbol a may depend on the configurationα . Clearly, a Zd n.n. SFT containing a safe symbol satisfies SSF. Also, it is easy tocheck that a Zd n.n. SFT Ω that satisfies SSF is strongly irreducible with gap g= 2(see Proposition 3.3.3).Example 3.2.2 (A Zd n.n. SFT that satisfies SSF without a safe symbol). The Zdq-colourings n.n. SFT Hom(Zd ,Kq) has no safe symbol for any q ≥ 2 and d ≥ 1.However, Hom(Zd ,Kq) satisfies SSF for q≥ 2d+1 (see [58]).3.3 Topological strong spatial mixingNow we introduce a property which is somehow a hybrid between the global andlocal combinatorial properties from last section. Because of its close relationshipwith topological Markov fields (see [20]), we prefer to use the word “topological”for naming it. This condition will be relevant to give a partial characterization ofsystems that admit measures satisfying SSM, and also to generalize results relatedwith pressure representation and approximation in Part II.Definition 3.3.1. A configuration space Ω is topologically strong spatial mixing(TSSM) with gap g ∈ N, if for any A,B,S b V with dist(A,B)≥ g, and for everyα ∈A A, β ∈A B, and σ ∈A S,[ασ ]Ω, [σβ ]Ω 6= /0 =⇒ [ασβ ]Ω 6= /0. (3.9)383.3. Topological strong spatial mixingNotice that TSSM implies strong irreducibility by taking S = /0 in Definition3.3.1. The difference here is that we allow an arbitrarily close globally admissibleconfiguration on S in between two sufficiently separated globally admissible con-figurations, provided that each of the two configurations is compatible with the oneon S individually. Clearly, TSSM with gap g implies TSSM with gap g+ 1. Wewill say that a configuration space satisfies TSSM (resp. strong irreducibility) if itsatisfies TSSM (resp. strong irreducibility) with gap g, for some g ∈ N.It can be checked that for a Zd n.n. SFT Ω,Safe symbol =⇒ SSF =⇒ TSSM =⇒ Strong irred. =⇒ Top. mixing, (3.10)and all implications are strict. See Section 3.5 for examples that illustrate the dif-ferences among some of these conditions.+Figure 3.3: The topological strong spatial mixing property.A useful tool when dealing with TSSM is the next lemma, which states that ifwe have the TSSM property when A and B are singletons, then we have it uniformly(in terms of separation distance) for any pair of finite sets A and B.Lemma 3.3.1. LetΩ be a configuration space and g∈N such that for any x,y∈ Vwith dist(x,y) ≥ g and S b V , we have that for every α ∈ A {x}, β ∈ A {y}, andσ ∈ A S with [ασ ]Ω, [σβ ]Ω 6= /0, then [ασβ ]Ω 6= /0. Then, Ω satisfies TSSM withgap g.Proof. We proceed by induction. The base case |A|+ |B| = 2 is given by the hy-pothesis of the lemma. Now, let’s suppose that the property is true for subsetsA,B b V such that |A|+ |B| ≤ n and let’s prove it for the case when |A|+ |B| =n+1.Given A,B,Sb V with dist(A,B)≥ g and |A|+ |B|= n+1, and given α ∈A A,393.3. Topological strong spatial mixingβ ∈A B, and σ ∈A S, we can write A = {x1, . . . ,xk} and B = {y1, . . . ,ym}, where|A|= k, |B|= m, and k+m = n+1, for k,m≥ 1. Let’s consider A′ = A\{xk} andB′ = B\{ym}, possibly empty sets (but not both empty at the same time, since wecan assume that |A|+ |B| > 2). Similarly, let’s consider the restrictions α ′ = α|A′and β ′ = β |B′ . By the induction hypothesis, we have that [α ′σβ ]Ω, [ασβ ′]Ω 6= /0(even in the case A′ or B′ being empty). Then, if we consider σ ′ = α ′σβ ′ onS′ = A′ ∪ S∪B′, we can apply the property for singletons with α|{xk} and β |{ym},and we conclude that /0 6= [α|{xk}σ ′ β |{ym}]Ω = [ασβ ]Ω.Proposition 3.3.2. Let Ω be a configuration space. The two following propertiesare equivalent:1. Ω satisfies TSSM with gap g.2. For every Γb V and η ,η ′ ∈LΓ(Ω) with ΣΓ(η ,η ′) = {x1, . . . ,xk}, there ex-ists a sequence η = η1,η2, . . . ,ηk+1 = η ′ ∈LΓ(Ω) such that ΣΓ(ηi,ηi+1)⊆ΣΓ(η ,η ′)∩Ng(xi), for all 1≤ i≤ k.Proof.(1) =⇒ (2). Take η1 = η and ηk+1 = η ′. Suppose that for some i ≤ k we havealready constructed a sequence η1, . . . ,ηi ∈LA(Ω) such thatΣA(η j,η j+1)⊆ ΣA(η ,η ′)∩Ng(x j),for all 1≤ j < i, and (3.11)ΣA(η j,η ′)⊆ {x j, . . . ,xk},for all 1≤ j ≤ i. (3.12)The base case i = 1 is clear. Now, let’s extend the sequence to i+1. Considerthe sets Ai = {xi}, Bi = ΣΓ(ηi,η ′)\Ng(xi), and Si = Γ\ΣΓ(ηi,η ′). Take the con-figurations αi ∈ A Ai , βi ∈ A Bi , and σi ∈ A Si defined as αi := η ′|Ai , βi := ηi|Bi ,and σi := ηi|Si = η ′|Si . Since dist(Ai,Bi)≥ g, /0 6= [η ′]Ω ⊆ [αiσi]Ω, and /0 6= [ηi]Ω ⊆[σiβi]Ω, by TSSM, we can take ωi ∈ [αiσiβi]Ω and consider ηi+1 := ωi|Γ ∈LΓ(Ω).Then, ΣΓ(ηi+1,η ′) ⊆ {xi+1, . . . ,xk} and ΣΓ(ηi,ηi+1) ⊆ ΣΓ(η ,η ′)∩Ng(xi), as wewanted. Iterating until i = k, we conclude.(2) =⇒ (1). Consider x,y ∈ V with dist(x,y) ≥ g, S b V , α ∈A {x}, β ∈A {y},and σ ∈A S. Suppose that [ασ ]Ω, [σβ ]Ω 6= /0. It suffices to prove that [ασβ ]Ω 6= /0and we conclude by applying Lemma 3.3.1. Let Γ= S∪{x,y} and take ω ∈ [ασ ]Ω,403.4. TSSM in Zd shift spacesυ ∈ [σβ ]Ω 6= /0, and let η1 = ω|Γ, η3 = υ |Γ. W.l.o.g., suppose that ΣΓ(η1,η3) ={x,y}. By (2), there exists a configuration η2 ∈LΓ(Ω) such that ΣΓ(η1,η2)⊆ {y}and ΣΓ(η2,η3) ⊆ {x}. Therefore, η2|S = σ , η2|{x} = α1|{x} = α , and η2|{y} =η3|{y} = β , so [ασβ ]Ω 6= /0. Notice that it is important that we took the orderx1 = y and x2 = x in the previous proof.Remark 3. Property (2) in Proposition 3.3.2 is a stronger version of the gener-alized pivot property (see [20]), related with connectedness of configurations’spaces (see [17]).Proposition 3.3.3. If an invariant n.n. configuration space Ω satisfies SSF, then itsatisfies TSSM with gap g = 2.Proof. Since Ω satisfies SSF, every locally admissible configuration is globallyadmissible. If we take g = 2, for all sets A,B,S b Zd such that dist(A,B) ≥ gand for every α ∈ A A, β ∈ A B, and σ ∈ A S, if [ασ ]Ω, [σβ ]Ω 6= /0, in particularwe have that ασ and σβ are locally admissible. Since dist(A,B) ≥ g = 2, ασβmust be locally admissible, too. Then, by SSF, ασβ is globally admissible and,therefore, [ασβ ]Ω 6= /0.3.4 TSSM in Zd shift spacesDefinition 3.4.1. Given a Zd shift space Ω and Γ⊆ Zd , a configuration η ∈A Γ iscalled a first offender for Ω if η /∈L (Ω) but η |S ∈L (Ω), for every S ( Γ. Wedefine the set of first offenders of Ω asO(Ω) :={η ∈ ∪~0∈ΓbZdA Γ∣∣η is a first offender for Ω} . (3.13)Note 2. When d = 1, a similar notion of first offender can be found in [56, Exercise1.3.8], where it is used to characterize a “minimal” family F inducing a Z SFT Ω.Proposition 3.4.1. Let Ω be a Zd shift space. Then Ω satisfies TSSM iff |O(Ω)|<∞.Proof. First, suppose that Ω satisfies TSSM with gap g, for some g ∈ N, and takeη ∈O(Ω)with shape ΓbZd such that~0∈Γ. By way of contradiction, assume that413.4. TSSM in Zd shift spacesdiam(Γ) ≥ g and let x,y ∈ Γ be such that dist(x,y) = diam(Γ). W.l.o.g., assumethat x 6= y. Then, for S = Γ \ {x,y} ( Γ and since η is a first offender, we havethat [η |{x} η |S]Ω, [η |S η |{y}]Ω 6= /0. Since dist(x,y) ≥ g, by TSSM, we have that[η ]Ω = [η |{x} η |S η |{y}]Ω 6= /0, which is a contradiction. Then, diam(Γ) < g and,since~0 ∈ Γ, we have that |O(Ω)|< ∞.Now, suppose that |O(Ω)|< ∞ and takeg∗ = 1+ max~0∈ΓbZd{dist(~0,y) : y ∈ Γ,A Γ∩O(Ω) 6= /0}< ∞, (3.14)which is well defined. Consider arbitrary x,y∈Zd and SbZd , with dist(x,y)≥ g∗,and take α ∈ A {x}, σ ∈ A S, β ∈ A {y} such that [ασ ]Ω, [σβ ]Ω 6= /0. W.l.o.g., byshift-invariance, we can take x =~0. Now, by contradiction, assume that [ασβ ]Ω =/0. Consider a minimal S′ ⊆ S such that [α σ |S′ β ]Ω = /0 but [α σ |S′′ β ]Ω 6= /0, forall S′′ ( S′ (this includes the case S′ = /0, where the condition over S′′ is vacuouslytrue). It is direct to check that α σ |S′ β is a first offender with shape Γ= {~0,y}∪S′.Then, since dist(~0,y) = dist(x,y)≥ g∗, we have a contradiction with the definitionof g∗. Therefore, thanks to Lemma 3.3.1, Ω satisfies TSSM with gap g∗.Notice that Ω=ΩO(Ω). Considering this, we have the following corollary.Corollary 1. Let Ω be a Zd shift space that satisfies TSSM. Then, Ω is a Zd SFT.Note 3. If Ω⊆A Zd is a Zd shift space that satisfies TSSM with gap g, then it canbe checked that Ω is a Zd SFT that can be defined by a family of configurationsF⊆A Ng .The next lemma provides another characterization of TSSM for Zd n.n. SFTs.Lemma 3.4.2. A Zd n.n. SFT Ω satisfies TSSM with gap g iff for all S⊆ Rg,∀(α,σ ,β ) ∈A {~0}×A S×A ∂Rg : [ασ ]Ω, [σβ ]Ω 6= /0 =⇒ [ασβ ]Ω 6= /0. (3.15)Proof. Let’s prove that if Ω satisfies Equation (3.15), then Ω satisfies TSSM withgap g. W.l.o.g., by Lemma 3.3.1 and shift-invariance, consider x,y ∈ Zd withdist(x,y)≥ g, x=~0, Sb Zd , and configurations α ∈A {x}, β ∈A {y}, and σ ∈A Ssuch that [ασ ]Ω, [σβ ]Ω 6= /0. Take ω ∈ [σβ ]Ω and consider α ′ = α , β ′ = ω|∂Rg\S,423.4. TSSM in Zd shift spacesand σ ′ = σ |Rg∩S. Then, /0 6= [ασ ]Ω ⊆ [α ′σ ′]Ω and /0 6= [ω|Rg ]Ω ⊆ [σ ′β ′]Ω, so[α ′σ ′β ′]Ω 6= /0, by Equation (3.15). Take υ ∈ [α ′σ ′β ′]Ω and notice that υ |∂Rg =ω|∂Rg . Then, since Ω is a Zd n.n. SFT, we conclude that ω˜ = υ |Rg ω|Zd\Rg ∈[ασβ ]Ω, so [ασβ ]Ω 6= /0 and Ω satisfies TSSM. The converse is immediate.3.4.1 Existence of periodic pointsProposition 3.4.3. Let Ω be a nonempty Zd shift space that satisfies TSSM withgap g. Then, Ω contains a periodic point of period 2g in every direction.Proof. Consider the hypercube Q= [1,2g]d∩Zd . Given `∈{0,1}d , denote Q(`)=g`+([1,g]d ∩Zd)⊆ Q. Notice that Zd =⊔x∈2gZd (x+Q) and Q =⊔`∈{0,1}d Q(`).ThenZd =⊔`∈{0,1}d⊔x∈2gZd(x+Q(`)) =⊔`∈{0,1}dW (`), (3.16)where W (`) =⊔x∈2gZd (x+Q(`)). Notice that dist(x+Q(`),y+Q(`))≥ g, for all` ∈ {0,1}d and x,y ∈ 2gZd such that x 6= y.Consider `0, `1, . . . , `2d−1 an arbitrary order in {0,1}d . Let α0 ∈LQ(`0)(Ω) andN ∈ N. By using repeatedly the TSSM property (in particular, strong irreducibil-ity), we can construct a point ωN0 ∈Ω with σ2gx(ωN0 )∣∣Q(`0)= α0, for all x such that‖x‖∞ ≤ N. By compactness of Ω, we can take the limit when N→ ∞ and obtain apoint ω0 ∈Ω such that σ2gx(ω0)|Q(`0) = α0, for all x ∈ Zd .Given 0 ≤ k < 2d − 1, suppose that there exists a point ωk ∈ Ω and αi ∈LQ(`i)(Ω), for i = 0,1, . . . ,k, such that σ2gx(ωk)|Q(`i) = αi, for all i ∈ {0,1, . . . ,k}and x ∈ Zd . Notice that if k = 2d − 1, the point ω2d−1 is periodic of period 2g inevery direction. Then, since we have already constructed ω0, it suffices to provethat we can construct ωk+1 from ωk.Take αk+1 = ωk|Q(`k+1). Notice that αk+1 = σ−2gx(ωk)|2gx+Q(`k+1) and thatσ−2gx(ωk) has the same property of ωk, i.e. σ2gy(σ−2gx(ωk))|Q(`i) = αi, for alli = 0,1, . . . ,k and y ∈ Zd .Consider an arbitrary enumeration of Zd = {x0,x1,x2, . . .}, with x0 =~0. Letω0k+1 = ωk and suppose that, given m ∈ N, there is a point ωmk+1 such that• σ2gx j(ωmk+1)∣∣Q(`i)= αi, for all i ∈ {0,1, . . . ,k} and j ∈ N, and433.4. TSSM in Zd shift spaces• σ2gx j(ωmk+1)∣∣Q(`k+1)= αk+1, for all 0≤ j ≤ m.Take A =⊔mj=0 2gx j +Q(`k+1), B = 2gxm+1 +Q(`k+1), and S =⊔kr=0W (`r).Notice that dist(A,B) ≥ g. Then take ωmk+1∣∣A ∈ A A, σ−2gxm+1(ωk)∣∣B ∈ A B, andωk|S ∈ A S. We have that ωmk+1∣∣A ωk|S is globally admissible by the hypothesisof the existence of ωmk+1 and ωk|S σ−2gxm+1(ωk)∣∣B is globally admissible thanks tothe observation about σ−2gx(ωk). Then, ωmk+1∣∣A ωk|S σ−2gxm+1(ωk)∣∣B is globallyadmissible by TSSM. Notice that here S is an infinite set and the TSSM property isfor finite sets. This is not a problem since we can consider the finite set S′ = S∩Bnand take the limit n→ ∞ for obtaining the desired point, by compactness.Figure 3.4: Construction of a periodic point using TSSM.Now, notice that any extension of ωmk+1∣∣A ωk|S σ−2gxm+1(ωk)∣∣B is a point withthe properties of ωm+1k+1 . Taking the limit m→ ∞, we obtain a point with the prop-erties of ωk+1. Since k was arbitrary, we can iterate the argument until k = 2d−1,for obtaining the point ω2d−1 which is periodic of period 2g in every canonicaldirection.Note 4. It is known that for d = 1,2 a non-empty strongly irreducible Zd SFTcontains a periodic point. The case d = 1 is easy once one knows how to representa Z SFT as the space of infinite paths in a finite directed graph. For the case d = 2,see [77, 55]. The case d ≥ 3 is still an open problem.Proposition 3.4.4. LetΩ be a nonemptyZd shift space that satisfies TSSM with gapg. Then, Ω contains a periodic point of periods k1+g, . . . ,kd +g in the directions~e1, . . . ,~ed , respectively, for every ki ≥ g. Moreover, the set of periodic points isdense in Ω.443.4. TSSM in Zd shift spacesProof. We can modify the proof of Proposition 3.4.3, replacing the hypercube Qby ∏di=1 ([1,ki+g]∩Z) and the sub-hypercube Q(`0) by ∏di=1 ([1,ki]∩Z). Thisgives the first part of the statement. For checking density of periodic points, noticethat in the proof of Proposition 3.4.3, α0 ∈LQ(`0)(Ω) was arbitrary.Remark 4. In particular, any globally admissible finite configuration with shapeS ⊆∏di=1 ([1,ki]∩Z) with ki ≥ g, can be embedded in a periodic point of periodsk1+g, . . . ,kd +g in directions~e1, . . . ,~ed , respectively.It is well known that in the case of Z SFTs (d = 1) the mixing hierarchy col-lapses, i.e. topologically mixing, strongly irreducible, and other intermediate prop-erties, such as block gluing and uniform filling, are all equivalent (for example, see[14]). In the nearest-neighbour case, we extend this to TSSM.Proposition 3.4.5. A Z n.n. SFT Ω satisfies TSSM iff it is topologically mixing.Proof. We prove that if Ω is topologically mixing, then it satisfies TSSM. Theother direction is obvious.It is known that a topologically mixing Z n.n. SFT Ω is strongly irreduciblewith gap g = g({0},{0}), where g({0},{0}) is the gap according to Definition3.2.1. Consider arbitrary x,y ∈ Z and S b Z \ {x,y} with dist(x,y) ≥ g. Takeα ∈ A {x}, β ∈ A {y}, and σ ∈ A S with [ασ ]Ω, [σβ ]Ω 6= /0. W.l.o.g., by shift-invariance, assume that x = 0 < y.First, consider the open interval (x,y) and suppose that S∩(x,y) = /0. By strongirreducibility, there is δ ∈L(x,y)∩Z(Ω) such that [αδβ ]Ω 6= /0. Considerω ∈ [ασ ]Ω,υ ∈ [σβ ]Ω, and τ ∈ [αδβ ]Ω. Then, ω|(−∞,x]∩Z τ|(x,y)∩Z υ |[y,∞)∩Z ∈ [ασβ ]Ω, so[ασβ ]Ω 6= /0. Now, suppose that S∩ (x,y) 6= /0. Take r ∈ S∩ (x,y) and ω ∈ [ασ ],υ ∈ [σβ ]. Then, ω|(−∞,r]∩Z υ |(r,∞)∩Z ∈ [ασβ ]Ω, and [ασβ ]Ω 6= /0. Finally, weconclude using Lemma Algorithmic resultsIn general, given a Zd n.n. SFT Ω for d ≥ 2, it is algorithmically undecidable toknow if a given configuration is in L (Ω) or not (see [12, 69]). We present nowsome algorithmic results related with TSSM. First, a lemma.453.4. TSSM in Zd shift spacesLemma 3.4.6. LetΩ be a nonempty Zd shift space that satisfies strong irreducibil-ity with gap g. Then, for all Ab Zd , α ∈A A, and ω ∈Ω, α is globally admissibleiff there exists υ ∈Ω such thatυ |A = α, and υ |Zd\Ng(A) = ω|Zd\Ng(A) . (3.17)Proof. This is a direct application of the definition of strong irreducibility for theconfigurations α and ω|Zd\Ng(A), considering that dist(A,Zd \Ng(A))≥ g.Corollary 2. Let Ω ⊆ A Zd be a nonempty Zd n.n. SFT that satisfies TSSM withgap g. Then, there is an algorithm to check whether α belongs to LA(Ω) or not,for every Ab Zd and α ∈A A, in time |A |O(|∂gA|).Proof. By Proposition 3.4.3, there exists a periodic point inΩ of period 2g in everydirection. Then, by checking all the possible configurations inA [0,2g]d, we can finda valid periodic point ω in time O(|A |(2g+1)d · d(2g+ 1)d |A |2). Given α ∈ A A,by Lemma 3.4.6, we only need to check that α and ω|∂Ng(A) can be extendedtogether to a locally admissible configuration onNg(A). It can be checked in timeO(d|A||A |2) whether α is locally admissible or not. On the other hand, it can bedecided in time O(|A ||∂gA| ·d|∂gA||A |2) if there exists a configuration σ ∈A ∂gAsuch that ασ ω|∂Ng(A) is locally admissible. This is enough for deciding if α isglobally admissible or not. Thanks to the discrete isoperimetric inequality |∂A| ≥2d|A| d−1d (this follows directly from the discrete Loomis and Whitney inequality[57]), we have that |A| = |A |O(|∂A|), and we conclude that the total time of thealgorithm is |A |O(|∂gA|).Remark 5. It is worthwhile to point that, when d ≥ 3, there are no known explicitbounds on the time for checking global admissibility in Zd n.n. SFTs that onlysatisfy strong irreducibility.Corollary 3. Let Ω ⊆ A Zd be a nonempty Zd n.n. SFT that satisfies strong ir-reducibility with gap g0. Then, for every g ≥ g0, there is an algorithm to checkwhether Ω satisfies TSSM with gap g or not, in time eO(gd log |A |).Proof. Given the set of n.n. constraints F such that Ω = ΩF, the algorithm wouldbe the following:463.5. Examples: Zd n.n. SFTs1. Look for the periodic point provided by Proposition 3.4.3. If such point doesnot exist, then Ω does not satisfy TSSM with gap g. If such point exists, let’sdenote it by ω . (This can be done in time eO(gd log |A |).)2. Fix a shape S⊆ Rg \{~0} and then fix configurations α ∈A {~0}, β ∈A ∂Rg\S,and σ ∈A S.(a) Using strong irreducibility with gap g0, check whether [ασ ]Ω, [σβ ]Ω,and [ασβ ]Ω are empty or not, by trying to embed ασ , σβ , and ασβin the periodic point ω in a locally admissible way (as in Corollary 2).(This can be done in time O(|Rg+g0 |)eO(|Rg+g0 | log |A |) = eO(|Rg| log |A |).)(b) If [ασ ]Ω = /0 or [σβ ]Ω = /0, continue.(c) If [ασ ]Ω, [σβ ]Ω 6= /0, but [ασβ ]Ω = /0, then Ω does not satisfy TSSMwith gap g.(d) If all the cylinders are nonempty, continue.3. If after checking all the configurations we have not found α , σ , and β suchthat [ασ ]Ω, [σβ ]Ω 6= /0, but [ασβ ]Ω = /0, then Ω satisfies TSSM with gap g(by Lemma 3.4.2).Then, since |Rn| ≤ (2n+1)d , the total time of this algorithm is eO(gd log |A |).3.5 Examples: Zd n.n. SFTsIn this chapter we exhibit examples of Zd n.n. SFTs which illustrate some of themixing properties discussed in the previous sections.3.5.1 A strongly irreducible Z2 n.n. SFT that is not TSSMClearly, the SSF property implies strong irreducibility (a way to see this is throughProposition 3.3.3). As it is mentioned in Example 3.2.2, the q-colourings Z2 n.n.SFT Hom(Z2,Kq) satisfies SSF iff q≥ 5. For the Z2 n.n. SFT Hom(Z2,K4), givenδ ∈ V4∂{~0} (where V4 := V(K4) = {1,2,3,4}) defined by δ (~e1) = 1, δ (~e2) = 2,δ (−~e1) = 3, and δ (−~e2) = 4, there is no a ∈ V4{~0} such that aδ remains locally473.5. Examples: Zd n.n. SFTsadmissible, so Hom(Z2,K4) does not satisfy SSF. However, inspired by the SSFproperty, we have the following definition.Definition 3.5.1. Given N ∈ N, a Zd n.n. SFT Ω satisfies N-fillability if, for everylocally admissible configuration δ ∈ A T , with T ⊆ Zd \ [1,N]d , there exists α ∈A [1,N]d∩Zd such that αδ is locally admissible.Remark 6. In the previous definition, since Ω is a Zd n.n. SFT, it is equivalent toconsider δ to have shape T ⊆ ∂ [1,N]d ∩Zd . In this sense, notice that 1-fillabilitycoincides with the notion of SSF (which only considers locally admissible configu-rations on ∂{~0}).Lemma 3.5.1. The Z2 n.n. SFT Hom(Z2,K4) satisfies 2-fillability.Proof. Consider an arbitrary locally admissible configuration δ ∈ V4T , with T ⊆Zd \ [1,2]2. We want to check if there is α ∈ V4[1,2]2∩Z2 such that αδ remainslocally admissible. W.l.o.g., we can assume that T = ∂ [1,2]2 ∩Z2, which is theworst case. Given a locally admissible boundary δ ∈ V4∂ [1,2]2∩Z2 and x ∈ [1,2]2∩Z2, let’s denote by W δx the set of values a ∈ V4{x} such that aδ remains locallyadmissible. Notice that |W δx | ≥ 2, for every x ∈ [1,2]2∩Z2 and for every such δ .W.l.o.g., assume that |W δx |= 2, W δ(1,1) = {1,2}, and consider α ∈ V4[1,2]2∩Z2 to bedefined.First, suppose that W δ(1,1)∩W δ(2,2) 6= /0 or W δ(2,1)∩W δ(1,2) 6= /0. By the symmetriesof [1,2]2∩Z2 and the constraints, we may assume that 1 ∈W δ(1,1)∩W δ(2,2) and takeα(1,1) = α(2,2) = 1, α(2,1) ∈W δ(2,1) \{1}, and α(1,2) ∈W δ(1,2) \{1}. It is easyto check that αδ is locally admissible.On the other hand, if W δ(1,1)∩W δ(2,2)= /0 and W δ(2,1)∩W δ(1,2)= /0, w.l.o.g. we havethat W δ(1,1) = {1,2} and W δ(2,2) = {3,4}. We consider two cases based on whether adiagonal and off-diagonal coincide or intersect in exactly one element:• If W δ(2,1) = {1,2} and W δ(1,2) = {3,4}, we can take α(1,1) = 1, α(2,1) = 2,α(1,2) = 3, and α(2,2) = 4.• If W δ(2,1) = {1,3} and W δ(1,2) = {2,4}, we can take α(1,1) = 1, α(2,1) = 3,α(1,2) = 2, and α(2,2) = 4.483.5. Examples: Zd n.n. SFTsIn both cases it can be checked that αδ is locally admissible. The remainingcases are analogous.Definition 3.5.2. Given N ∈ N, a set A ⊆ Zd is called an N-shape if it can bewritten as a union of translations of [1,N]d ∩Zd , i.e. if there exists a set Γ ⊆ Zdsuch that A =⋃x∈Γ(x+([1,N]d ∩Zd)). A set is called a co-N-shape if it is thecomplement of an N-shape. Notice that every shape is a 1-shape and co-1-shape.Lemma 3.5.2. If a Zd n.n. SFT Ω satisfies N-fillability then, for any N-shapeA⊆ Zd and every locally admissible configuration δ ∈A T , with T ⊆ Zd \A, thereexists α ∈A A such that αδ is locally admissible.Proof. Let A⊆Zd be an N-shape and δ ∈A T , for T ⊆Zd \A, a locally admissibleconfiguration. Consider a minimal Γ⊆Zd such that A=⋃x∈Γ (x+ ([1,N]d ∩Zd)),in the sense that⋃x∈Γ′(x+([1,N]d ∩Zd))( A, for every Γ′ ( Γ. Take an arbitraryx∗ ∈ Γ. By N-fillability, consider β ∈ A (x∗+([1,N]d∩Zd)) such that βδ is locallyadmissible (notice that T ⊆ Zd \ (x∗+[1,N]d).Now, take the set A′ =⋃x∈Γ\{x∗}(x+([1,N]d ∩Zd)). Notice that A′ is alsoan N-shape. By minimality of Γ, we have that /0 6= A\A′ ⊆ (x∗+ ([1,N]d ∩Zd)).Define δ ′ = β |A\A′ δ and T ′ = (A\A′)∪T . Then, A′ is an N-shape and δ ′ ∈A T′is a locally admissible configuration, with T ′ ⊆ Zd \A′ as in the beginning, butA′ ( A.Now, given M ∈ N and iterating the previous argument, we can always findα ∈A A∩BM such that αδ is locally admissible. Since M is arbitrary and A Zd is acompact space, then there must exist α ∈ A A such that αδ is locally admissible.Definition 3.5.3. Given N ∈ N, a Zd shift space Ω is said to be N-strongly irre-ducible with gap g if, for any A,Bb Zd with dist(A,B)≥ g and such that AunionsqB isa co-N-shape, we have∀(α,β ) ∈A A×A B : [α]Ω, [β ]Ω 6= /0 =⇒ [αβ ]Ω 6= /0. (3.18)Proposition 3.5.3. If a Zd n.n. SFT Ω satisfies N-fillability, then it is N-stronglyirreducible with gap g = 2.493.5. Examples: Zd n.n. SFTsProof. Let A,Bb Zd with dist(A,B)≥ 2 and such that AunionsqB is a co-N-shape, andtake α ∈ A A, β ∈ A B such that [α]Ω, [β ]Ω 6= /0. Then consider δ = αβ ∈ A AunionsqBand W = (AunionsqB)c. Notice that δ is a locally admissible configuration (α and β areglobally admissible and dist(A,B) ≥ 2), and W is an N-shape. Then, by Lemma3.5.2, there exists η ∈ A W such that ω = ηδ is locally admissible. Then, ω is alocally admissible point (therefore globally admissible) such that ω ∈ [αβ ]Ω.Proposition 3.5.4. If a Zd shift space Ω is N-strongly irreducible with gap g, thenΩ is strongly irreducible with gap g+2N.Proof. Let A,B b Zd with dist(A,B) ≥ g+ 2N, and α ∈ A A, β ∈ A B such that[α]Ω, [β ]Ω 6= /0. Consider the partition Zd = ⊔x∈NZd (x+ ([1,N]d ∩Zd)), and thesetsS1 :={x ∈ NZd :(x+[1,N]d)∩A 6= /0}, (3.19)S2 :={x ∈ NZd :(x+[1,N]d)∩B 6= /0}. (3.20)Consider A′ :=⊔x∈S1 x+([1,N]d ∩Zd) and B′ :=⊔x∈S2 x+ ([1,N]d ∩Zd). No-tice that A⊆ A′ and B⊆ B′. Take ω ∈ [α]Ω and υ ∈ [β ]Ω, and consider the config-urations α ′ = ω|A′ and β ′ = υ |B′ . Then, we have that [α ′]Ω, [β ′]Ω 6= /0, A′∪B′ is aco-N-shape and dist(A′,B′)≥ dist(A,B)−2N ≥ (g+2N)−2N = g so, by N-strongirreducibility, we conclude that /0 6= [α ′β ′]Ω ⊆ [αβ ]Ω.Corollary 4. If a Zd n.n. SFTΩ satisfies N-fillability, then it is strongly irreduciblewith gap 2(N+1).Corollary 5. The Z2 n.n. SFT Hom(Z2,K4) is strongly irreducible with gap g= 6.We have concluded Hom(Z2,Kq) is strongly irreducible iff q ≥ 4 (the casesk = 2,3 do not even satisfy the D-condition due to the existence of frozen config-urations; see Definition 5.2.3 and [58]). On the other hand, Hom(Z2,Kq) satisfiesTSSM (in particular, SSF) iff q ≥ 5. In particular, TSSM fails when q = 4, as thenext result shows.Proposition 3.5.5. The Z2 n.n. SFT Hom(Z2,K4) does not satisfy TSSM.503.5. Examples: Zd n.n. SFTsProof. Take g ∈ N, consider the sets A = {(−2g,0)}, B = {(2g,0)}, and S =([−2g,2g]∩Z)×{−1,1}, and the configurations α ∈ V4A, β ∈ V4B, and σ ∈ V4S(see Figure 3.5) defined by α = 3A, β = 4B, andσ((i, j)) =1 if ( j = 1 and i ∈ 2Z) or ( j =−1 and i /∈ 2Z),2 if ( j = 1 and i /∈ 2Z) or ( j =−1 and i ∈ 2Z). (3.21)Then, if we denoteΩ=Hom(Z2,K4), it can be checked that [ασ ]Ω, [σβ ]Ω 6= /0.However, for all ω ∈ [ασ ]Ω and υ ∈ [σβ ]Ω we have that ω((0,0)) = 3 6= 4 =υ((0,0)). Therefore, [ασβ ]Ω = /0. Since g was arbitrary and dist(A,B) = 4g≥ g,we conclude that Ω does not satisfy TSSM.1 2 1 2 1 2 1 2 1 2 1 2 132 1 2 1 2 1 2 1 2 1 2 1 21 2 1 2 1 2 1 2 1 2 1 2 13 4 3 4 3 4 3 4 3 4 3 4 32 1 2 1 2 1 2 1 2 1 2 1 21 2 1 2 1 2 1 2 1 2 1 2 142 1 2 1 2 1 2 1 2 1 2 1 21 2 1 2 1 2 1 2 1 2 1 2 14 3 4 3 4 3 4 3 4 3 4 3 42 1 2 1 2 1 2 1 2 1 2 1 21 2 1 2 1 2 1 2 1 2 1 2 13 42 1 2 1 2 1 2 1 2 1 2 1 21 2 1 2 1 2 1 2 1 2 1 2 13 4 3 4 3 4 3 4 3 4 3 42 1 2 1 2 1 2 1 2 1 2 1 2?Figure 3.5: Proof that Ω= Hom(Z2,K4) does not satisfy TSSM nor SSM.A by-product of the construction from the previous counterexample is the fol-lowing result, which also illustrates how TSSM is related with SSM.Proposition 3.5.6. Take Ω = Hom(Z2,K4) and let pi be any n.n. Gibbs (Ω,Φ)-specification. Then, pi cannot satisfy SSM.Proof. Let’s suppose that there is a n.n. Gibbs (Ω,Φ)-specification pi for Ω =Hom(Z2,K4) that satisfies SSM with decay function f . Take n0 ∈ N such thatf (n)< 1, for all n≥ n0. Consider the set Γ= ([−2n0+1,2n0−1]∩Z)×{0}bZ2and its boundary ∂Γ = ([−2n0,2n0]∩Z)×{−1,1}∪ {(−2n0,0)}∪ {(2n0,0)} =A∪B∪S, where A, B, and S are as in Proposition 3.5.5. Take δ1,δ2 ∈V4∂Γ definedby δ1|S = δ2|S = σ (where σ is also as in Proposition 3.5.5), δ1|A = δ1|B = 3, and513.5. Examples: Zd n.n. SFTsδ2|A = δ2|B = 4. It is easy to see that δ1 and δ2 are both globally admissible and,in particular, there exists ω1,ω2 ∈Ω such that ωi|∂Γ = δi, for i = 1,2. Now, if weconsider the configuration β = 3{(0,0)}, we have that1 = |1−0|= ∣∣piω1Γ (β )−piω2Γ (β )∣∣≤ f (2n0)< 1, (3.22)which is a contradiction. Then, pi cannot satisfy SSM.Remark 7. It has been suggested (see [74]) that the uniform Gibbs specificationsupported on Hom(Z2,K4) satisfies exponential WSM. Here we have proven thatSSM is not possible for any n.n. Gibbs specification supported on Hom(Z2,K4) andfor any rate, not necessarily exponential. The counterexample in Proposition 3.5.6corresponds to a family of very particular shapes where SSM fails and not what wecould call a “common shape” (like Bn, for example), but is enough for discardingthe possibility of SSM if we stick to its definition. We also have to consider that thisfamily of configurations (and other variations, with different colours and differentnarrow shapes) can appear as sub-configurations in more general shapes and stillproduce combinatorial long-range correlations.3.5.2 A TSSM Z2 n.n. SFT that is not SSFThe Iceberg model was considered in [19] as an example of a strongly irreducibleZ2 n.n. SFT with multiple measures of maximal entropy. Given M ≥ 2, and thealphabet AM = {−M, . . . ,−1,+1, . . . ,+M}, the Iceberg model IM is defined asIM :={ω ∈A Z2M : ω(x) ·ω(x+~ei)≥−1, for all x ∈ Zd , i = 1, . . . ,d}. (3.23)In the following, we show that for every M ≥ 2, the Iceberg model satisfiesTSSM, but not SSF. In particular, this provides an example of a Z2 n.n. SFTsatisfying TSSM with multiple measures of maximal entropy.It is easy to see that IM does not satisfy SSF, since +M and −M cannot be atdistance less than 3. In particular, we can take the configuration δ ∈A ∂{~0}M givenby δ (~e1) = δ (~e2) = +M, and δ (−~e1) = δ (−~e2) = −M, which does not remainlocally admissible for any a ∈A {~0}M . On the other hand, IM satisfies TSSM, as itis shown in the next proposition.523.5. Examples: Zd n.n. SFTsProposition 3.5.7. For every M ≥ 2, the Iceberg model IM satisfies TSSM withgap g = 3.Proof. Consider Lemma 3.3.1 and take x,y ∈ Z2 with dist(x,y) ≥ 3 and S b Z2.Given α ∈A {x}M , and β ∈A {y}M , and σ ∈A SM, suppose that [ασ ]IM , [σβ ]IM 6= /0.Next, take ω ∈ [σβ ]IM and define a new point υ given byυ(z) =ω(z) if z ∈ S∪{y},+1 if z ∈ (S∪{y})c and ω(x) ∈ {+1, . . . ,+M},−1 if z ∈ (S∪{y})c and ω(x) ∈ {−M, . . . ,−1}.(3.24)It is not hard to see that υ is a valid point in [σβ ]IM . Now, let’s construct apoint τ ∈ [ασβ ]IM from υ .Case 1: α(x) = ±1. W.l.o.g., suppose that α(x) = +1. Now, since [ασ ]IM 6= /0,all the values of υ |∂{x}∩S must belong to {−1,+1, . . . ,+M}. On the other hand,since ∂{x}\S ⊆ (S∪{y})c, all the values in υ |∂{x}\S belong to {−1,+1}. Then,all the values in υ |∂{x} belong to {−1,+1, . . . ,+M}, and we can replace υ |{x} by+1 in order to get a valid point τ from υ such that τ ∈ [ασβ ]IM .Case 2: α(x) 6= ±1. W.l.o.g., suppose that α(x) = +M. Then, all the valuesin υ |∂{x}∩S belong to {+1, . . . ,+M}. We claim that we can switch every −1 in∂{x}\S to a +1. If it is not possible to do this for some site x∗ ∈ ∂{x}\S, thenits neighbourhood ∂{x∗} contains a site with value in {−M, . . . ,−2} and, in par-ticular, different from +1 and −1. Then, ∂{x∗} necessarily intersects S (and not{y}, because dist(x,y) ≥ 3). Then, a site in ∂{x∗}∩ S 6= /0 is fixed to some valuein {−M, . . . ,−2} and then the site x∗ must take a value in {−M, . . . ,−1}, givenσ . Therefore, x cannot take a value in {+2, . . . ,+M}, contradicting the fact that[ασ ]IM 6= /0. Therefore, we can set all the values in υ |∂{x}\S to +1. Let’s call thatpoint υ ′. Finally, if we replace υ ′(x) = +1 by +M, we obtain a valid point τ fromυ ′ such that τ ∈ [ασβ ]IM .Then, we conclude that IM satisfies TSSM with gap g = 3, for every M ≥2.Remark 8. In particular, Proposition 3.5.7 provides an alternative way of checking533.5. Examples: Zd n.n. SFTsthe well-known fact that IM is strongly irreducible.3.5.3 Arbitrarily large gap, arbitrarily high rateNotice that the Iceberg model can be regarded as a Zd shift space where two types(positives and negatives) coexist separated by a boundary of ±1s. Now we intro-duce a variation of the Iceberg model that extends the idea of two types coexistingto an arbitrary number of them. First, we will see that this variation gives a familyof Zd n.n. SFTs satisfying TSSM with gap g but not g− 1, for arbitrary g ∈ N.Second, these models admits the existence of a n.n. Gibbs specification satisfyingexponential SSM with arbitrarily high decay rate, showing in particular (as far aswe know, for the first time) that there are systems that satisfy SSM and TSSM,without satisfying any of the other stronger combinatorial mixing properties, likehaving a safe symbol or satisfying SSF.Given g,d ∈ N, consider the alphabet Ag = {0,1, . . . ,g} and the Zd n.n. SFTdefined byΩdg :={ω ∈A Zdg : |ω(x)−ω(x+~ei)| ≤ 1, for all x ∈ Zd , i = 1, . . . ,d}. (3.25)Notice that Ωd0 ={0Zd}(a fixed point) and Ωd1 =AZd1 (a 2 symbols full shift),so both satisfy TSSM with gap g = 0 and g = 1, respectively. Also, notice that 1 isa safe symbol for Ωd2 (and therefore, by Proposition 3.3.3, it satisfies TSSM withgap g = 2).Proposition 3.5.8. The Zd n.n. SFT Ωdg satisfies TSSM with gap g but not g−1.Proof. First, let’s see that Ωdg does not satisfy TSSM with gap g−1. In fact, recallthat TSSM with gap g−1 implies strong irreducibility with the same gap. However,if we consider two configurations on single sites with values 0 and g, respectively,they cannot appear in the same point if they are separated by a distance less orequal to g−1, since the value of consecutive sites can only increase or decrease byat most 1. Therefore, Ωdg is not TSSM with gap g−1.In order to prove that Ωdg satisfies TSSM with gap g, by way of Lemma 3.3.1,let x,y∈Zd with dist(x,y)≥ g, and SbZd\{x,y}. Given α ∈A {x}g , β ∈A {y}g , andσ ∈A Sg , suppose that [ασ ]Ωdg , [σβ ]Ωdg 6= /0. We want to prove that [ασβ ]Ωdg 6= /0.543.5. Examples: Zd n.n. SFTsSince [σβ ]Ωdg 6= /0, we can consider a point ω ∈ [σβ ]Ωdg . If ω(x) = α(x), weare done. W.l.o.g., suppose that ω(x)< α(x) (the case ω(x)> α(x) is analogous).We proceed by finding a valid point ω ′ such that ω ′|S = σ , ω ′(y) = β (y), andω ′(x) = ω(x)+1. Iterating this process |α(x)−ω(x)| times, we conclude. Noticethat the only obstruction to an increase by 1 of ω(x) are the values of neighboursof x strictly below ω(x).We introduce an auxiliary digraph of descending paths Dω,x = (Vgω,x,Egω,x),where V0ω,x = {x}, E0ω,x = /0, and, for n≥ 1,Vn+1ω,x = Vnω,x∪⋃y′∈∂Vnω,x,ω(y′)=ω(x)−n{y′}, (3.26)En+1ω,x = Enω,x∪⋃y′∈∂Vnω,x,ω(y′)=ω(x)−n⋃x′∈Vnω,x,x′∼y′{(x′,y′)}. (3.27)Notice that, sinceω(x)< g, the recurrence stabilizes for some n< g, i.e. Vnω,x =Vg−1ω,x and Enω,x = Eg−1ω,x , for every n ≥ g. In particular, the sites that Dω,x reachesare sites at distance at most g−1 from x, and the site y cannot belong to the graph.Now, suppose that a site from S belongs to Dω,x. If that is the case, the value at xof any point in [σ ]Ωdg would be forced to be at most ω(x) (since the graph is strictlydecreasing from x to S), which contradicts the fact that [ασ ]Ωdg 6= /0.Then neither y nor any element of S belongs toDω,x, so if we modify the valuesof Dω,x in a valid way, we will still obtain a valid point ω ′ such that ω ′|S = σ andω ′(y) = β (y). Now, take the set D=Vgω,x ⊆Zd and consider the point ω ′ such thatω ′∣∣D = ω′∣∣D+1D, and ω ′∣∣Zd\D = ω|Zd\D , (3.28)where ω(D)+1D represents the configuration obtained from ω|D after adding 1 inevery site. We claim that ω ′ is a valid point. To see this, we only need to checkthat the difference between values in the endpoints of an arbitrary edge is at most1. If both endpoints are in D or in Zd\D, it is clear that the edge is valid since theoriginal point ω was a valid point, and adding 1 to both endpoints does not affectthe difference. If one endpoint is in x1 ∈ D and the other one is in x2 ∈ Zd\D,then ω(x1) ≤ ω(x2), necessarily (if not, ω(x1) > ω(x2), and x2 would be part of553.6. Relationship between spatial and combinatorial mixing propertiesthe digraph of descending paths). Since |ω(x1)−ω(x2)| ≤ 1 and ω(x1) ≤ ω(x2),then ω(x2)−ω(x1) ∈ {0,1}. Therefore, ω ′(x1)−ω ′(x2) = (ω(x1)+1)−ω(x2) =1− (ω(x2)−ω(x1)) ∈ {1,0}, so |ω ′(x1)−ω ′(x2)| ≤ 1. Then, we conclude thatω ′ ∈Ωdg and ω ′|S = σ , ω ′(y) = β (y), and ω ′(x) = ω(x)+1, as we wanted.The next result is a special case of a more general result, whose full proof willbe given in the following chapters.Proposition 3.5.9. For any g,d ∈ N and γ > 0, there exists a n.n. Gibbs (Ωdg ,Φ)-specification that satisfies exponential SSM with decay rate γ .Proof. This result follows from Proposition 4.4.2, Corollary 7, Proposition 4.6.2,and from noticing that Ωdg is a homomorphism space of the form Hom(Zd ,T ), withT a looped tree.Note 5. The ingredients of the proof of Proposition 3.5.9 can be easily adapted tothe Zd hard-core lattice gas model case.3.6 Relationship between spatial and combinatorialmixing propertiesIn this section we establish some connections between spatial and combinatorialmixing properties. In particular, we show that TSSM is a property that arises natu-rally for the support of Gibbs specifications satisfying (exponential) SSM, at leastwhen the decay rate is high enough.3.6.1 SSM criterionLet Ω be a Zd n.n. SFT and Φ a shift-invariant n.n. interaction. For the n.n. Gibbs(Ω,Φ)-specification pi , we defineQ(pi) := maxω1,ω2∈ΩdTV(piω1{~0},piω2{~0}). (3.29)In the following, let pc(Zd) to denote the critical value of Bernoulli site per-colation on Zd (see [41]). The following result is essentially in [11].563.6. Relationship between spatial and combinatorial mixing propertiesTheorem 3.6.1. Let Ω be a Z2 n.n. SFT, Φ a shift-invariant n.n. interaction, andpi the corresponding n.n. Gibbs (Ω,Φ)-specification. Suppose that Ω has a safesymbol. If Q(pi)< pc(Z2), then pi satisfies exponential SSM.Proof. Take µ ∈ G(pi). Since Ω has a safe symbol, µ is fully supported, i.e.supp(µ) =Ω (very special case of [70, Remark 1.14]). Given a Z2-MRF µ , defineQ(µ) := maxδ1,δ2dTV(µ(·|δ1)|{~0} , µ(·|δ2)|{~0}), (3.30)where δ1 and δ2 range over all configurations on ∂{~0} such that µ(δ1)µ(δ2) > 0.Then, Q(µ)≤Q(pi)< pc(Z2), so by [11, Theorem 1] and shift-invariance of Φ, µsatisfies exponential SSM as a MRF (see [59, Theorem 3.10]). Finally, since µ isfully supported, we can conclude that pi satisfies exponential SSM.3.6.2 Uniform bounds of conditional probabilitiesWe relate TSSM is closely related with bounds on elements of Gibbs specificationssatisfying SSM. We have the following lemma.Lemma 3.6.2. Let G be a board of bounded degree ∆ and pi a Gibbs (Ω,Φ)-specification. Then, for every ω ∈Ω and Ab V ,piωA (ω|A)≥ exp(−4∆|A|(Φmax+ log |A |)). (3.31)Proof. Notice thatH ΦA,ω(ω|A) = ∑x∈A∪∂AΦ(ω|{x})+ ∑{x,y}∈E [A∪∂A]:{x,y}∩A 6= /0Φ(ω|{x,y}) (3.32)≤ (|A|+∆|A|)Φmax ≤ 2∆|A|Φmax, (3.33)where we use that |{E [A∪∂A] : {x,y}∩A 6= /0}| ≤ ∆|A|. Similarly,H ΦA,ω(ω|A)≥−2∆|A|Φmax, so ZΦA,ω ≤ |A ||A| exp(2∆|A|Φmax). Therefore,piΦA,ω(ω|A) =1ZΦA,ωexp(−H ΦA,ω(ω|A)) (3.34)≥ |A |−|A| exp(−2∆|A|Φmax)exp(−2∆|A|Φmax) (3.35)573.6. Relationship between spatial and combinatorial mixing properties≥ exp(−4∆|A|(Φmax+ log |A |)). (3.36)Given a a Gibbs (Ω,Φ)-specification pi , define the function cpi : Ω→ [0,1]given bycpi(ω) := infAbVinfx∈ApiωA (ω|{x}), (3.37)for ω ∈Ω, and denotecpi := infω∈Ωcpi(ω) ∈ [0,1]. (3.38)Notice that for a board G of bounded degree ∆ we have that, for every x ∈ Vand g ∈ N, |Ng(x)| ≤ ∆g+1. Considering this, we have the next proposition.Proposition 3.6.3. Let pi be a Gibbs (Ω,Φ)-specification, with G any board ofbounded degree ∆. Suppose that Ω satisfies TSSM with gap g. Then,cpi ≥ exp(−5∆1+g(Φmax+ log |A |))> 0. (3.39)Proof. Consider a point ω ∈ Ω, a set A b V , and a site x ∈ A. W.l.o.g., supposethat A is connected. If not, let Cx be the connected component of A containing x,and notice that piωA (ω|{x}) = piωCx(ω|{x}).Define the set Ag,x to be the connected component of A∩Ng−1(x) containing x,and let B := A∩∂Ag,x. Notice that B ⊆Ng(x) and |Ng(x)| ≤ ∆g+1. First, assumethat B = /0. If this is the case, then ∂Ag,x ⊆ Ac, so A ⊆Ng−1(x). Therefore, byLemma 3.6.2,piωA (ω|{x})≥ piΦA,ω(ω|A) (3.40)≥ exp(−4∆|A|(Φmax+ log |A |)) (3.41)≥ exp(−4∆1+g(Φmax+ log |A |)). (3.42)where have used that |A| ≤ |Ng−1(x)| ≤ ∆g.On the other hand, suppose that B 6= /0. By a counting argument, there mustexist β ∈A B such that piωA (β )≥ |A |−|B|. In particular, this says that [βω|Ac ]Ω 6= /0.Since [ω|Ac ω|{x}]Ω 6= /0 and dist(x,B)≥ g, we have [β ω|Ac ω|{x}]Ω 6= /0, by TSSM.583.6. Relationship between spatial and combinatorial mixing propertiesNow, take ω ′ ∈ [β ω|Ac ω|{x}]Ω. Then, by the MRF property,piωA (ω|{x})≥ piωA (ω ′∣∣Ag,x) (3.43)≥ piωA (ω ′∣∣Ag,x| ω ′∣∣B)piωA (ω ′∣∣B) (3.44)= piω′Ag,x(ω′∣∣Ag,x)piωA (β ) (3.45)≥ exp(−4∆|Ag,x|(Φmax+ log |A |))|A |−|B|. (3.46)Notice that |Ag,x| ≤ ∆g and |B| ≤ ∆g, sopiωA (ω|{x})≥ e−4∆1+g(Φmax+log |A |)|A |−∆g ≥ e−5∆1+g(Φmax+log |A |). (3.47)Since ω , A, and x were all arbitrary, we conclude.Proposition 3.6.4. Let pi be a Gibbs (Ω,Φ)-specification that satisfies SSM withdecay function f and such that cpi > 0. Then, Ω satisfies TSSM with gap g =min{n ∈ N : f (n)< cpi}.Proof. Take n0 ∈N such that f (n)< cpi , for all n≥ n0 (recall that a decay functionis decreasing). We claim that Ω satisfies TSSM with gap n0. By contradiction,and considering Lemma 3.3.1, suppose that there exist x,y ∈ V with dist(x,y) ≥n0 and S b V , α ∈ A {x}, β ∈ A {y}, and σ ∈ A S such that [ασ ]Ω, [σβ ]Ω 6= /0,but [ασβ ]Ω = /0. Take ω1 ∈ [ασ ]Ω and ω2 ∈ [σβ ]Ω. Notice that ω1 6= ω2 andω1|S = ω2|S = σ . Consider the set A =Nn0(x)\S. We have that piω1A (α)≥ cpi andpiω2A (α) = 0. Then,cpi ≤ piω1A (α) = |piω1A (α)−piω2A (α)| (3.48)≤ f (dist(x,Σ∂A(ω1,ω2)))≤ f (n0)< cpi , (3.49)which is a contradiction.Notice that no assumption on the degree of the board was necessary in the proofof Proposition 3.6.4.Corollary 6. Let pi be Gibbs (Ω,Φ)-specification that satisfies SSM, with G aboard of bounded degree. Then, Ω satisfies TSSM iff cpi > 0.593.6. Relationship between spatial and combinatorial mixing propertiesProof. The result follows directly from Proposition 3.6.3 and Proposition SSM with high decay rateWe have the following theorem.Theorem 3.6.5. Let Ω⊆A Z2 be a Z2 n.n. SFT and let pi be a n.n. Gibbs (Ω,Φ)-specification that satisfies exponential SSM with decay function f (n) = Ce−γn,where γ > 4log |A |. Then, Ω satisfies TSSM with gapg = min{n ∈ N : γ−4log |A |> 1nlog(4Cn)}. (3.50)Proof. We will prove that cpi > 0 and then conclude by Proposition 3.6.4. Takeω ∈ Ω, A b Z2, and x ∈ A. Our goal is to bound piωA (ω|{x}) away from zero,uniformly in ω , A, and x.Let n ∈ N be such thatγ−4log |A |> 1nlog(4Cn), (3.51)and define An−1,x to be the connected component of A∩Nn−1(x) containing x, andB := A∩∂An−1,x. Notice that B⊆ A∩∂Nn−1(x), |An−1,x| ≤ |Nn−1(x)| ≤ 2n2, and|B| ≤ |∂Nn−1(x)|= 4n.If B = /0, then ∂An−1,x ⊆ Ac, and by Lemma 3.6.2,piωA (ω|{x}) = piωAn−1,x(ω|{x}) (3.52)≥ exp(−4∆|An−1,x|(Φmax+ log |A |)) (3.53)≥ exp(−32n2(Φmax+ log |A |)). (3.54)Now, let’s suppose that B 6= /0. By a counting argument, there must exist β ∈A B such that piωA (β )≥ |A |−|B| and, in particular, [ω|Ac β ]Ω 6= /0.By contradiction, let’s suppose that [ω|{x} ω|Ac β ]Ω = /0. Then, piωA\{x}(β ) = 0.On the other hand, since piωA (β )≥ |A |−|B|, by taking weighted averages over con-figurations on {x}, there must exist α ∈ A {x} such that piωA (β |α) = piυA\{x}(β ) ≥603.6. Relationship between spatial and combinatorial mixing properties|A |−|B|, for υ ∈ [α ω|Ac β ]Ω 6= /0. Notice that x /∈ B and dist(B,Σ∂A\{x}(ω,υ)) ≤dist(B,x) = n. Then, we have that|A |−|B| ≤∣∣∣piωA\{x}(β )−piυA\{x}(β )∣∣∣≤ |B|Ce−γn, (3.55)by the MRF and SSM properties. Since B ⊆ ∂Nn−1(x), then |B| ≤ |∂Nn−1(x)|,and|A |−|∂Nn−1(x)| ≤ |A |−|B| ≤ |B|Ce−γn ≤ |∂Nn−1(x)|Ce−γn. (3.56)Figure 3.6: Representation of the proof of Theorem 3.6.5.By taking logarithms, −4n log |A | ≤ log(4n)+ logC− γn, soγ ≤ 1nlog(4Cn)+4log |A |= 4log |A |+o(1), (3.57)which is a contradiction with the fact that γ > 4log |A | for sufficiently large n(notice that the difference between γ and 4log |A | determines the size of |B| and itsdistance to x). We conclude that[ω|{x} ω|Ac β]Ω6= /0. Therefore, by consideringω ′ ∈ [ω|{x} ω|Ac β ]Ω 6= /0 and repeating the argument in the proof of Proposition613.6. Relationship between spatial and combinatorial mixing properties3.6.3, we have thatpiωA (ω|{x})≥ piω′An,x(ω′∣∣An,x)piωA (β ) (3.58)≥ exp(−4∆|An,x|(Φmax+ log |A |))|A |−|B| (3.59)≥ exp(−32n2(Φmax+ log |A |))|A |−4n. (3.60)Since this lower bound is positive and independent ofω , A, and x, taking the in-fimum we have that cpi ≥ exp(−32n2(Φmax+ log |A |))|A |−4n > 0 and, by Corol-lary 6, we conclude that Ω exhibits TSSM.Notice that Proposition 3.5.9 gives us an alternative way to prove TSSM forΩ2g,since the decay rate γ can be arbitrarily large (in particular, larger than 4log(g+1))and Theorem 3.6.5 applies.Remark 9. Recall that TSSM implies strong irreducibility, so in view of the preced-ing result SSM with high exponential rate implies strong irreducibility. In general,it is not known whether SSM implies strong irreducibility.Note 6. If pi were a Zd Gibbs (Ω,Φ)-specification satisfying SSM with decay func-tion f (n) = Ce−γnd−1 , we could modify the previous proof to conclude that Ω ex-hibits TSSM for sufficiently large decay rate γ . The reason why exponential SSM isnot enough in this proof for an arbitrary d, is that only in Z2 the boundary of a ballgrows linearly with the radius. Consequently, the previous proof should work inany board where the boundary of neighbourhoods grows linearly with the radius,probably under a change of the bound for the decay rate γ .62Chapter 4SSM in homomorphism spacesGiven a board G , a constraint graph H, and a constrained energy function φ ,we are interested in studying Gibbs (Hom(G ,H),Φ)-specifications, where Φ de-notes is the induced n.n. interaction. Whenever we talk about a Gibbs (G ,H,φ)-specification, we will understand that it is in the previous sense. Also, when talkingabout cylinder sets [α]Ω, for Ω = Hom(G ,H), we will sometimes denote them by[α]GH in order to emphasize the board and constraint graph.One of the main purposes in this chapter is to understand the combinatorialproperties that constraint graphs H and homomorphism spaces Hom(G ,H) shouldsatisfy in order to admit the existence of Gibbs (G ,H,φ)-specifications pi withspatial mixing properties. The Gibbs (G ,H,φ)-specifications with a unique Gibbsmeasure have been already studied and, to some extent, characterized in [17]. Weaim to develop a somewhat analogous framework and sufficiently general condi-tions under which Gibbs specifications satisfy SSM.4.1 Dismantlable graphs and homomorphism spacesThe next definition is a structural description of a class of graphs introduced in[65], and heavily studied and characterized in [17].Definition 4.1.1. Given a constraint graph H and u,v ∈ V such that N(u)⊆ N(v),a fold is a homomorphism α : H→ H[V\{u}] such that α(u) = v and α|V\{u} isthe identity id|V\{u}.A constraint graph H is dismantlable if there is a sequence of folds reducingH to a graph with a single vertex (with or without a loop).Notice that a fold α : H→ H[V \ {u}] amounts to just removing u and edgescontaining it from the graph H, as long as a suitable vertex v exists which can634.1. Dismantlable graphs and homomorphism spaces“absorb” u.The following proposition is a good example of the kind of results that we aimto achieve.Proposition 4.1.1. Let H be a constraint graph. Then, H is dismantlable iff forevery board G of bounded degree and γ > 0, there exists a constrained energyfunction φ such that the Gibbs (G ,H,φ)-specification satisfies exponential WSMwith decay rate γ .See Proposition 4.2.1 for a proof of this result. As we remark there, it is es-sentially due to Brightwell and Winkler [17]. One of our goals is to prove similarstatements in which “WSM” is replaced by “SSM.”As in Proposition 4.1.1, it will be very common to have results where when aproperty is satisfied for every board G , sometimes one can conclude facts about H.Another simple example is the following.Proposition 4.1.2. Let H be a constraint graph such that Hom(G ,H) satisfies SSF,for every board G . Then, H has a safe symbol.Proof. Let G = S|H| (the n-star graph with n = |H|) and let x ∈ V to be the centralvertex with boundary ∂{x} = {y1, . . . ,y|V|}. Write V = {v1, · · · ,v|V|} and takeβ ∈ V∂{x} such that β (yi) = vi, for 1≤ i≤ |V|. Then, by SSF, there exists a graphhomomorphism α : G [∂{x}∪{x}]→ H such that α|∂{x} = β . Since α is a graphhomomorphism, x ∼G yi =⇒ α(x) ∼H α(yi). Therefore, α(x) ∼H vi, for every i,so α(x) is a safe symbol for H.Notice that the converse also holds, i.e. if a constraint graph H has a safesymbol, then Hom(G ,H) satisfies SSF, for every board G .Summarizing, given a homomorphism space Hom(G ,H), we have the follow-ing implications:H has a safe symbol =⇒ Hom(G ,H) satisfies SSF (4.1)=⇒ Hom(G ,H) satisfies TSSM (4.2)=⇒ Hom(G ,H) is strongly irreducible, (4.3)644.2. Dismantlable graphs and WSMand all implications are strict in general (even if we fix G to be some particularboard, like G = Z2). See Section 3.5 and [58, 15] for examples that illustrate thedifferences among some of these conditions.4.2 Dismantlable graphs and WSMDismantlable graphs are closely related with Gibbs measures, as illustrated by thefollowing proposition, parts of which appeared in [17].Proposition 4.2.1 ([17]). Let H be a constraint graph. Then, the following areequivalent:1. H is dismantlable.2. Hom(G ,H) is strongly irreducible with gap 2|H|+1, for every board G .3. Hom(G ,H) is strongly irreducible with some gap, for every board G .4. For every board G of bounded degree, there exists a n.n. Gibbs (G ,H,φ)-specification that admits a unique n.n. Gibbs measure µ .5. For every board G of bounded degree and γ > 0, there exists a n.n. Gibbs(G ,H,φ)-specification that satisfies exponential WSM with decay rate γ .The equivalence in Proposition 4.2.1 of (1), (2), (3), and (4) is proven in [17].However, in that work the concept of WSM (a priori, stronger than uniqueness)is not considered. Since WSM implies uniqueness, (5) =⇒ (4) is trivial. In theremaining part of this section, we introduce the necessary background to provethe missing implications, for which it is sufficient to show that (1) =⇒ (5) (seeProposition 4.2.5). This and a subsequent proof (see Proposition 4.4.2) will havea similar structure to the proof of (1) =⇒ (4) from [17, Theorem 7.2]. However,some coupling techniques will need to be modified, plus other combinatorial ideasneed to be considered.Given a dismantlable graph H, the only case where a sequence of folds reducesH to a vertex without a loop is when H is a set of isolated vertices without loops.In this case, we call H trivial (see [17, p. 6]). If v∗ ∈ V has a loop and there is asequence of folds reducing H to v∗, then we call v∗ a persistent vertex of H.654.2. Dismantlable graphs and WSMLemma 4.2.2 ([17, Lemma 5.2]). Let H be a nontrivial dismantlable constraintgraph and v∗ a persistent vertex of H. Let G be a board, A b V , and ω ∈Hom(G ,H). Then there exists υ ∈ Hom(G ,H) such that1. υ(x) = ω(x), for every x ∈ V \N|H|−2(A),2. υ(x) = v∗, for every x ∈ A, and3. ω−1(v∗)⊆ υ−1(v∗).Now, given a constraint graph H, a persistent vertex v∗, and λ > 1, define φλto be the constrained energy function given byφλ (v∗) =− logλ , and φλ |V\{v∗}∪E ≡ 0. (4.4)We have the following lemma.Lemma 4.2.3. Let H be dismantlable constraint graph and let G be a board ofbounded degree ∆. Given λ > 1, consider the Gibbs (G ,H,φλ )-specification pi , apoint ω ∈ Hom(G ,H), and B ⊆ A b V such that N|H|−2(B) ⊆ A. Then, for anyk ∈ N,piωA ({α : |{y ∈ B : α(y) 6= v∗}| ≥ k})≤ |H||B|∆|H|−1λ−k. (4.5)Proof. Let’s denote Ω = Hom(G ,H) and K = N|H|−2(B). W.l.o.g., consider anarbitrary configuration α ∈ VA such that α ω|Ac ∈ Ω (and, in particular, such thatpiωA (α)> 0). Denote ω1 = α ω|Ac . By Lemma 4.2.2, there exists ω2 ∈Ω such thatω1|V \K = ω2|V \K , ω2(x) = v∗ for every x ∈ B, and ω−11 (v∗)⊆ ω−12 (v∗).Notice that ω1|A ω|Ac , ω2|A ω|Ac ∈Ω and ω1|A\K = ω2|A\K . Now, given somek ≤ |B|, suppose that α is such that |{y ∈ B : α(y) 6= v∗}| ≥ k. Then, by the defini-tion of Gibbs specification and the fact that ω−11 (v∗)⊆ ω−12 (v∗),piωA(ω2|K∣∣∣ω1|A\K)piωA(ω1|K∣∣∣ω1|A\K) =piω1K (ω2|K)piω1K (ω1|K)≥ λ k. (4.6)Therefore,piωA(α|K∣∣∣α|A\K)= piωA (ω1|K∣∣∣ω1|A\K)≤ λ−k. (4.7)664.2. Dismantlable graphs and WSMNext, by taking weighted averages over all configurations β ∈LA\K(Ω) suchthatβ α|K ω|Ac ∈Ω, (4.8)we havepiωA (α|K) =∑βpiωA (α|K |β )piωA (β )≤∑βλ−kpiωA (β ) = λ−k. (4.9)Notice that |K|= ∣∣N|H|−2(B)∣∣≤ |B|∆|H|−1. In particular,|LK (Ω)| ≤ |H||B|∆|H|−1 . (4.10)Then, since α was arbitrary,piωA ({α : |{y ∈ B : α(y) 6= v∗}| ≥ k})≤ ∑α|K :α∈LA(Ω),|{y∈B:α(y)6=v∗}|≥kpiωA (α|K) (4.11)≤ |H||B|∆|H|−1λ−k. (4.12)As mentioned before, we essentially use some coupling techniques from [17],with slight modifications. We will use the following theorem.Theorem 4.2.4 ([11, Theorem 1]). Given a n.n. Gibbs (G ,H,φ)-specification pi ,Ab V , and ω1,ω2 ∈Hom(G ,H), there exists a coupling ((α1(x),α2(x)),x ∈ A) ofpiω1A and piω2A (whose distribution we denote by Pω1,ω2A ), such that for each x ∈ A,α1(x) 6= α2(x) iff there is a path of disagreement from x to Σ∂A(ω1,ω2), Pω1,ω2A -a.s.In general, by a path of disagreement from A to B, we mean that there is apath P from A to B such that α1(y) 6= α2(y), for all y ∈ P. We denote this event by{A6=←→ B}.Remark 10. The result in [11, Theorem 1] is for MRFs, but here we state it forn.n. Gibbs specifications.Proposition 4.2.5. Let G be a board of bounded degree ∆ and H a dismantlableconstraint graph. Then, for all γ > 0, there exists λ0 = λ0(γ, |H|,∆) such that for674.2. Dismantlable graphs and WSMevery λ > λ0, the Gibbs (G ,H,φλ )-specification pi satisfies exponential WSM withdecay rate γ .Proof. Let Ab V , B⊆ A, β ∈VB, and ω1,ω2 ∈Hom(G ,H). W.l.o.g. (since C canbe taken arbitrarily large in the desired decay function Ce−γn), we may supposethatdist(B,∂A) = n > |H|−2. (4.13)By Theorem 4.2.4, we have∣∣piω1A (β )−piω2A (β )∣∣= ∣∣Pω1,ω2A (α1|B = β )−Pω1,ω2A (α2|B = β )∣∣ (4.14)≤ Pω1,ω2A (α1|B 6= α2|B) (4.15)≤ ∑x∈BPω1,ω2A (α1(x) 6= α2(x)) (4.16)= ∑x∈BPω1,ω2A(x6=←→ Σ∂A(ω1,ω2))(4.17)≤ ∑x∈BPω1,ω2A(x6=←→ ∂A)(4.18)≤ ∑x∈BPω1,ω2A(x6=←→N|H|−2(∂A)). (4.19)When considering a path of disagreement P from x to N|H|−2(∂A), we canassume thatN|H|−2(P)⊆ A. Then, by Lemma 4.2.3,piωA ({α : |{y ∈ P : α(y) 6= v∗}| ≥ k})≤ |H||P|∆|H|−1λ−k. (4.20)In addition, |P| ≥ n−|H|+2 and, for every y ∈ P, we have that α1(y) 6= α2(y),so α1(y) and α2(y) cannot be both v∗ at the same time, and either α1|P or α1|Pmust have |P|/2 sites different from v∗. In consequence,Pω1,ω2A(x6=←→N|H|−2(∂A))(4.21)≤∞∑k=n−|H|+2∑|P|=kpiω1A({α1 : |{y ∈ P : α1(y) 6= v∗}| ≥ k2})(4.22)+∞∑k=n−|H|+2∑|P|=kpiω2A({α2 : |{y ∈ P : α2(y) 6= v∗}| ≥ k2})684.2. Dismantlable graphs and WSM≤ 2∞∑k=n−|H|+2∑|P|=k|H|k∆|H|−1λ− k2 (4.23)≤ 2∞∑k=n−|H|+2∆(∆−1)k(|H|∆|H|−1λ 1/2)k(4.24)= 2∆∞∑k=n−|H|+2((∆−1)|H|∆|H|−1λ 1/2)k. (4.25)Figure 4.1: Decomposition in the proofs of Lemma 4.2.3 and Proposition 4.2.5.Finally, we have∣∣piω1A (β )−piω2A (β )∣∣≤ ∑x∈BPω1,ω2A(x6=←→N|H|−2(∂A))(4.26)≤ 2|B|∆∞∑k=n−|H|+2((∆−1)|H|∆|H|−1λ 1/2)k, (4.27)so, in order to have exponential decay, it suffices to take(∆−1)2|H|2∆|H|−1 < λ , (4.28)and we note that any decay rate γ is achievable by taking λ sufficiently large.694.2. Dismantlable graphs and WSMProof of Proposition 4.2.1. The implication (1) =⇒ (5) follows from Proposition4.2.5. Since WSM implies uniqueness, we have (5) =⇒ (4). The implications(4) =⇒ (3) =⇒ (2) =⇒ (1) can be found in [17, Theorem 4.1].A priori, one would be tempted to think that the proof of Proposition 4.2.5could give SSM instead of just WSM, since the coupling in Theorem 4.2.4 in-volves a path of disagreement from B to Σ∂A(ω1,ω2) and not just to ∂A, just asin the definition of SSM. One of our motivations is to illustrate that this is notalways the case (see the examples in Section 4.7), mainly due to combinatorialobstructions. We will see that in order to have an analogous result for the SSMproperty, H must satisfy even stronger conditions than dismantlability, which guar-antees WSM by Proposition 4.2.1. One of the main issues is that our proof requiredthat dist(B,∂A) > |H|− 2, so B cannot be arbitrarily close to ∂A, which is in op-position to the spirit of SSM.Notice that, by Proposition 4.2.1, Hom(G ,H) is strongly irreducible for ev-ery board G iff H is dismantlable. Clearly, the forward direction still holds if“strongly irreducible” is replaced by “TSSM,” since TSSM implies strongly irre-ducible. Later, we will address the question of whether the reverse direction holdswith this replacement.For a dismantlable constraint graph H and a particular or arbitrary board G , weare interested in whether or not Hom(G ,H) is TSSM and whether or not there existsa Gibbs (G ,H,φ)-specification pi that satisfies exponential SSM with arbitrarilyhigh decay rate. Here we restate Theorem 3.6.5 in the context of homomorphismspaces, which shows that these two desired conclusions are related.Theorem 4.2.6. Let pi be a Gibbs (Z2,H,φ)-specification that satisfies exponentialSSM with decay rate γ > 4log |H|. Then, Hom(Z2,H) satisfies TSSM.One of our main goals is to look for conditions on Hom(G ,H) suitable forhaving a Gibbs specification that satisfies SSM. SSM seems to be related withTSSM, as illustrated in the previous results. In the following section, we exploresome other properties related with TSSM.704.3. The unique maximal configuration property4.3 The unique maximal configuration propertyFix a constraint graph H and consider an arbitrary board G . Given a linear order on the set of vertices V, we consider the partial order (that, in a slight abuse ofnotation, we also denote by ) on VV obtained by extending coordinate-wise thelinear order to subsets of V , i.e. given α1,α2 ∈VA, for some A⊆ V , we say thatα1  α2 iff α1(x)  α2(x), for all x ∈ A. If α1(x)  α2(x) but α1(x) 6= α2(x), wewrite α1(x) ≺ α2(x). In addition, if two vertices u,v ∈ V are such that u ∼ v andu v, we will denote this by u- v.Definition 4.3.1. Given g ∈ N, we say that Hom(G ,H) satisfies the unique maxi-mal configuration (UMC) property with distance g if there exists a linear order on V such that, for every Ab V ,(M1) for every α ∈LA(Hom(G ,H)), there is a unique point ωα ∈ [α]GH such thatω  ωα , for every point ω ∈ [α]GH , and(M2) for any two α1,α2 ∈LA(Hom(G ,H)), ΣV (ωα1 ,ωα2)⊆Ng(ΣA(α1,α2)).If Hom(G ,H) satisfies the UMC property, then for any α ∈LA(Hom(G ,H))and β = α|B with B ⊆ A, it is the case that ωα  ωβ . This is natural, since wecan see the configurations α and β as “restrictions” to be satisfied by ωα andωβ , respectively. In addition, observe that condition (M2) in Definition 4.3.1 im-plies that ΣV (ωα ,ωβ ) ⊆Ng(A \B). In particular, by taking B = /0, we see that ifHom(G ,H) satisfies the UMC property, then there must exist a greatest elementω∗ ∈ Hom(G ,H) such that ω  ω∗, for every ω ∈ Hom(G ,H), and ΣV (ωα ,ω∗)⊆Ng(A), for every α ∈LA(Hom(G ,H)).The following proposition establishes that the UMC property is related withTSSM.Proposition 4.3.1. Suppose that Hom(G ,H) satisfies the UMC property with dis-tance g. Then, Hom(G ,H) satisfies TSSM with gap 2g+1.Proof. Consider a pair of sites x,y ∈ V with dist(x,y) ≥ 2g+ 1, a set S b V ,and configurations α ∈ V{x}, β ∈ V{y}, and σ ∈ VS such that [ασ ]GH , [σβ ]GH 6= /0.Take the corresponding maximal configurations ωσ , ωασ , and ωσβ . Notice that714.4. SSM and the UMC propertyΣV (ωασ ,ωσ ) ⊆Ng(x) and ΣV (ωσ ,ωσβ ) ⊆Ng(y). Since dist(x,y) ≥ 2g+ 1, wecan conclude thatNg(x)c∩Ng(y) = /0 andNg(x)∩Ng(y)c = /0. Therefore,ωσ |Ng(x)c∩Ng(y)c = ωασ |Ng(x)c∩Ng(y)c = ωσβ∣∣Ng(x)c∩Ng(y)c , (4.29)and since Hom(G ,H) is a topological MRF (see [20]), we haveωασ |Ng(x) ωσ |Ng(x)c∩Ng(y)c ωσβ∣∣Ng(y)∈ Hom(G ,H), (4.30)so [ασβ ]GH 6= /0. Using Lemma 3.3.1, we conclude.4.4 SSM and the UMC propertyGiven a constraint graph H, a linear order  in V = {v1, . . . ,vk, . . . ,v|H|} suchthat v1 ≺ ·· · ≺ vk ≺ ·· · ≺ v|H|, and λ > 1, define φλ to be the constrained energyfunction given byφλ (vk) =−k logλ , and φλ∣∣E ≡ 0. (4.31)We have the following lemma.Lemma 4.4.1. Suppose that Hom(G ,H) satisfies the UMC property with distanceg, for G a board of bounded degree ∆. Given λ > 1, consider the Gibbs (G ,H,φλ )-specification pi , a point ω ∈Hom(G ,H), and sets B⊆ Ab V . Then, for any k ∈N,piωA ({α : |{y ∈ B : α(y)≺ ωδ (y)}| ≥ k})≤ |H||B|∆g+1λ−k, (4.32)where δ = ω|∂A.Proof. Let’s denote Ω= Hom(G ,H). Consider an arbitrary configuration α ∈ VAsuch that α ω|Ac ∈ Ω (so, in particular, piωA (α) > 0). Take the set K = A∩Ng(B)and decompose its boundary K into the two subsets A∩∂K and ∂A∩∂K. ConsiderD = A∩ ∂K and name η = α|D. Since δη ∈ L∂A∪D(Ω), there exists a uniquemaximal configuration ωδη . Clearly, α(x)  ωδη(x), for every x ∈ A. Moreover,ωδη∣∣B = ωδ |B, since ΣV (ωδη ,ωδ )⊆Ng(D) and dist(B,D)> g, soNg(D)∩B= /0.724.4. SSM and the UMC propertyNow, suppose that α is such that |{y ∈ B : α(y)≺ ωδ (y)}| ≥ k, for some k ≤|B|. Then, ∣∣{y ∈ B : α(y)≺ ωδη(y)}∣∣ ≥ k and, by the (topological and measure-theoretical) MRF property,piωA(ωδη∣∣K∣∣∣α|A\K)piωA(α|K∣∣∣α|A\K) =piωδηK(ωδη∣∣K)piωδηK (α|K)≥ λ k. (4.33)Therefore,piωA(α|K∣∣∣α|A\K)≤ λ−k. (4.34)Next, by taking weighted averages over all configurations β ∈LA\K(Ω) suchthatβ α|K ω|Ac ∈Ω, (4.35)we havepiωA (α|K) =∑βpiωA (α|K |β )piωA (β )≤∑βλ−kpiωA (β ) = λ−k. (4.36)Notice that |K| ≤ |Ng(B)| ≤ |B|∆g+1, so |LK (Ω)| ≤ |H||B|∆g+1 . Then, since αwas arbitrary,piωA ({α : |{y ∈ B : α(y)≺ ωδ (y)}| ≥ k})≤ ∑α|K :α∈LA(Ω),|{y∈B:α(y)≺ωδ (y)}|≥kpiωA (α|K) (4.37)≤ |H||B|∆g+1λ−k. (4.38)Proposition 4.4.2. Suppose that Hom(G ,H) satisfies the UMC property with dis-tance g, for G a board of bounded degree ∆. Then, for all γ > 0, there existsλ0 = λ0(γ, |H|,∆,g) such that for every λ > λ0, the Gibbs (G ,H,φλ )-specificationpi satisfies exponential SSM with decay rate γ .Proof. Let A b V , x ∈ A, β ∈ V{x}, and ω1,ω2 ∈ Hom(G ,H). W.l.o.g., we maysuppose thatdist(x,Σ∂A(ω1,ω2)) = n > g. (4.39)734.4. SSM and the UMC propertyBy Theorem 4.2.4, and similarly to the proof of Proposition 4.2.5, we have∣∣piω1A (β )−piω2A (β )∣∣≤ Pω1,ω2A (α1(x) 6= α2(x)) (4.40)= Pω1,ω2A(x6=←→ Σ∂A(ω1,ω2))(4.41)≤ Pω1,ω2A(x6=←→Ng(Σ∂A(ω1,ω2))). (4.42)When considering a path of disagreement P from x to Ng(Σ∂A(ω1,ω2)), wecan assume (by truncating if necessary) that P ⊆ A \Ng(Σ∂A(ω1,ω2)) and |P| ≥n− g. By the UMC property, if we take δ1 = ω1|∂A and δ2 = ω2|∂A, we haveΣV (ωδ1 ,ωδ2) ⊆ Ng(Σ∂A(ω1,ω2)) = Ng(Σ∂A(δ1,δ2)), so ωδ1∣∣P= ωδ2∣∣P=: θ ∈LP(Hom(G ,H)). Since P is a path of disagreement, for every y ∈ P we haveα1(y) ≺ α2(y)  θ(y) or α2(y) ≺ α1(y)  θ(y). In consequence, using Lemma4.4.1 yieldsPω1,ω2A(x6=←→Ng(Σ∂A(ω1,ω2)))(4.43)≤∞∑k=n−g∑|P|=kPω1,ω2A (P is a path of disagr. from x toNg(Σ∂A(δ1,δ2))) (4.44)≤∞∑k=n−g∑|P|=kpiω1A({α1 : |{y ∈ P : α1(y)≺ θ(y)}| ≥ k2})(4.45)+∞∑k=n−g∑|P|=kpiω2A({α2 : |{y ∈ P : α2(y)≺ θ(y)}| ≥ k2})≤ 2∞∑k=n−g∑|P|=k|H|k∆g+1λ− k2 ≤ 2∆∞∑k=n−g((∆−1)|H|∆g+1λ 1/2)k. (4.46)Then, by Lemma 3.1.1, exponential SSM holds whenever(∆−1)2|H|2∆g+1 < λ , (4.47)and any decay rate γ may be achieved by taking λ large enough.Notice that here λ0 is defined in terms of γ , |H|, ∆ and g. In the WSM proof, gimplicitly depended on |H|, but here the two parameters could be, a priori, virtually744.5. UMC and chordal/tree decomposable graphsindependent.4.5 UMC and chordal/tree decomposable graphs4.5.1 Chordal/tree decompositionsDefinition 4.5.1. A simple constraint graph H is said to be chordal if all cycles offour or more vertices have a chord, which is an edge that is not part of the cyclebut connects two vertices of the cycle.Definition 4.5.2. A perfect elimination ordering in a simple constraint graphH = (V,E) is an ordering v1, . . . ,vn of V such that H[vi∪ ({vi+1, . . . ,vn}∩N(vi))]is a complete graph, for every 1≤ i≤ n = |H|.Proposition 4.5.1 ([30]). A simple constraint graph H is chordal iff it has a perfectelimination ordering.Definition 4.5.3. A constraint graph H = (V,E) will be called loop-chordal ifLoop(H) = V and H′ = (V,E\{{v,v} : v ∈V}) (i.e. H′ is the version of H withoutloops) is chordal.Proposition 4.5.2. Given a loop-chordal constraint graph H = (V,E), there ex-ists an ordering v1, . . . ,vn of V such that H[vi∪ ({vi+1, . . . ,vn}∩N(vi))] is a loop-complete graph, for every 1≤ i≤ n = |H|.Proof. This follows immediately from Proposition 4.5.1.Proposition 4.5.2 can also be thought of as saying that a graph G = (V,E) isloop-chordal iff Loop(G) =V and there exists an order v1 ≺ ·· · ≺ vn such thatvi - v j ∧ vi - vk =⇒ v j ∼ vk. (4.48)Proposition 4.5.3. A connected loop-chordal graph G is dismantlable.Proof. Let v1, . . . ,vn be the ordering of V given by Proposition 4.5.2 and take v ∈N(v1). Clearly, v ∈ {v2, . . . ,vn} and then we have G[v1∪ ({v2, . . . ,vn}∩N(v1))] =G[N(v1)] is a loop-complete graph and v ∈ N(v1). Therefore, N(v1) ⊆ N(v) and754.5. UMC and chordal/tree decomposable graphsthere is a fold from G to G[V \ {v1}]. It can be checked that G[V \ {v1}] is alsoloop-chordal, so we apply the same argument to G[V \ {v1}] and so on, until weend with only one vertex (with a loop).CTJT1T2T3J1J2Figure 4.2: A chordal/tree decomposition.We say that a constraint graph H = (V,E) has a chordal/tree decompositionor is chordal/tree decomposable if we can write V = CunionsqTunionsq J such that1. H[C] is a nonempty loop-chordal graph,2. T = T1 unionsq ·· · unionsqTm and, for every 1 ≤ j ≤ m, H[T j] is a tree such that thereexist unique vertices r j ∈T j (the root of T j) and c j ∈C such that {r j,c j} ∈E,3. J = J1unionsq ·· · unionsq Jn and, for every 1 ≤ k ≤ n, H[Jk] is a connected graph with aunique vertex ck ∈ C such that {u,ck} ∈ E, for every u ∈ Jk, and4. E[T : J] = E[T j1 : T j2 ] = E[Jk1 : Jk2 ] = /0, for every j1 6= j2 and k1 6= k2.Notice that, for every k, the vertex ck ∈ C is a safe symbol for H[{ck}∪ Jk].764.5. UMC and chordal/tree decomposable graphs4.5.2 A natural linear orderGiven a chordal/tree decomposable constraint graph H, we define a linear order on V as follows:• If w ∈ J and t ∈ T, then w≺ t.• If t ∈ T and c ∈ C, then t ≺ c.• If w ∈ Jk and w′ ∈ Jk′ , for some 1≤ k < k′ ≤ n, then w≺ w′.• If t ∈ T j and t ′ ∈ T j′ , for some 1≤ j < j′ ≤ m, then t ≺ t ′.• Given 1≤ k ≤ n, we fix an arbitrary order in Jk.• Given 1≤ j ≤ m and t1, t2 ∈ T j, then– if dist(t1,r j)< dist(t2,r j), then t1 ≺ t2,– if dist(t1,r j)> dist(t2,r j), then t2 ≺ t1, and– for each i, we arbitrarily order the set of vertices t with dist(t,r j) = i.• If c1,c2 ∈ C, then c1 and c2 are ordered according to Proposition 4.5.2.Proposition 4.5.4. If a constraint graph H has a chordal/tree decomposition, thenH is dismantlable.Proof. W.l.o.g., suppose that |H| ≥ 2 (the case |H| = 1 is trivial). Let G be anarbitrary board. In the following theorem (Theorem 4.5.6), it will be proven that ifH is chordal/tree decomposable, then Hom(G ,H) satisfies the UMC property withdistance |H|−2. Therefore, by Proposition 4.3.1, Hom(G ,H) satisfies TSSM withgap 2(|H|− 2)+ 1 and, in particular, Hom(G ,H) is strongly irreducible with gap2|H|+ 1. Since the gap is independent of G , we can apply Proposition 4.2.1 toconclude that H must be dismantlable.Proposition 4.5.5. If a constraint graph H has a safe symbol, then H is chordal/treedecomposable.Proof. This follows trivially by considering C = {s}, T = /0 and J = V\{s}, withs a safe symbol for H.774.5. UMC and chordal/tree decomposable graphsWe show that chordal/tree decomposable graphs H induce combinatorial prop-erties on homomorphism spaces Hom(G ,H).Theorem 4.5.6. Let H be a chordal/tree decomposable constraint graph. Then,Hom(G ,H) has the UMC property with distance |H|−2, for any board G .Before proving Theorem 4.5.6, we introduce some useful tools. From now on,we fix Hom(G ,H) and x1,x2, . . . to be an arbitrary order of V . We also fix thelinear order  on V as defined above. For i ∈ {1, . . . , |H|}, define the sets Di :={(ω1,ω2,x) ∈ Hom(G ,H)×Hom(G ,H)×G : vi = ω1(x)≺ ω2(x)}, and considerD(`)=⋃`i=1 Di, for 1≤ `≤ |H|, and D :=D(|H|). Notice that D|H|= /0. In addition,given (ω1,ω2,x) ∈ D, define the setN−(ω1,x) := {y ∈ N(x) : ω1(y)≺ ω1(x)}, (4.49)and the partitionN−(ω1,x) = N−≺(ω1,ω2,x)unionsqN−(ω1,ω2,x)unionsqN−=(ω1,ω2,x), (4.50)whereN−≺(ω1,ω2,x) := {y ∈ N−(ω1,x) : ω1(y)≺ ω2(y)}, (4.51)N−(ω1,ω2,x) := {y ∈ N−(ω1,x) : ω1(y) ω2(y)}, and (4.52)N−=(ω1,ω2,x) := {y ∈ N−(ω1,x) : ω1(y) = ω2(y)}. (4.53)LetP : D→ D be the function that, given (ω1,ω2,x) ∈ D, returns1. (ω1,ω2,y), if N−≺(ω1,ω2,x) 6= /0,2. (ω2,ω1,y), if N−≺(ω1,ω2,x) = /0 and N−(ω1,ω2,x) 6= /0, or3. (ω1,ω2,x), if N−≺(ω1,ω2,x) = N−(ω1,ω2,x) = /0,where y is the minimal element in N−≺(ω1,ω2,x) or N−(ω1,ω2,x), respectively.Here the minimal element y is taken according to the previously fixed order ofV . We chose y to be minimal just to have P well-defined; it will not be oth-erwise relevant. Notice that if (ω1,ω2,x) ∈ D` and P(ω1,ω2,x) 6= (ω1,ω2,x),784.5. UMC and chordal/tree decomposable graphsthen P(ω1,ω2,x) ∈ D(`− 1). This implies that every element in D1 must bea fixed point. Moreover, for every (ω1,ω2,x) ∈ D, the (|H| − 2)-iteration of Pis a fixed point (though not necessarily in D1), i.e. P(P |H|−2(ω1,ω2,x))=P |H|−2(ω1,ω2,x).We have the following lemma.Lemma 4.5.7. Let (ω1,ω2,x) ∈ D be such that P(ω1,ω2,x) = (ω1,ω2,x). Then,there exists u ∈ V such that ω1(x)≺ u, and the point ω˜1 defined asω˜1(y) =u if y = x,ω1(y) if y 6= x, (4.54)is globally admissible. In particular, ω1 ≺ ω˜1.Proof. Notice that if (ω1,ω2,x) is a fixed point, N−(ω1,x) = N−=(ω1,ω2,x). Wehave two cases:Case 1: N−(ω1,x) = /0. If this is the case, then ω1(y)  ω1(x), for all y ∈ N(x).Notice that ω1(x)≺ ω2(x) v|H|, so ω1(x)≺ v|H|. Then we have three sub-cases:Case 1.a: ω1(x) ∈ Jk for some 1 ≤ k ≤ n. Since {v,ck} ∈ E for all v ∈ Jk, we canmodify ω1 at x in a valid way by replacing ω1(x) ∈ Jk with u = ck.Case 1.b: ω1(x) ∈ T j for some 1≤ j ≤ m. Since ω1(y) ω1(x), for all y ∈ N(x),but ω1(x) ∈ T j and T j does not have loops, we have ω1(y)  ω1(x), for all y ∈N(x). Call t = ω1(x). Then, there are three possibilities: t = r j, dist(t,r j) = 1, ordist(t,r j)> 1.If t = r j, then ω1(y) = c j for all y ∈N(x), where c j  r j is the unique vertex inC connected with r j. Since c j must have a loop, we can replace ω1(x) by u = c j ω1(x) in ω1.If dist(t,r j) = 1, then ω1(y) = r j for all y∈N(x), and, similarly to the previouscase, we can replace ω1(x) by u = c j  ω1(x) in ω1.Finally, if dist(t,r j)> 1, then ω1(y) = f  t, for all y ∈ N(x), where f ∈ T j isthe parent of t in the r j-rooted tree H[T j]. Then, since dist(t,r j) > 1, there mustexist h ∈ T j that is the parent of f , so we can replace ω1(x) by u = h in ω1.794.5. UMC and chordal/tree decomposable graphsCase 1.c: ω1(x) ∈ C. If this is the case, and since ω1(x) ≺ v|H|, there must exist1 ≤ i < |C| such that ω1(x) = ci and N(ci)∩{ci+1, . . . ,c|C|} is nonempty. Now,ω1(y)  ω1(x), for all y ∈ N(x), so ω1(y) ∈ {ci}∪(N(ci)∩{ci+1, . . . ,c|C|}), forall y ∈ N(x). Since H[{ci}∪(N(ci)∩{ci+1, . . . ,c|C|})] is a loop-complete graphwith two or more elements, then we can replace ω1(x) by any element u ∈ N(ci)∩{ci+1, . . . ,c|C|} in ω1.Case 2: N−(ω1,x) 6= /0. In this case, ω1(x)≺ ω2(x) and, sinceN−(ω1,x) 6= /0 and N−(ω1,x) = N−=(ω1,ω2,x), (4.55)there must exist y∗ ∈ N(x) such that ω1(y∗)≺ ω1(x) and ω1(y∗) = ω2(y∗). Noticethat in this case, ω1(x) cannot belong to T, because ω1(x) ≺ ω2(x) and both areconnected to ω1(y∗); this would imply that, for some 1 ≤ j ≤ m, either (a) T jdoes not induce a tree, or (b) more than one vertex in T j is adjacent to a vertexin C. Therefore, we can assume that ω1(x) belongs to C (and therefore, sinceω2(x) ω1(x), also ω2(x) belongs to C).We are going to prove that, for every y ∈ N(x), we have ω1(y)∼ ω2(x), so wecan replace ω1(x) by ω2(x) in ω1. Since ω1(y) = ω2(y), for every y ∈ N−(ω1,x),we only need to prove that ω1(y) ∼ ω2(x), for every y ∈ N(x) such that ω1(y) ω1(x).Take any y ∈N−=(ω1,ω2,x). Then, ω2(y) = ω1(y)- ω1(x) and ω2(y)- ω2(x),so ω1(x) ∼ ω2(x), since H is loop-chordal. Consider now an arbitrary y ∈ N(x)such that ω1(y)  ω1(x). If ω1(y) = ω1(x), we have ω1(y) = ω1(x) ∼ ω2(x), sowe can assume that ω1(y)  ω1(x). Then, ω1(x) - ω1(y) and ω1(x) - ω2(x), soω1(y) ∼ ω2(x), again by loop-chordality of H. Then, ω1(y) ∼ ω2(x), for everyy ∈ N(x), and we can replace ω1(x) by u = ω2(x) in ω1, as desired.Now we are in a good position to prove Theorem 4.5.6.Proof of Theorem 4.5.6. Fix an arbitrary set Ab V and α ∈LA(Hom(G ,H)). Weproceed to prove the conditions (M1) (i.e. existence and uniqueness of a maximalpoint ωα ) and (M2).Condition (M1). Choose an ordering x1,x2, . . . of V \A and, for n ∈ N, defineAn :=A∪{x1, . . . ,xn}. Let α0 :=α and suppose that, for a given n and all 0< i≤ n,804.5. UMC and chordal/tree decomposable graphswe have already constructed a sequence αi ∈LAi(Hom(G ,H)) such that αi|Ai−1 =αi−1 and β (xi) αi(xi), for any β ∈LAi(Hom(G ,H)) such that β |Ai−1 = αi−1.Next, look for the globally admissible configuration αn+1 such that αn+1|An =αn and β (xn+1)αn+1(xn+1), for any β ∈LAn+1(Hom(G ,H)) such that β |An =αn.Iterating and by compactness of Hom(G ,H), we conclude the existence of a uniquepoint ωˆ ∈⋂n∈N[αn]GH .We claim that ωˆ is independent of the ordering x1,x2, . . . of V \A.By contradiction, suppose that given two orderings of V \ A we can obtaintwo different configurations ωˆ1 and ωˆ2 with the properties described above. Takex ∈ V \A such that ωˆ1(x) 6= ωˆ2(x). W.l.o.g., suppose that ωˆ1(x)≺ ωˆ2(x). Then wehave that (ωˆ1, ωˆ2,x) ∈ D and P |H|−2(ωˆ1, ωˆ2,x) is a fixed point for P . W.l.o.g.,suppose that P |H|−2(ωˆ1, ωˆ2,x) = (ωˆ1, ωˆ2, x˜), where x˜ ∈ V \A (note that x˜ is notnecessarily equal to x). By an application of Lemma 4.5.7, ωˆ1(x˜) can be replacedin a valid way by a vertex u ∈ V such that ωˆ1(x˜) ≺ u. If we let n be such thatx˜ = xn for the ordering corresponding to ωˆ1, we have a contradiction with themaximality of ωˆ1, since we could have chosen u instead of ωˆ1(xn) in the nth stepof the construction of ωˆ1.Therefore, there exists a particular ωˆ common to any ordering x1,x2, . . . ofV \A. We claim that taking ωα = ωˆ proves (M1). In fact, suppose that thereexists ω ∈ [α]GH and x∗ ∈ V \A such that ωˆ(x∗)≺ω(x∗). We can always choose anordering of V \A such that x1 = x∗. Then, according to such ordering, β  ωˆ|A1for any β ∈ LA1(Hom(G ,H)) such that β |A = α . In particular, if we take β =α ω|{x∗}, we have a contradiction.Condition (M2). Notice that if (ω ′1,ω ′2,y) =P(ω1,ω2,x), then x = y or x ∼ y.In addition, since the (|H| − 2)-iteration of P is a fixed point, if (ω ′1,ω ′2,y) =P |H|−2(ω1,ω2,x), then dist(x,y) ≤ |H| − 2. In order to prove condition (M2),consider two configurations α1,α2 ∈LA(Hom(G ,H)), and the set ΣV (ωα1 ,ωα2).We want to prove thatΣV (ωα1 ,ωα2)⊆N|H|−2(ΣA(α1,α2)). (4.56)W.l.o.g., suppose that α1 6= α2 and take x ∈ ΣV (ωα1 ,ωα2) 6= /0. It suffices tocheck that dist(x,ΣA(α1,α2))≤ |H|−2. Suppose for the sake of contradiction that814.6. The looped tree casedist(x,ΣA(α1,α2)) > |H| − 2 and let (ω ′α1 ,ω ′α2 ,y) =P |H|−2(ωα1 ,ωα2 ,x). Noticethat, by definition ofP , y also belongs to ΣV (ωα1 ,ωα2), and since dist(x,y)≤ |H|,we have y /∈ ΣA(α1,α2). Then, there are two possibilities: (a) y ∈ A \ΣA(α1,α2),or (b) y ∈ V \A.If y ∈ A\ΣA(α1,α2), then ωα1(y) = α1(y) = α2(y) = ωα2(y), and that contra-dicts the fact that y ∈ ΣV (ωα1 ,ωα2).If y ∈ V \ A and, w.l.o.g., ωα1(y) ≺ ωα2(y), we can apply Lemma 4.5.7 tocontradict the maximality of ωα1 .4.6 The looped tree caseA looped tree T will be called trivial if |T | = 1 and nontrivial if |T | ≥ 2. Weproceed to define a family of graphs that will be useful in future proofs.0 1 2 n n+1. . .Figure 4.3: The n-barbell graph Bn.Definition 4.6.1. Given n∈N, define the n-barbell graph as Bn = (V (Bn),E(Bn)),whereV (Bn) = {0,1, . . . ,n,n+1} , (4.57)andE(Bn) = {{0,0},{0,1}, . . . ,{n,n+1},{n+1,n+1}} . (4.58)Notice that a looped tree with a safe symbol must be an n-star (see Equation(2.7)) with a loop at the central vertex, possibly along with other loops. The graphHϕ can be seen as a very particular case of a looped tree with a safe symbol. Formore general looped trees, we have the next result.Proposition 4.6.1. Let T be a finite nontrivial looped tree. Then, the following areequivalent:(1) T is chordal/tree decomposable.(2) T is dismantlable.824.6. The looped tree case(3) Loop(T ) is connected in T and nonempty.Proof. We have the following implications.(1) =⇒ (2): This follows from Theorem 4.6.3, which is for general constraintgraphs.(2) =⇒ (3): Assume that T is dismantlable. First, suppose Loop(T ) = /0. Then,in any sequence of foldings of T , in the next to last step, we must end with a graphconsisting of just two adjacent vertices vn−1 and vn, without loops. However, thisis a contradiction, because N(u)(N(v) and N(v)(N(u), so such graph cannot befolded into a single vertex. Therefore, Loop(T ) is nonempty.Next, suppose that Loop(T ) is nonempty and not connected. Then, T musthave an n-barbell as a subgraph, for some n ≥ 1. Therefore, in any sequence offoldings of T , there must have been a vertex in the n-barbell that was folded first.Let’s call such vertex u and take v ∈V with N(u)⊆N(v). Then, v is another vertexin the n-barbell or it belongs to the complement. Notice that v cannot be in the n-barbell, because no neighbourhood of vertex in the n-barbell (even restricted to thebarbell itself) contains the neighbourhood of another vertex in the n-barbell. Onthe other hand, v cannot be in the complement of n-barbell, because v would haveto be connected to two or more vertices in the n-barbell (u and its neighbours), andthat would create a cycle in T . Therefore, Loop(T ) is connected.(3) =⇒ (1): Define C := Loop(T ). Then C is connected in T and nonempty.Then, if we denote by T its complement V \C and define J= /0, we have that V canbe partitioned into the three subsets CunionsqTunionsq J, which corresponds to a chordal/treedecomposition.Corollary 7. Let T be a finite nontrivial looped tree. Then, the following areequivalent:(1) T is chordal/tree decomposable.(2) Hom(G ,T ) has the UMC property, for every board G .(3) Hom(G ,T ) satisfies TSSM, for every board G .(4) Hom(G ,T ) is strongly irreducible, for every board G .834.6. The looped tree case(5) T is dismantlable.Proof. By Theorem 4.6.3, we have (1) =⇒ (2) =⇒ (3) =⇒ (4) =⇒ (5). Theimplication (5) =⇒ (1) follows from Proposition 4.6.1.Sometimes, given a constraint graph H, if a property for homomorphism spacesholds for a certain distinguished board or family of boards, then the property holdsfor any board G . For example, this is proven in [17] for a dismantlable graph Hand the strong irreducibility property, when G ∈ {Td}d∈N. The next result givesanother example of this phenomenon.Proposition 4.6.2. Let T be a finite looped tree. Then, the following are equivalent:(1) Hom(G ,T ) satisfies TSSM, for every board G ..(2) Hom(Z2,T ) satisfies TSSM.(3) There exist Gibbs (Z2,T,φ)-specifications which satisfy exponential SSMwith arbitrarily high decay rate.Proof. We have the following implications.(1) =⇒ (2): Trivial.(2) =⇒ (1): Let’s suppose that Hom(Z2,T ) satisfies TSSM. If T is trivial, thenHom(G ,T ) is a single point or empty, depending on whether the unique vertex in Thas a loop or not. In both cases, Hom(G ,T ) satisfies TSSM, for every board G . IfT is nontrivial, then Loop(T ) must be nonempty. To see this, by contradiction, firstsuppose that T is nontrivial and Loop(T ) = /0. Take an arbitrary vertex u ∈ V (T )and a neighbour v ∈ N(u). Notice that Hom(Z2,T ) is nonempty, since the pointωu,v defined asωu,v(x) =u if x1+ x2 = 0 mod 2,v if x1+ x2 = 1 mod 2, (4.59)is globally admissible. Now, if we interchange the roles of u and v, and consider the(globally admissible) pointωv,u, we haveωu,v((0,0))= u andωv,u((2g+1,0))= u,for an arbitrary g ∈N. However, if this is the case, Hom(Z2,T ) cannot be strongly844.6. The looped tree caseirreducible with gap g, for any g (and therefore, cannot be TSSM), because[ωu,v|(0,0) ωv,u|(2g+1,0)]Z2T= /0. (4.60)A way to check this is by considering the fact that both T and Z2 are bipartitegraphs. Therefore, we can assume that Loop(T ) 6= /0.  0   0   0   0  0   0   0  0   0n+1 n+1 n+1 n+1 n+1 n+1 n+1  0   0  0  1  2 n-1  n n-2n+1 n+1 n+1 n+1n+1 n-1  n  1  2  3  0  1  2 n-1  n n-2n+1 n-1  n  1  2  3  0  1  2 n-1  n n-2n+1 n-1  n  1  2  3  0  1  2 n-1  n n-2n+1 n-1  n  1  2  3  0  1  2 n-1  n n-2n+1 n-1  n  1  2  3  0  1  2 n-1  n n-2  0   0   0   0  0   0   0  0   0  0   0n+1 n-1  n  1  2  3  0  1  2 n-1  n n-2n+1 n-1  n  1  2  3  0  1  2 n-1  n n-2  0  1  2 n-1  n n-2n+1 n-1  n  1  2  3  0  1  2 n-1  n n-2n+1 n-1  n  1  2  3n+1 n-1  n  1  2  3  0  1  2 n-1  n n-2n+1 n-1  n  1  2  3n+1 n+1 n+1 n+1 n+1 n+1 n+1 n+1 n+1 n+1 n+1. . . . . .. . . . . .. . .. . .. . .. . .. . .. . .. . . . . .. . . . . .. . . . . .. . . . . .. . .. . .. . .. . .. . .. . .. . . . . .. . . . . .012n-2n+1. . .n-1n12. . .n-1n3n+10Figure 4.4: A “channel” in Hom(Z2,T ) with two incompatible extremes.Now, suppose that Loop(T ) 6= /0 and Loop(T ) is not connected in T . If thisis the case, T must have an n-barbell as an induced subgraph, for some n ≥ 1.Then, we would be able to construct configurations in L (Hom(Z2,T )) as shown854.6. The looped tree casein Figure 4.4. Note that vertices in the n-barbell can reach each other only throughthe path determined by the n-barbell, since T does not contain cycles. In Figure 4.4are represented the cylinder sets [ασ ]Z2T (top left), [σβ ]Z2T (top right) and [ασβ ]Z2T(bottom), where:1. α is the vertical left-hand side configuration in red, representing a sequenceof nodes in the n-barbell that repeats 0 but not n+1,2. β is the vertical right-hand side configuration in red, representing a sequenceof nodes in the n-barbell that repeats n+1 but not 0, and3. σ is the horizontal (top and bottom) configuration in black, representingloops on the vertices 0 and n+1, respectively.It can be checked that [ασ ]Z2T and [σβ ]Z2T are nonempty. However, the cylin-der set [ασβ ]Z2T is empty for every even separation distance between α and β ,since α and β force incompatible alternating configurations inside the “channel”determined by σ . Therefore, Hom(Z2,T ) cannot satisfy TSSM, which is a contra-diction.We conclude that Loop(T ) is nonempty and connected in T , and by Proposition4.6.1 T is chordal/tree decomposable. Finally, by Proposition 7, we conclude thatHom(G ,T ) satisfies TSSM, for every board G .(3) =⇒ (2): This follows from Theorem 4.2.6.(1) =⇒ (3): Since Hom(G ,T ) satisfies TSSM for every board G , Hom(G ,T ) hasthe UMC property for all board G (see Corollary 7). In particular, Hom(Z2,T ) hasthe UMC property. Then, (3) follows from Proposition 4.4.2.We have the following summary.Theorem 4.6.3. Fix a constraint graph H.• Then,H has a safe symbol ⇐⇒ Hom(G ,H) is SSF, for all G=⇒ H is chordal/tree decomposable864.7. Examples: Homomorphism spaces=⇒ Hom(G ,H) has the UMC property, for all G=⇒ Hom(G ,H) satisfies TSSM, for all G=⇒ Hom(G ,H) is strongly irreducible, for all G⇐⇒ H is dismantlable.• Let G a fixed board. Then,Hom(G ,H) has the UMC property =⇒ Hom(G ,H) satisfies TSSM=⇒ Hom(G ,H) is strongly irreducible.• Let G a fixed board with bounded degree. Then,Hom(G ,H) has the UMC property =⇒ For all γ > 0, there exists a Gibbs(G ,H,φ)-specification that satisfiesexponential SSM with decay rate γ⇓H is dismantlable =⇒ For all γ > 0, there exists a Gibbs(G ,H,φ)-specification that satisfiesexponential WSM with decay rate γ.Proof. The first chain of implications and equivalences follows from Proposition4.1.2, Proposition 4.5.5, Theorem 4.5.6, Proposition 4.3.1, Equation (4.3), andProposition 4.2.1. The second one, from Proposition 4.3.1 and Equation (4.3). Thelast one, from Proposition 4.4.2, Proposition 4.2.1, and the fact that SSM alwaysimplies WSM.4.7 Examples: Homomorphism spacesProposition 4.7.1. There exists a homomorphism space Hom(G ,H) that satisfiesTSSM but not the UMC property.Proof. The homomorphism space Hom(Z2,K5) satisfies TSSM (in fact, it satisfies874.7. Examples: Homomorphism spacesSSF) but not the UMC property. If Hom(Z2,K5) satisfies the UMC property, thenthere must exist an order≺ and a greatest element ω∗ ∈Hom(Z2,K5) according tosuch order (see Section 4.3). Denote V (K5) = {1,2, . . . ,5} and, w.l.o.g., assumethat 1 ≺ 2 ≺ ·· · ≺ 5. Then, because of the constraints imposed by K5, there mustexist x ∈ Z2 such that ω∗(x)≺ ω∗(x+(1,0)). Now, consider the point ω˜ such thatω˜(x) = ω∗(x+(1,0)) for every x ∈ Z2, i.e. a shifted version of ω∗ (in particu-lar, ω˜ also belongs to Hom(Z2,K5)). Then ω(x) ≺ ω∗(x+(1,0)) = ω˜(x), whichcontradicts the maximality of ω∗.Note 7. We are not aware of a homomorphism space Hom(G ,H) that satisfies theUMC property with H not a chordal/tree decomposable graph.a bc dFigure 4.5: A dismantlable graph H such that Hom(Z2,H) is not TSSM.Proposition 4.7.2. There exists a dismantlable graph H such that1. Hom(Z2,H) does not satisfy TSSM.2. There is no constrained energy function φ such that the Gibbs (Z2,H,φ)-specification satisfies SSM.3. For every board of bounded degree G and γ > 0, there exists a constrainedenergy function φ such that the Gibbs (G ,H,φ)-specification satisfies expo-nential WSM with decay rate γ .Proof. Consider the constraint graph H = (V,E) given by V = {a,b,c,d}, andE = {{a,a},{b,b},{c,c},{a,b},{a,c},{b,d},{c,d}} . (4.61)884.7. Examples: Homomorphism spacesIt is easy to check that H is dismantlable (see Figure 4.5).By Proposition 4.2.1, we know that (3) holds. However, if we consider theconfigurations α and β (the pairs bb and cc in red, respectively) and the fixedconfiguration σ (the diagonal alternating configurations adad · · · in black) shownin Figure 4.6, we have that [ασ ]Z2H , [σβ ]Z2H 6= /0, but [ασβ ]Z2H = /0, and TSSM cannothold. This construction works in a similar way to the construction in the proof((2) =⇒ (1)) of Proposition 4.6.2.abaaaddddbbbabaaaddddbbb bbbbbbbbacaaaddddcccabaaaddddccc cccccccc. . .. . .. . .. . .abaaaddddbbbacaaaddddccccccccbbbb. . .. . .?Figure 4.6: Two incompatible configurations α and β (both in red).Now assume the existence of a Gibbs (Z2,H,φ)-specification pi satisfying SSMwith decay function f . Call A⊆ Z2 the shape enclosed by the two diagonals madeby alternating sequence of a’s and d’s shown in Figure 4.7 (in grey), and let xland xr be the sites (in red) at the left and right extreme of A, respectively. If wedenote by σ the boundary configuration of the a and d symbols on ∂A\{xr}, andα1 = b{xr} and α2 = c{xr}, it can be checked that [σα1]GH , [σα2]GH 6= /0. Then, takeω1 ∈ [σα1]GH , ω2 ∈ [σα2]GH , and call B = {xl} and β = bB.aaaaddddaaaadddb/c. . .daa. . .Figure 4.7: A scenario where SSM fails for any Gibbs (Z2,H,φ)-specification.894.7. Examples: Homomorphism spacesNotice that, similarly as before, the symbols b and c force repetitions of them-selves, respectively, from xr to xl along A. Then, we have that piω1A (β ) = 1 andpiω2A (β ) = 0. Now, since we can always take an arbitrarily long set A, suppose thatdist(xl,xr)≥ n0, with n0 such that f (n0)< 1. Therefore,1 = |1−0|= ∣∣piω1A (β )−piω2A (β )∣∣ (4.62)≤ |B| f (dist(B,Σ∂A(ω1,ω2)))≤ f (n0)< 1, (4.63)which is a contradiction.Proposition 4.7.3. There exists a dismantlable graph H and a constant γ0 > 0,such that1. the set of constrained energy functions φ for which the Gibbs (Z2,H,φ)-specification satisfies exponential SSM is nonempty,2. there is no constrained energy function φ for which the Gibbs (Z2,H,φ)-specification satisfies exponential SSM with decay rate greater than γ0,3. Hom(Z2,H) satisfies SSF (in particular, Hom(Z2,H) satisfies TSSM), and4. for every γ > 0, there exists a constrained energy function φ for which theGibbs (Z2,H,φ)-specification satisfies exponential WSM with decay rate γ .Moreover, there exists a family {Hq}q∈N of dismantlable graphs with theseproperties such that |Hq| → ∞ as q→ ∞.Proof. By Theorem 3.6.1, we know that if pi is such that Q(pi) < pc(Z2), then pisatisfies exponential SSM, where pc(Z2) denotes the critical value of Bernoulli sitepercolation on Z2 and Q(pi) is defined asQ(pi) = maxω1,ω2dTV(piω1{~0},piω2{~0}). (4.64)Given q ∈N, consider the graph Hq as shown in Figure 4.8. The graph Hq con-sists of a complete graph Kq+1 and two other extra vertices a and b, both adjacentto every vertex in the complete graph. In addition, a has a loop, and a and b are notadjacent. Notice that Hq is dismantlable (we can fold a into b and then we can fold904.7. Examples: Homomorphism spacesevery vertex in Kq+1 into a), and a is a persistent vertex for Hq. Since Hq is dis-mantlable, by Proposition 4.2.1, for every γ > 0, there exists a constrained energyfunction φ for which the Gibbs (Z2,H,φ)-specification satisfies exponential WSMwith decay rate γ .Take pi to be the uniform Gibbs specification on Hom(Z2,H) (i.e. Φ≡ 0). Thedefinition of piω{~0}(u) implies that, whenever piω{~0}(u) 6= 0 for u ∈ V,1q+2≤ piω{~0}(u)≤1q−1 . (4.65)Notice that piω{~0}(a) = 0 iff b appears in ω|∂{~0}. Similarly, piω{~0}(b) = 0 iff aor b appear in ω|∂{~0}, and for u 6= a,b, piω{~0}(u) = 0 iff u appears in ω|∂{~0}. Since|∂{~0}|= 4, at most 8 terms vanish in the definition of Q(pi) (4 for each ωi, i= 1,2).Then, since |Hq|= q+3 and q≥ 1,dTV(piω1{~0},piω2{~0})=12 ∑u∈V∣∣∣piω1{~0}(u)−piω2{~0}(u)∣∣∣ (4.66)≤ 12(81q−1 +(q+3)∣∣∣∣ 1q−1 − 1q+2∣∣∣∣)≤ 6q−1 . (4.67)Kq+1ba10254 3Figure 4.8: The graph Hq, for q = 5.Then, if q > 1+ 6pc(Z2) , we have that Q(pi)< pc(Z2), so pi satisfies exponentialSSM. Since pc(Z2)> 0.556 (see [10, Theorem 1]), it suffices to take q≥ 12. In par-914.7. Examples: Homomorphism spacesticular, the set of constrained energy functions Φ for which the Gibbs (Z2,Hq,φ)-specification satisfies exponential SSM is nonempty if q > 12.Now, let pi be an arbitrary Gibbs (Z2,Hq,φ)-specification that satisfies SSMwith decay function f (n) =Ce−γn, for some C and γ that could depend on q and φ .For now, we fix q and an arbitrary φ .Consider a configuration like the one shown in Figure 4.9. Define V˜ = V \{0,a,b}, E˜ = E[V˜], and let H˜q = Hq[V˜]. Notice that H˜q is isomorphic to Kq.Construct the auxiliary constrained energy function φ˜ : V˜∪ E˜→ (−∞,0] given byφ˜(u) = φ(u)+φ(u,0)+φ(u,b), for every u ∈ V˜ (representing the interaction withthe “wall” · · ·0b0b · · · ), and φ˜ ≡ φ |E˜. The constrained energy function φ˜ inducesa Gibbs (Z, H˜q, φ˜)-specification p˜i that inherits the exponential SSM property frompi with the same decay function f (n) =Ce−γn. It follows that there is a unique (andtherefore, stationary) n.n. Gibbs measure µ for p˜i , which is a Markov measure withsome symmetric q× q transition matrix M with zero diagonal (see [35, Theorem10.21] and [21]).b00bj1b00b0bb00bb00b0bb00bb0k. . .. . .. . .. . .n nFigure 4.9: A Markov chain embedded in a Z2 Markov random field.Let 1 = λ1 ≥ λ2 ≥ ·· · ≥ λq be the eigenvalues of M. Since tr(M) = 0, wehave that ∑qi=1λi = 0. Let λ∗ = max{|λ2|, |λq|}. Then, since λ1 = 1, we have that1≤ ∑qi=2 |λi| ≤ (q−1)λ∗. Therefore, λ∗ ≥ 1q−1 .Since M is stochastic, M~1 =~1 and, since M is primitive, λ∗ < 1 (see [63,Section 3.2]). W.l.o.g., suppose that |λ2| = λ∗ and let ~` be the left eigenvectorassociated to λ2 (i.e. ~`M = λ2~`). Then ~` ·~1 = 0, because λ2~` ·~1 = (~` ·M) ·~1 =~` · (M ·~1) = ~` ·~1, so (1−λ2)~` ·1 = 0. Then, ~` ∈〈~e2−~e1,~e3−~e1, . . . ,~eq−~e1〉R, sowe can write ~`=∑qk=2 ck(~ek−~e1), where {~ek}qk=1 denotes the canonical basis ofRqand ck ∈R. We conclude that λ n∗ ~`= ~` ·Mn =∑qk=2 ck(~ek−~e1) ·Mn =∑qk=2 ck(Mnk•−Mn1•), where Mni• is the vector given by the ith row of M.924.7. Examples: Homomorphism spacesConsider j ∈ {1, . . . ,q} such that ~` j > 0. Then, λ n∗ = ∑qk=2 ck~` j (Mnk j−Mn1 j) and∣∣Mnk j−Mn1 j∣∣= ∣∣∣µ ( j{0}∣∣∣k{−n})−µ ( j{0}∣∣∣1{−n})∣∣∣ (4.68)≤∣∣∣µ ( j{0}∣∣∣k{−n})−µ ( j{0}∣∣∣k{−n},1{n})∣∣∣ (4.69)+∣∣∣µ ( j{0}∣∣∣k{−n},1{n})−µ ( j{0}∣∣∣1{−n},k{n})∣∣∣ (4.70)+∣∣∣µ ( j{0}∣∣∣1{−n},k{n})−µ ( j{0}∣∣∣1{−n})∣∣∣ (4.71)≤ 3Ce−γn, (4.72)by the exponential SSM property of p˜i and using that µ(j{0}∣∣k{−n}) is a weightedaverage ∑m∈V˜ µ(j{0}∣∣k{−n},m{n})µ (m{n}∣∣k{−n}), along with a similar decompo-sition of µ(j{0}∣∣1{−n}). Therefore, λ n∗ ≤ 3C(q− 1)maxk |ck||~` j|e−γn. By taking loga-rithms and letting n→ ∞, we conclude that γ ≤− logλ∗ ≤ log(q−1). Then, sinceφ was arbitrary, there is no constrained energy function φ for which the Gibbs(Z2,Hq,φ)-specification satisfies exponential SSM with decay rate greater thanγ0 := log(q−1).Finally, it is easy to see that if q≥ 4, Hom(Z2,H) satisfies SSF. Therefore, byProposition 3.3.3, Hom(Z2,Hq) satisfies TSSM (with gap g = 2).93Part IIRepresentation and poly-timeapproximation for pressure94Chapter 5Entropy and pressureIn the context of Zd shift spaces, we consider (topological) pressure and its particu-lar case, topological entropy. The two appear in several subjects and both somehowtry to capture the complexity of a given system by associating to it a nonnega-tive real number. Furthermore, and as an additional motivation, sometimes thesequantities can help us to distinguish between systems that are not isomorphic (orconjugate) in the sense described below.5.1 Topological entropyA natural way to transform one Zd shift space to another is via a particular class ofmaps compatible with the shift action called sliding block codes.Definition 5.1.1. A sliding block code between two Zd shift spacesΩ1⊆A Zd1 andΩ2 ⊆A Zd2 is a map J :Ω1→Ω2 for which there is N ∈ N and j :LBN (Ω1)→A2such thatJ(ω)(x) = j(σx(ω)|BN ), (5.1)for all x ∈ Zd and ω ∈ Ω1. A conjugacy is an invertible sliding block code, andtwo Zd shift spacesΩ1 andΩ2 are said to be conjugate (denotedΩ1 ∼=Ω2) if thereis a conjugacy from one to the other.Example 5.1.1. Given N ∈ N and a Zd shift space Ω ⊆ A Zd , a natural slidingblock code is the higher block code JN : Ω→(A BN)Zd defined byJN(ω)(x) = σx(ω)|BN . (5.2)We call the image JN(Ω), a higher block code representation of Ω. Noticethat the alphabet of JN(Ω) is a subset of A BN .955.1. Topological entropyTwo Zd shift spaces are often regarded as being the same if they are conju-gate. Properties preserved by conjugacies are called conjugacy invariants. Forexample, the property of being a Zd SFT is a conjugacy invariant: if a Zd shiftspace Ω is conjugate to an SFT, then Ω itself is a Zd SFT. The topologically mix-ing and strong irreducibility properties are invariants, too (see Subsection 3.2.1).Another important invariant, and one of the main objects of study in this work, isthe following.Definition 5.1.2. The topological entropy of a Zd shift space Ω is defined ash(Ω) := infnlog |LBn(Ω)||Bn| = limn→∞log |LBn(Ω)||Bn| . (5.3)Topological entropy is a conjugacy invariant, i.e. if Ω1 ∼= Ω2, then h(Ω1) =h(Ω2). The limit always exists because {|LBn(Ω)|}n is a (coordinate-wise) sub-additive sequence and a well-known multidimensional extension of Fekete’s sub-additive lemma applies (see [5]). Notice that topological entropy can be regardedas the exponential growth rate of globally admissible configurations on Bn.It is important to point out that for every Zd SFT there is a Zd n.n. SFT higherblock code representation, which makes Zd n.n. SFTs a sufficiently rich family tostudy. For this reason, from now on we restrict our attention to Zd n.n. SFTs.Example 5.1.2 (TSSM is not a conjugacy invariant). Given A = {0,1,2} and thefamily of configurations F= {00,102,201}, consider the Z SFT Ω=ΩF. It can bechecked that Ω is strongly irreducible with gap g = 3. However, Ω is not TSSM. Infact, given g ∈ N, consider S = {x ∈ Z : 0 < x < 2g,x odd} and the configurationsσ = 0S, α = 1{0}, and β = 2{2g}. Then, [ασ ]Ω, [σβ ]Ω 6= /0, because ασ can beextended with 1s in Z\(S∪{0}) and σβ can be extended with 2s in Z\(S∪{2g}).However, [ασβ ]Ω = /0, since the 1 in α forces any point in [ασ ]Ω to have value1 in ((0,2g)∩Z) \ S, but the 2 in β forces any point in [σβ ]Ω to have value 2 in((0,2g)∩Z)\S. Therefore, since g was arbitrary, Ω is not TSSM for any gap g.Now, if we define Ω′ := β1(Ω), where β1 is the higher block code with N = 1(see Example 5.1.1), thenΩ′ is aZ n.n. SFT conjugate toΩ (Ω∼=Ω′), and thereforestrongly irreducible (which is a conjugacy invariant). Then, by Proposition 3.4.5,we have that Ω′ is TSSM, while Ω is not.965.1. Topological entropyThis example can be extended to any dimension d by considering the con-straints F in only one canonical direction. In other words, TSSM is not a conjugacyinvariant for any d.When Ω is a Zd n.n. SFT, there is a simple algorithm for computing h(Ω)when d = 1, because h(Ω) = logλM, for λM the largest eigenvalue of the adjacencymatrix M of the edge shift representation of Ω [58]. However, for d ≥ 2, there isin general no known closed form for topological entropy. Only in a few specificcases a closed form is known (e.g. dimer model [46], square ice-type model [54]).Example 5.1.3. ForΩ1ϕ =Hom(Z,Hϕ), it is easy to see that h(Ω1ϕ) = logϕ , whereϕ = 1+√52 = 1.61803 . . . is the golden ratio. On the other hand, for d≥ 2, no closedform is known for the value of h(Ωdϕ).For d ≥ 2, one can hope to approximate the value of the topological entropyof a Zd n.n. SFT, whether by using its definition and truncating the limit or byalternative methods. A relevant fact to remember (see 3.4.2) is that, for d ≥ 2, it isalgorithmically undecidable to know if a given configuration is inL (Ω) or not. Inthis sense, it is useful to define an alternative, still meaningful, set of configurations.Given a set of n.n. constraints F and A b Zd , we will denote L l.a.A (F) the set oflocally admissible configurations in A. Considering this, we have the followingresult.Theorem 5.1.1 ([29, 44]). Given a finite set of n.n. constraints F,h(ΩF) = infnlog∣∣L l.a.Bn (F)∣∣|Bn| = limn→∞log∣∣L l.a.Bn (F)∣∣|Bn| , (5.4)i.e. the topological entropy h(Ω) of the Zd n.n. SFT Ω = ΩF can be computed bycounting locally admissible configuration rather than globally admissible ones.Since counting locally admissible configurations is tractable, it can be said thatTheorem 5.1.1 already provides an approximation algorithm for the topologicalentropy of a Zd n.n. SFT. Formally, a real number h is right recursively enu-merable if there is a Turing machine which, given an input n ∈ N, computes arational number r(n) ≥ h such that r(n)↘ h as n→ ∞. Given Theorem 5.1.1 and975.1. Topological entropythe fact that such limit is also an infimum, we can see that h(Ω) is right recursivelyenumerable for any Zd n.n. SFT Ω. In fact, the converse is also true due to thefollowing celebrated result from M. Hochman and T. Meyerovitch.Theorem 5.1.2 ([44]). The class of nonnegative real right recursively enumerablenumbers is exactly the class of topological entropies of Zd n.n. SFTs.A real number h is computable if there is a Turing machine which, givenan input n ∈ N, computes a rational number r(n) such that |h− r(n)| < 1n . Forexample, every algebraic number is computable, since there are numerical methodsfor approximating the roots of an integer polynomial. This is a strictly strongernotion than right recursively enumerable (for more information, see [50]). It canbe shown that, under extra assumptions (e.g. mixing properties) on a Zd n.n. SFTΩ, the topological entropy h(Ω) turns out to be computable.Theorem 5.1.3 ([44]). If a Zd n.n. SFT Ω is strongly irreducible, then h(Ω) iscomputable.Moreover, the difference |h− r(n)| can be thought as a function of n, intro-ducing a refinement of the classification of entropies by considering the speed ofapproximation. A relevant case for us is when the time to compute an approxima-tion r(n) such that |h− r(n)|< 1n is bounded by a polynomial in 1n .Example 5.1.4 ([67, 32]). The topological entropy h(Ω2ϕ) of the hard square shiftΩ2ϕ is a computable number that can be approximated in polynomial time.In Example 5.1.4, which is basically a combinatorial result, the proofs from[67] and [32] are almost entirely based on probabilistic and measure-theoretic tech-niques. This motivates the following definition, which is a notion of entropy forshift-invariant Borel probability measures.Definition 5.1.3. The measure-theoretic entropy of µ ∈M1,σ (A Zd ) is definedash(µ) := limn→∞−1|Bn| ∑α∈A Bnµ(α) log(µ(α)), (5.5)where 0log0 = 0.985.2. Topological pressureA fundamental relationship between topological and measure-theoretic entropyis the following.Theorem 5.1.4 (Variational Principle [64]). Given a Zd n.n. SFT Ω,h(Ω) = supµ∈M1,σ (Ω)h(µ) = maxµ∈M1,σ (Ω)h(µ). (5.6)Remark 11. The measure(s) that achieve the maximum are called measures ofmaximal entropy (m.m.e.) for Ω. Notice that if µ is an m.m.e. for Ω, thenh(Ω) = h(µ).5.2 Topological pressureNow we proceed to define topological pressure of a continuous function f ∈C (Ω),which can be regarded as a generalization of topological entropy.Definition 5.2.1. Given a Zd n.n. SFT Ω and f ∈ C (Ω), the topological pressureof f on Ω isPΩ( f ) := supµ∈M1,σ (Ω)(h(µ)+∫f dµ). (5.7)In this case, the supremum is also always achieved and any measure µ whichachieves the supremum is called an equilibrium state for Ω and f . Notice thatin the special case when f ≡ 0, and thanks to Theorem 5.1.4, P(0) coincides withthe topological entropy h(Ω), and equilibrium states are the same as measures ofmaximal entropy.Note 8. The preceding definition is a characterization of topological pressure interms of a variational principle, but can also be regarded as its definition (see [70,Theorem 6.12]). Informally, topological pressure can be thought as an exponentialgrowth rate, where the configurations are “weighted” by the given function f . Thisidea is formalized for a more particular case in the next paragraphs.We define the pressure of a shift-invariant n.n. interaction Φ on a Zd n.n. SFTΩ.995.2. Topological pressureDefinition 5.2.2. Given a Zd n.n. SFT Ω and a shift-invariant n.n. interaction Φon Ω, the pressure of Φ is defined asPΩ(Φ) := limn→∞1|Bn| logZΦBn . (5.8)The pressure coincides (up to a sign) with the specific Gibbs free energy of Φ,a more common name in statistical mechanics. We write P(Φ) (resp. P( f )) insteadof PΩ( f ) (resp. PΩ( f )) if Ω is understood. If Φ is induced by a Zd lattice energyfunction φ , we can also define an analogous version ZˆΦBn of the partition functionZΦBn over locally admissible configurations in Bn rather than globally admissibleones, by extendingH ΦBn in the obvious way, i.e.ZˆΦBn := ∑α∈L l.a.Bn (F)exp(−H ΦBn (α)) . (5.9)Notice that ZˆΦBn ≥ ZΦBn . The following result (analogous to Theorem 5.1.1)states that in the normalized limit, both quantities coincide.Theorem 5.2.1 ([70, Theorem 3.4], see also [29, Theorem 2.5]). Given a set ofn.n. constraints F, a Zd lattice energy function φ , and the corresponding Zd n.n.SFT ΩF and n.n. interaction Φ on ΩF,PΩF(Φ) = limn→∞1|Bn| logZˆΦBn . (5.10)One can ask whether is necessary to take increasing sequences of n-blocks Bnin the previous definitions instead of any other sequence of sets increasing to Zd .Given a sequence of sets {An}n with An b Zd , we say that {An}n tends to infinityin the sense of van Hove, denoted An ↗ ∞, if |An| → ∞ as n→ ∞ and, for eachx ∈ Zd ,limn→∞|An4(x+An)||An| = 0, (5.11)where 4 denotes the symmetric difference. It is well-known (see [70, Corollary3.13]) that for any sequence such that An ↗ ∞, we can replace Bn by An in the1005.2. Topological pressuredefinition of pressure, i.e.P(Φ) = limn→∞1|An| logZΦAn . (5.12)In order to discuss connections between the definition of P(Φ) and topologicalpressure for functions f ∈ C (Ω), we need a mechanism for turning an interaction(which is a function on finite configurations) into a continuous function on theinfinite configurations in Ω. We do this for the special case of n.n. interactions Φas follows. Define the (continuous) function AΦ : Ω→ R, given byAΦ(ω) :=−Φ(ω|{~0})−d∑i=1Φ(ω|{~0,~ei}). (5.13)A version of the Variational Principle states that the pressure of an interactionhas a variational characterization in terms of shift-invariant measures. We state thevariational principle below for the case of a n.n. interaction Φ.Theorem 5.2.2 (Variational Principle [47, 64, 70]). Given a n.n. interaction Φ ona Zd n.n. SFT Ω, we have that P(Φ) = P(AΦ), i.e.P(Φ) = supµ∈M1,σ (Ω)(h(µ)+∫AΦdµ). (5.14)In particular, if µ is an equilibrium state for AΦ, thenP(Φ) = h(µ)+∫AΦdµ. (5.15)The following property will be useful.Definition 5.2.3. AZd n.n. SFTΩ satisfies the D-condition if there exist sequencesof finite subsets {An}n, {Sn}n of Zd such that An↗ ∞, An ⊆ Sn, |An||Sn| → 1, and forany α ∈LAn(Ω), and β ∈LB(Ω), with Bb Scn, we have that [α]Ω∩ [β ]Ω 6= /0.Given a Zd n.n. SFT Ω, a shift-invariant n.n. interaction Φ on Ω, and thecorresponding Gibbs (Ω,Φ)-specification pi , we can relate equilibrium states forAΦ and shift-invariant n.n. Gibbs measures for pi . In fact, if Ω satisfies the D-1015.3. Pressure representationcondition,µ is an equilibrium state for AΦ and Ω ⇐⇒ µ ∈ G(pi)∩M1,σ (Ω). (5.16)The “only if” direction is always true in the n.n. case (see [70, Theorem 3]).Another relevant consequence of the D-condition is the following.Proposition 5.2.3 ([70, Remark 1.14]). LetΩ be a Zd n.n. SFT,Φ a shift-invariantn.n. interaction on Ω, and pi the corresponding Gibbs (Ω,Φ)-specification. If Ωsatisfies the D-condition, then supp(µ) =Ω, for every µ ∈ G(pi).Note 9. Notice that strong irreducibility implies the D-condition. In [70, Remark1.14] a property even weaker than the D-condition is shown to imply supp(µ) =Ω.5.3 Pressure representationWe fix the following elements:• A Zd n.n. SFT Ω that satisfies the D-condition,• a shift-invariant n.n. interaction Φ on Ω,• the corresponding n.n. Gibbs (Ω,Φ)-specification pi , and• µ ∈ G(pi)∩M1,σ (Ω).Given a set Sb Zd\{~0}, define pµ,S : Ω→ [0,1] to bepµ,S(ω) := µ(ω|{~0}∣∣∣ω|S) . (5.17)Notice that pµ,S(ω) is a value that depends only on ω|S∪{~0}.When dealing with pressure representation, it is useful to consider an orderin the lattice. Recall the definitions of lexicographic order ≺ and past P fromSubsection 2.1.2. Given n ∈ N, define the setPn :=P ∩Bn, and letpµ(ω) := limn→∞ pµ,Pn(ω), (5.18)1025.3. Pressure representationwhich exists µ-a.s. by Le´vy’s zero-one law (see [47, Theorem 3.1.10]). In otherwords, pµ is the µ-probability of taking the value ω(~0) at the origin conditionedon boxes growing to the lexicographic pastP .The information function Iµ is µ-a.s. defined asIµ(ω) :=− log pµ(ω). (5.19)It is well-known (see [35, p. 318, Equation (15.18)] or [51, Theorem 2.4, p.283]) that the measure-theoretic entropy of µ (in fact, this is true for any shift-invariant measure, not necessarily an equilibrium state) can be expressed ash(µ) =∫Iµdµ. (5.20)Therefore, since µ is an equilibrium state for AΦ, Equation (5.20) implies thatP(Φ) =∫ (Iµ +AΦ)dµ, (5.21)so the pressure can be represented as the integral with respect to µ of a functiondetermined by an equilibrium state µ and Φ.5.3.1 A generalization of previous resultsFor certain classes of equilibrium states and Gibbs measures, sometimes there areeven simpler representations for the pressure. A recent example of this was givenby D. Gamarnik and D. Katz in [32, Theorem 1], who showed that if in addition pisatisfies the SSM property and Ω has a safe symbol s, thenP(Φ) = Iµ(sZd)+AΦ(sZd). (5.22)Here, sZd ∈A Zd is the fixed point which is s at every site of Zd . Notice thatIµ(sZd)+AΦ(sZd)=∫ (Iµ +AΦ)dδsZd , (5.23)where δsZd is the δ -measure supported on sZd . They used this simple representa-tion to give a polynomial time approximation algorithm for P(Φ) in certain cases1035.3. Pressure representation(the hard-core model, in particular; see Example 5.1.4). Later, B. Marcus and R.Pavlov [58] weakened the hypothesis and extended their results for pressure repre-sentation, obtaining the following corollary.Corollary 8 ([58]). Let Ω be a Zd n.n. SFT, Φ a shift-invariant n.n. interaction onΩ, and pi the corresponding n.n. Gibbs (Ω,Φ)-specification such that• Ω satisfies SSF, and• pi satisfies SSM.Then,P(Φ) =∫ (Iµ +AΦ)dν , (5.24)for every ν ∈M1,σ (Ω).Corollary 8 relied on a more technical theorem from [58, Theorem 3.1]. Toreview this theorem, we first need a couple of definitions.We denotelimS→Ppµ,S(ω) (5.25)if there exists L ∈ R such that for all ε > 0, there is n ∈ N (that may depend on ω)such thatPn ⊆ SbP =⇒ |pµ,S(ω)−L|< ε. (5.26)If such L exists, L = pµ(ω) by definition. In addition, given the equilibriumstate µ , we definecµ := infω∈ΩinfSbZd\{~0}pµ,S(ω), (5.27)and, for ν ∈M1,σ (Ω),c−µ (ν) := infω∈supp(ν)infSbPpµ,S(ω). (5.28)Recall from Equation 3.38 the definition of cpi . It can be checked that c−µ (ν)≥cµ = cpi . Considering all this, we have the following theorem.Theorem 5.3.1 ([58]). Let Ω be a Zd n.n. SFT and Φ a shift-invariant n.n. in-teraction on Ω. Consider µ an equilibrium state for AΦ, and ν ∈M1,σ (Ω) suchthat1045.3. Pressure representationA1. Ω satisfies the D-condition,A2. limS→P pµ,S(ω) = pµ(ω) uniformly over ω ∈ supp(ν), andA3. c−µ (ν)> 0.Then, P(Φ) =∫ (Iµ +AΦ)dν .Considering Theorem 5.3.1 and the TSSM property, we have the followinggeneralization of Corollary 8.Corollary 9. Let Ω be a Zd n.n. SFT, Φ a shift-invariant n.n. interaction on Ω,and pi the corresponding n.n. Gibbs (Ω,Φ)-specification such that• Ω satisfies TSSM, and• pi satisfies SSM.Then,P(Φ) =∫ (Iµ +AΦ)dν , (5.29)for every ν ∈M1,σ (Ω).Proof. This follows from Theorem 5.3.1: (A1) is implied by TSSM, since TSSMimplies the D-condition; (A2) is implied by SSM (see [58, Proposition 2.14]); and(A3) is implied by TSSM (see Proposition 3.6.3), considering that c−µ (ν)≥ cpi .Corollary 10. Let Ω ⊆ A Z2 be a Z2 n.n. SFT, Φ a shift-invariant n.n. inter-action on Ω, and pi the corresponding n.n. Gibbs (Ω,Φ)-specification. Supposethat pi satisfies exponential SSM with decay rate γ > 4log |A |. Then, P(Φ) =∫ (Iµ +AΦ)dν , for every ν ∈M1,σ (Ω).Proof. This follows from Theorem 3.6.5 and Corollary 9.Notice that, in contrast to preceding results, no mixing condition on the supportis explicitly needed in Corollary 10.1055.3. Pressure representation5.3.2 The function pˆi and a new pressure representation theoremWith the exception of Theorem 5.3.1, all the previous pressure representation re-sults involved the SSM property, and therefore they are bound to fail when thereis a phase transition (i.e. multiple equilibrium states). Here we show that in suchcase, and assuming some extra conditions, the pressure can still be represented asthe integral of a function similar to Iµ +AΦ, with respect to any shift-invariantmeasure ν . This will turn out to be useful for approximation of pressure when ν isan atomic measure supported on a periodic configuration (see Chapter 7).Given a n.n. Gibbs (Ω,Φ)-specification pi , for Ω a Zd n.n. SFT and a shift-invariant n.n. interaction Φ on Ω, we introduce some useful functions from Ω toR. First, given~0 ∈ Ab Zd and ω ∈Ω, we define piA : Ω→ [0,1] to bepiA(ω) := piωA (ω|{~0}). (5.30)Recall that, for y,z ∈ Zd such that y,z ≥~0, we have defined the set Qy,z as{x <~0 : −y ≤ x ≤ z}. Now, given y,z ≥~0 and ω ∈ Ω, define piy,z(ω) := piQy,z(ω)and, given n ∈ N, abbreviate pin(ω) := pi~1n,~1n(ω). Considering this, we also definethe limit pˆi(ω) := limn→∞pin(ω), whenever it exists. If such limit exists, we willalso denote Ipi(ω) := − log pˆi(ω), in a similar fashion as the information functionIµ .It is not difficult to prove that under some mixing assumptions, namely the D-condition and the SSM property, one has that the original information function Iµfor an equilibrium state µ coincides with Ipi in Ω. This new definition provides ageneralization of previous results, since Iµ may not be defined in the same pointsas Ipi .We say thatlimy,z→∞piy,z(ω) = pˆi(ω), (5.31)if for all ε > 0, there is k ∈ N such thaty,z≥~1k =⇒ |piy,z(ω)− pˆi(ω)|< ε. (5.32)1065.3. Pressure representationIn addition, we introduce the boundcpi(ν) := infω∈supp(ν)inf~0∈AbZdpiA(ω). (5.33)Notice that, by shift-invariance, cpi(ν) = infω∈supp(ν) cpi(ω). We leave it to thereader to verify that cpi(ν)≥ cµ(ν), for any µ ∈ G(pi)∩M1,σ (Ω). In particular, byProposition 3.6.3, if Ω satisfies TSSM, cpi(ν)> 0, for any ν ∈M1,σ (Ω).Definition 5.3.1. A Zd n.n. SFT Ω satisfies the square block D-condition if thereexists a sequence of integers {rn}n≥1 such that rnn → 0 as n→ ∞ and, for any finiteset Bb Bcn+rn , α ∈LBn(Ω) and β ∈LB(Ω), we have that [α]Ω∩ [β ]Ω 6= /0.This condition is a strengthened version of the D-condition. Notice that, asfor the standard D-condition, strong irreducibility also implies the square block theD-condition.The pressure representation results in [58, Theorems 3.1 and 3.6] are not ade-quate for the application to the specific models we consider here (see Section 6).Instead we will use the following result, whose proof is adapted from the proofof [58, Theorem 3.1], as well as an idea of [58, Theorem 3.6]. In contrast to theresults of [58], this result makes assumptions on the Gibbs specification rather thanan equilibrium state.Theorem 5.3.2. LetΩ⊆A Zd be aZd n.n. SFT,Φ a shift-invariant n.n. interactionon Ω, and pi the corresponding n.n. Gibbs (Ω,Φ)-specification. Suppose that, forν ∈M1,σ (Ω),B1. Ω satisfies the square block D-condition,B2. limy,z→∞piy,z(ω) = pˆi(ω) uniformly over ω ∈ supp(ν), andB3. cpi(ν)> 0.Then,P(Φ) =∫(Ipi +AΦ)dν . (5.34)1075.3. Pressure representationProof. Given n ∈ N, let rn be as in the definition of the square block D-conditionand consider the sets Bn and Λn := Bn+rn . We begin by proving that1|Bn|(logZΦBn + logpiωΛn(ω|Bn)+H ΦBn (ω|Bn))→ 0, (5.35)uniformly in ω ∈ Ω. For this, we will only use the square block D-condition. Wefix n ∈ N, ω ∈ supp(ν), and let mn := |Λn|− |Bn|. Let Cd ≥ 1 be a constant suchthat for any Ab Zd , the total number of sites and bonds contained in A is boundedfrom above by Cd |A|. Then,piωΛn(ω|Bn)≥ piωΛn(ω|Λn) (5.36)=exp(−H ΦΛn,ω(ω|Λn))∑α∈A Λn :α ω|Λcn∈Ω exp(−HΦΛn,ω(α))(5.37)≥ exp(−HΦBn (ω|Bn)−CdmnΦmax)∑β∈LBn (Ω) exp(−H ΦBn (β ))|A |Cdmn exp(CdmnΦmax)(5.38)=exp(−H ΦBn (ω|Bn))ZΦBnexp(−Cdmn(2Φmax+ log |A |)). (5.39)Now, if τmax achieves the maximum of piωΛn(ω|Bn τ) over τ ∈A Λn\Bn , thenpiωΛn(ω|Bn) = ∑τ∈A Λn\BnpiωΛn(ω|Bn τ) (5.40)≤ |A |mnpiωΛn(ω|Bn τmax) (5.41)= |A |mn exp(−HΦΛn,ω(ω|Bn τmax))ZΦΛn,ω(5.42)≤ |A |mn exp(−HΦBn (ω|Bn)+CdmnΦmax)∑β∈LBn (Ω) exp(−H ΦBn (β ))exp(−CdmnΦmax)(5.43)≤ exp(−HΦBn (ω|Bn))ZΦBnexp(Cdmn(2Φmax+ log |A |)), (5.44)where the square block D-condition has been used in Equation (5.43). Therefore,θ−mn ≤ piωΛn(ω|Bn)ZΦBn exp(H ΦBn (ω|Bn))≤ θmn , (5.45)1085.3. Pressure representationwhere θ := exp(Cd(2Φmax+ log |A |)). Since mn|Bn| → 0, we have obtained Equation(5.35). We use Equation (5.35) to represent pressure:P(Φ) = limn→∞logZΦBn|Bn| = limn→∞∫ logZΦBn|Bn| dν (5.46)= limn→∞∫ − logpiωΛn(ω|Bn)−H ΦBn (ω|Bn)|Bn| dν . (5.47)(Here the second equality comes from the fact thatlogZΦBn|Bn| is independent of ω ,and the third from Equation (5.35).) Since ν is shift-invariant, it can be checkedthatlimn→∞∫ −H ΦBn (ω|Bn)|Bn| dν =∫AΦdν , (5.48)and so we can writeP(Φ) =∫AΦdν− limn→∞∫ logpiωΛn(ω|Bn)|Bn| dν . (5.49)It remains to show thatlimn→∞∫ − logpiωΛn(ω|Bn)|Bn| dν =∫Ipidν . (5.50)Fix ω ∈ supp(ν) and denote c := cpi(ν). We will decompose piωΛn(ω|Bn) as aproduct of conditional probabilities. By (B2), for any ε > 0, there exists k = k(ε)so that for y,z ≥~1k, |piy,z(ω)− pˆi(ω)| < ε for all ω ∈ supp(ν). For x ∈ Bn−1, wedenote B−n (x) := {y ∈ Bn−1 : y≺ x}. Then, we can decompose piωΛn(ω|Bn) aspiωΛn(ω|Bn) = piωΛn(ω|∂Bn)∏x∈Bn−1piωΛn(ω|{x}∣∣∣ω|B−n (x)∪∂Bn) (5.51)= piωΛn(ω|∂Bn)∏x∈Bn−1piy(x),z(x)(σx(ω)), (5.52)where y(x) :=~1n+ x and z(x) :=~1n− x, thanks to the MRF property and station-arity of the Gibbs specification.1095.3. Pressure representationLet’s denote Rn,k :=Bn\Bn−k. Then, Bn = ∂BnunionsqBn−k−1unionsqRn−1,k, and we havec|∂Bn|+|Rn−1,k| ∏x∈Bn−k−1piy(x),z(x)(σx(ω))≤ piωΛn(ω|Bn) (5.53)≤ ∏x∈Bn−k−1piy(x),z(x)(σx(ω)). (5.54)Taking − log(·), we have that0≤− logpiωΛn(ω|Bn)− ∑x∈Bn−k−1− logpiy(x),z(x)(σx(ω)) (5.55)≤ (|∂Bn|+ |Rn−1,k|) log(c−1). (5.56)So, by the choice of k, for x ∈ Bn−k−1,∣∣piy(x),z(x)(σx(ω))− pˆi(σx(ω))∣∣< ε, (5.57)and since piy(x),z(x)(σx(ω)), pˆi(σx(ω))≥ c > 0, by the Mean Value Theorem,∣∣− logpiy(x),z(x)(σx(ω))− Ipi(σx(ω))∣∣< εc−1. (5.58)It follows from (B2) that pˆi is the uniform limit of continuous functions onsupp(ν). In addition, pˆi(ω) ≥ c > 0, for all ω ∈ supp(ν). Therefore, we canintegrate with respect to ν and obtain∣∣∣∣∫ − logpiy(x),z(x)(σx(ω))dν−∫ Ipi(ω)dν∣∣∣∣< εc−1. (5.59)We now combine the previous equations to see that∣∣∣∣∫ − logpiωΛn(ω|Bn+1)dν−∫ Ipi(ω)dν |Bn−k−1|∣∣∣∣ (5.60)≤ |Bn−k−1|εc−1+(|∂Bn|+ |Rn−1,k|) log(c−1). (5.61)Notice that, for a fixed k, limn→∞|∂Bn|+|Rn−1,k||Bn| = 0 and limn→∞|Bn−k−1||Bn| = 1.1105.3. Pressure representationTherefore,−εc−1+∫Ipi(ω)dν ≤ liminfn→∞∫ − logpiωΛn(ω|Bn)|Bn| dν (5.62)≤ limsupn→∞∫ − logpiωΛn(ω|Bn)|Bn| dν (5.63)≤∫Ipi(ω)dν+ εc−1. (5.64)By letting ε → 0, we see thatlimn→∞∫ − logpiωΛn(ω|Bn)|Bn| dν =∫Ipi(ω)dν , (5.65)completing the proof.111Chapter 6Classical lattice models andrelated properties6.1 Three Zd lattice modelsIn this chapter we introduce three families of classical lattice models (see alsoExample 2.3.1). The first one will be the Potts model, which can be regarded as ageneralization of the Ising model by considering more than two types of particles.The second one, the Widom-Rowlinson model, is also a multi-type particle systembut with hard-core exclusion between particles of different type. The third one isthe classical hard-core lattice gas model.Recall the definition of Gibbs (G ,H,φ)-specifications in the context of homo-morphism spaces from Chapter 4, where G denotes a board, H a constraint graph,and φ a constrained energy function. In the context of Zd lattice models, besidesA, B, etc., we will sometimes use the uppercase Greek letters Λ, ∆, Θ to denotesubsets of the lattice.6.1.1 The (ferromagnetic) Potts modelGiven d,q ∈ N and β > 0, the Zd (ferromagnetic) Potts model with q types andinverse temperature β is given by the Gibbs (Zd ,Kq ,βφFP)-specification piFPβ ,whereφFP∣∣V ≡ 0, and φFP({u,v}) =−1{u=v}. (6.1)We will denote the alphabet V(Kq ) of the Potts model with q types byAFP,q ={1, . . . ,q}, ΩFP = Hom(Zd ,Kq ) its support (assuming d and q are understood),1126.1. Three Zd lattice modelsand Φβ the induced n.n. interaction.The Zd (ferromagnetic) Potts model is given by the Gibbs specification piFPβ ={piωβ ,Λ : ω ∈ΩFP,Λb Zd}, where neighbouring sites preferably align to each otherwith the same type or “colour”.Notice that when q = 2, we recover the classical ferromagnetic Ising model.Theorem 6.1.1 ([9]). For the Z2 (ferromagnetic) Potts model with q types and in-verse temperature β , there exists a critical inverse temperature βc(q) := log(1+√q) such that uniqueness of Gibbs measures holds for β < βc(q) and for β > βc(q)there is a phase transition.6.1.2 The (multi-type) Widom-Rowlinson modelGiven d,q ∈ N and ζ > 0, the Zd (multi-type) Widom-Rowlinson model with qtypes and activity ζ is given by the the Gibbs (Zd ,Sq ,− log(ζ )φWR)-specifica-tion piWRζ , whereφWR(v) =−1{v6=0}, and φWR∣∣E ≡ 0. (6.2)We will denote the alphabet V(Sq ) of the Widom-Rowlinson model with qtypes by AWR,q = {0,1, . . . ,q}, ΩWR = Hom(Zd ,Sq ) its support (assuming d andq are understood), and Φζ the induced n.n. interaction.The Zd (multi-type) Widom-Rowlinson model is given by the Gibbs specifi-cation piWRζ ={piωζ ,Λ : ω ∈ΩWR,Λb Zd}, where neighbouring sites are forced toalign to each other with the same type or “colour” or with the central symbol 0.Theorem 6.1.2 ([71], see also [38]). For the Z2 (multi-type) Widom-Rowlinsonmodel with q types and activity ζ , uniqueness of Gibbs measures holds for suffi-ciently small ζ and there is a phase transition for sufficiently large ζ .6.1.3 The hard-core lattice gas modelGiven d ∈ N and λ > 0, the Zd hard-core lattice gas model with activity λ isgiven by the Gibbs (Zd ,Hϕ ,φHCλ )-specification piHCλ , whereφHC(0) = 0, φHC(1) =−1, and φHC∣∣E ≡ 0. (6.3)1136.1. Three Zd lattice modelsWe will denote the alphabet V(Hϕ) of the hard-core model by AHC = {0,1},ΩHC = Hom(Zd ,Hϕ) its support (assuming d is understood), and Φλ the inducedn.n. interaction.The Zd hard-core lattice gas model is given by the Gibbs specification piHCλ ={piωλ ,Λ : ω ∈ΩHC,Λb Zd}, where neighbouring sites cannot be both 1.Theorem 6.1.3 ([36, Theorem 3.3]). For the Z2 hard-core lattice gas model withactivity λ , uniqueness of Gibbs measures holds for sufficiently small λ and there isa phase transition for sufficiently large λ .In fact, a major open problem is the uniqueness of the phase transition pointfor the Z2 hard-core lattice model.For both the Potts and Widom-Rowlinson models we will also distinguish aparticular type of particle or colour in the alphabet. W.l.o.g., we can take the typeq in AFP,q or AWR,q \ {0}, respectively. Given this colour, we will denote by ωqthe fixed point qZd. For the hard-core lattice gas model, we will consider the twospecial points ω(e) and ω(o), given byω(e)(x) :=0 if ∑i xi is even,1 if ∑i xi is odd, (6.4)and ω(o) = σ~e1(ω(e)).Notice that the Potts, Widom-Rowlinson, and hard-core lattice gas models havea safe symbol (any a∈AFP,q, 0∈AWR,q, and 0∈AHC, respectively). In particular,we have that the supports of the three models satisfy the square block D-condition,and cpi > 0 for pi the corresponding n.n. Gibbs specification.The Potts and Widom-Rowlinson models have interpretations in terms of arandom-cluster representation. The Potts model is related to a random-clustermodel on bonds (via the so-called Edwards-Sokal coupling), while the Widom-Rowlinson is naturally related to a random-cluster model on sites.From now on, when talking about Gibbs specifications for the Potts, Widom-Rowlinson, and hard-core lattice gas models, we will distinguish them (in a slightabuse of notation) by the subindex corresponding to the parameter β , ζ , or λ of themodel, i.e. piωβ ,Λ should be understood as a probability measure in the Potts model,1146.2. The bond random-cluster modelpiωζ ,Λ in the Widom-Rowlinson, and piωλ ,Λ in the hard-core lattice gas model, and piβ ,piζ and piλ will denote the corresponding Gibbs specifications. Also, we will writepiβΛ , pˆiβ , and Iβpi for the functions piΛ, pˆi , and Ipi in the Potts model, and short-handnotations when Λ= Qn or Qy,z such aspiβn (ω) = piωβ ,Qn(ω|{~0}). (6.5)The analogous notation will be used for the Widom-Rowlinson and hard-corelattice gas cases, but using the parameters ζ and λ , respectively.6.2 The bond random-cluster modelWe will make use of the bond random-cluster model. One the main results, Part 1of Theorem 6.5.1, is proven using arguments based on this model. This model isa two-parameter family of dependent bond percolation models (on a finite graph).We are mainly interested in finite subgraphs of Z2.Fix a set of sites Λ. Let E 0(Λ) denote the set of bonds with both endpointsin Λ (i.e. if e = {x,y} is a bond, then both x and y belong to Λ), and E 1(Λ)the set of bonds with at least one endpoint in Λ. We will consider configurationsw ∈ {0,1}E i(Λ), for i = 0,1. We speak of a bond e as being open if w(e) = 1, andas being closed if w(e) = 0.We describe the model with boundary conditions indexed by i = 0,1. We willgive special attention to the case d = 2. A set A⊆ Z2 is simply lattice-connectedif A and Ac are both connected.Definition 6.2.1. Given a finite simply lattice-connected set Λ, and parameters p∈[0,1] and q> 0, we define the free (i= 0) and wired (i= 1) bond random-clusterdistributions on E i(Λ) (i= 0,1) as the measures ϕ(i)p,q,Λ that to each w ∈ {0,1}Ei(Λ)assigns probability proportional toϕ(i)p,q,Λ(w) ∝{∏e∈E i(Λ)pw(e)(1− p)1−w(e)}qkiΛ(w) =(p1− p)#1(w)qkiΛ(w), (6.6)where #1(w) is the number of open bonds in w, and k0Λ(w) and k1Λ(w) are the1156.2. The bond random-cluster modelnumber of connected components (including isolated sites) in the graphs (Λ,{e ∈E 0(Λ) : w(e) = 1}) and (Z2,E 0(Z2 \Λ)∪{e ∈ E 1(Λ) : w(e) = 1}), respectively.Notice that when q = 1, we recover the ordinary Bernoulli bond percolationmeasure ϕp,Λ, while other choices of q lead to dependence between bonds. Forgiven p and q, one can also define bond random-cluster measures ϕ(i)p,q on Z2 as alimit of finite-volume measures ϕ(i)p,q,Λ (i = 0,1).Theorem 6.2.1 ([36, Lemma 6.8]). For p ∈ [0,1] and q ∈N, the limiting measuresϕ(i)p,q = limn→∞ϕ(i)p,q,Λn , i ∈ {0,1}, (6.7)exist and are shift-invariant, where {Λn}n is any increasing sequence of finite sim-ply lattice-connected sets that exhausts Z2.General bond random-cluster measures on {0,1}Z2 can be defined using ananalogue of the DLR equation (see [42, Definition 4.29]). For q ≥ 1, there is avalue pc(q) that delimits exactly the transition for existence of an infinite opencluster for these measures. It is known (see [42, p. 107] and [24]) that for q ≥ 1and p 6= pc(q), there is a unique such measure which we denote by ϕp,q, and thatcoincides with ϕ(0)p,q and ϕ(1)p,q. It was recently proven (see [9]) that pc(q) =√q1+√q ,for every q≥ 1.Let p = 1− e−β . The free Edwards-Sokal coupling P(0)p,q,Λ (see [42]) is acoupling between the free-boundary Potts measure pi( f )β ,Λ and ϕ(0)p,q,Λ. The wiredEdwards-Sokal coupling P(1)p,q,Λ is a coupling between piωqβ ,Λ and ϕ(1)p,q,Λ. Noticethat pc(q) = 1− e−βc(q).These couplings are measures on pairs of site configurations and correspondingbond configurations. The projection to site configurations is the free-boundary/ωq-boundary Potts measure, and the projection to bond configurations is the free/wiredbond random-cluster measure, respectively.Theorem 6.2.2 ([42, Theorem 1.13]). Let ΛbZ2 be a finite simply lattice-connec-ted set, q ∈ N, and let p ∈ [0,1] and β > 0 be such that p = 1− e−β . Then:1. Given w ∈ {0,1}E 1(Λ), the conditional measure P(1)p,q,Λ(·∣∣AFP,qΛ×{w}) onAFP,qΛ is obtained by putting random colours on entire clusters of w not1166.2. The bond random-cluster modelconnected with Z2 \Λ (of which there are k1Λ(w)− 1) and colour q on theclusters connected with Z2 \Λ. These colours are constant on given clus-ters, are independent between clusters, and the random ones are uniformlydistributed on the set AFP,q.2. Given α ∈AFP,qΛ, the conditional measure P(1)p,q,Λ(·∣∣∣{α}×{0,1}E 1(Λ)) on{0,1}E 1(Λ) is obtained as follows. Consider the extended configuration α =αq∂Λ and an arbitrary bond e = {x,y} ∈ E 1(Λ). If α(x) 6= α(y), we setw(e) = 0. If α(x) = α(y), we setw(e) =1 with probability p,0 otherwise, (6.8)the values of different w(e) being (conditionally) independent random vari-ables.The couplings can be used to relate probabilities and expectations for the Pottsmodel to corresponding events and expectations in the associated bond random-cluster model. A main example is a relation between the two-point correlationfunction in the Potts model and the connectivity function in the bond random-cluster model (see [42, Theorem 1.16]).By considering a displaced version of Z2, namely 12~1+Z2 (the dual lattice),we can define a notion of duality for bond configurations w. Notice that everybond e ∈ E (Z2) (if we think of bonds as unitary vertical and horizontal straightsegments) is intersected perpendicularly by one and only one dual bond e∗ ∈E (12~1+Z2), so there is a clear correspondence between E (Z2) and E (12~1+Z2).We are mainly interested in wired bond random-cluster distributions on the set ofsites B˜n := [−n+1,n]2∩Z2. Given n ∈N, if we consider the set of bonds E 1(B˜n),it is easy to check that there is a correspondence e 7→ e∗ between this set and the setof bonds from 12~1+Z2 with both endpoints in [−n,n]2∩(12~1+Z2), which can beidentified with the set E 0(Bn). Then, given a bond configuration w ∈ {0,1}E 1(B˜n)we can associate a dual bond configuration w∗ ∈ {0,1}E 0(Bn) such that w∗(e∗) = 0iff w(e) = 1. Considering this, we have the following equality.1176.3. The site random-cluster modelProposition 6.2.3 ([42, Equation (6.12) and Theorem 6.13]). Given n ∈ N, p ∈[0,1], and q ∈ N,ϕ(1)p,q,B˜n(w) = ϕ(0)p∗,q,Bn(w∗), (6.9)for any bond configuration w ∈ {0,1}E 1(B˜n), where B˜n = [−n+ 1,n]2 ∩Z2 andp∗ ∈ [0,1] is the dual value of p, which is given byp∗1− p∗ =q(1− p)p. (6.10)The previous duality result can be generalized to more arbitrary shapes andit has also a counterpart from free-to-wired boundary conditions, instead of fromwired-to-free.The unique fixed point of the map p 7→ p∗ defined by Equation (6.10) is√q1+√qand, as mentioned before, is known to coincide with the critical point pc(q) forthe existence of an infinite open cluster for the bond random-cluster model (see [9,Theorem 1]). It is easy to see that p > pc(q) iff p∗ < pc(q).6.3 The site random-cluster modelIn a similar fashion to the bond random-cluster model, we can perturb Bernoullisite percolation, where the probability measure is changed in favour of configura-tions with many (for q> 1) or few (for q< 1) connected components. The resultingmodel is called the site random-cluster model.Definition 6.3.1. GivenΛbZ2, and parameters p∈ [0,1] and q> 0, the wired siterandom-cluster distribution ψ(1)p,q,Λ is the probability measure on {0,1}Λ which toeach θ ∈ {0,1}Λ assigns probability proportional toψ(1)p,q,Λ(θ) ∝{∏x∈Λpθ(x)(1− p)1−θ(x)}qκΛ(θ) = ζ #1(θ)qκΛ(θ), (6.11)where ζ = p1−p , #1(θ) is the number of 1’s in θ , and κΛ(θ) is the number ofconnected components in {x ∈ Λ : θ(x) = 1} that do not intersect ∂Λ.1186.3. The site random-cluster modelThe free site random-cluster measure ψ(0)p,q,Λ is defined as in Equation (6.11)by replacing κΛ(θ)with the total number of connected components in Λ. However,we will not require that measure in this work. In any case, taking q= 1 gives the or-dinary Bernoulli site percolation ψp,Λ, while other choices of q lead to dependencebetween sites, similarly to the bond random-cluster model.Proposition 6.3.1. Given a set Λb Z2, and parameters ζ > 0 and q ∈N, considerthe (multi-type) Widom-Rowlinson model with q types distribution and monochro-matic boundary condition piωqζ ,Λ. Now, let f :AWR,qΛ→{0,1}Λ be defined site-wiseas( f (α))(x) =0 if α(x) = 0,1 if α(x) 6= 0, (6.12)for α ∈AWR,qΛ and x ∈ Λ, and let p = ζ1+ζ . Then,f∗piωqζ ,Λ = ψ(1)p,q,Λ, (6.13)where f∗piωqζ ,Λ(·) := piωqζ ,Λ( f−1(·)) denotes the push-forward measure on {0,1}Λ.The requirement that κΛ(·) does not count connected components that intersectthe inner boundary of Λ in the site random-cluster model, corresponds to the factthat non 0 sites adjacent to the monochromatic boundary ωq∣∣∂Λ in the Widom-Rowlinson model must have the same colour q.For q = 2, Proposition 6.3.1 is proven in [43, Lemma 5.1 (ii)], and the proofextends easily for general q. Proposition 6.3.1 can be regarded as a coupling be-tween piωqζ ,Λ and ψ(1)p,q,Λ, because a push-forward measure can be naturally coupledwith the original measure.It is important to notice that ψ(1)p,q,Λ is itself not an MRF: given sites on a “ring”C, the inside and outside of C are generally not conditionally independent, becauseknowledge of sites outside C could cause connected components of 1’s in C to“amalgamate” into a single component, which would affect the conditional dis-tribution of configurations inside C (the same for the bond random-cluster modelϕ(1)p,q,Λ). The following lemma shows that in certain situations, when conditioningon a cycle C labeled entirely by 1’s, this kind of amalgamation does not occur.1196.3. The site random-cluster modelLemma 6.3.2. Let /0 6= Θ ⊆ Λ b Z2 be such that Λc ∪Θ? is connected. Take∆ := ∂ ?Θ∩Λ. Consider an event A ∈FΘ and a configuration τ ∈ {0,1}Σ, whereΣ⊆ Λ\Θ?. Then,ψ(1)p,q,Λ(A|1∆τ) = ψ(1)p,q,Λ(A|1∆0Λ\Θ?). (6.14)Proof. W.l.o.g., we may assume that A is a cylinder event [θ ] with θ ∈ {0,1}Θ (bylinearity) and Σ= Λ\Θ? (by taking weighted averages).Now, Σ= Λ\Θ? can be written as a disjoint union of ?-connected componentsΣ = K1 unionsq ·· · unionsqKn. For every i, ∂ ?Ki ⊆ Λc ∪Θ? (in fact, ∂ ?Ki ⊆ Λc ∪∆). SinceΛc∪Θ? is connected and Λ is finite, for every site in ∂ ?Ki there is a path to infinitythat does not intersect Ki.Then, by application of a result of Kesten (see [48, Lemma 2.23]), ∂ ?Ki isconnected, for every i. In addition, we have that Λ=Θunionsq∆unionsqΣ and ∂ ?Ki ⊆ Λc∪∆.We claim thatκΛ(υ) = κΛ(υ |Θ 1∆0Σ)+n∑i=1κKi(υ |Ki) = κΛ(υ |Θ 1∆0Σ)+κΣ(τ), (6.15)for any υ ∈ {0,1}Λ such that υ |∆ = 1∆ and υ |Σ = τ .To see this, given such υ , we exhibit a bijection r between the connected com-ponents of υ that do not intersect ∂Λ and the union of: (a) the connected compo-nents of υ |Θ 1∆0Σ that do not intersect ∂Λ, and (b) the connected components ofυ |Ki that do not intersect ∂Ki, for all i. Namely, if C⊆Λ is a connected componentof υ , then r is defined as followsr(C) =C∩Θ? if C∩Θ? 6= /0,C if C ⊆ Σ.(6.16)In order to see that r is well-defined, note that if C intersects Θ? and Σ, theset C∩Θ? is still connected since ∂ ?Ki is connected and υ |∆ = 1∆. To see that ris onto, observe that if C′ is a connected component of υ |Θ 1∆0Σ, then there is aunique component C of υ such that C∩Θ? =C′, due again to the fact that ∂ ?Ki isconnected. And r is clearly injective because two distinct connected components1206.3. The site random-cluster modelcannot intersect.Finally, we conclude from Equation (6.15) thatψ(1)p,q,Λ(θ | 1∆τ) =ζ #1(θ1∆τ)qκΛ(θ1∆τ)∑υ∈{0,1}Λ:υ |∆=1∆,υ |Σ=τ ζ#1(υ)qκΛ(υ)(6.17)=ζ #1(θ1∆)+#1(τ)qκΛ(θ1∆0Σ)+κΣ(τ)∑υ∈{0,1}Λ:υ |Σ=τ ζ#1(υ(Θ)1∆)+#1(τ)qκΛ(υ(Θ)1∆0Σ)+κΣ(τ)(6.18)=ζ #1(θ1∆)qκΛ(θ1∆0Λ\Θ?)∑θ˜∈{0,1}Θ ζ #1(θ˜1∆)qκΛ(θ˜1∆0Λ\Θ?)= ψ(1)p,q,Λ(θ |1∆0Λ\Θ?), (6.19)as we wanted.Figure 6.1: A ?-connected set Θ (in black), ∆= ∂ ?Θ∩Λ (in dark grey), and Λc (inlight grey) for Λ= Qy,z.Remark 12. We claim that if /0 6= Θ ⊆ Λ b Z2 are such that Λc is connected, Θis ?-connected and Θ? ∩ ∂Λ 6= /0, then Λc ∪Θ? is connected, which is the mainhypothesis of Lemma 6.3.2. This follows from the easy fact that the ?-closure of a?-connected set is connected.1216.4. Additional properties6.4 Additional properties6.4.1 Spatial mixing propertiesWe now introduce some extra concepts related with spatial mixing.Definition 6.4.1. ([4, p. 445]) A Zd-MRF µ satisfies the ratio strong mixingproperty for a class of sets C if there exists C,γ > 0 such that for any Λ ∈ C , any∆,Θ⊆ Λ and δ ∈A ∂Λ with µ(δ )> 0,sup{∣∣∣∣ µ(A∩B|δ )µ(A|δ )µ(B|δ ) −1∣∣∣∣ : A ∈F∆,B ∈FΘ,µ(A|δ )µ(B|δ )> 0}≤C ∑x∈∆,y∈Θe−γdist(x,y). (6.20)Proposition 6.4.1. Let pi be a n.n. Gibbs specification with Ω = A Z2 , the fullshift. Consider µ ∈ G(pi) that satisfies the ratio strong mixing property for theclass of finite simply lattice-connected sets. Then, pi satisfies exponential SSM forthe family of sets {Qy,z}y,z≥0.Proof. Fix y,z ≥ 0 and the corresponding set A = Qy,z b Z2. Let B ⊆ A, β ∈A B,and ω1,ω2 ∈Ω. Consider1. the sets Θ := Σ∂A(ω1,ω2) and Λ := A∪Θ,2. an arbitrary point ω˜ ∈Ω such that ω˜|∂A\Θ = ω1|∂A\Θ (= ω2|∂A\Θ),3. the configuration δ˜ = ω˜|∂Λ, and4. the events A := [β ] ∈FB and Bi := [ωi|Θ] ∈FΘ, for i = 1,2.Notice that Λ is a finite simply lattice-connected set. Since supp(µ) = A Zd ,we can be sure that µ(ω1|∂A)µ(ω2|∂A)µ(δ˜ )> 0. Then,∣∣piω1A (β )−piω2A (β )∣∣= |µ (β |ω1|∂A)−µ (β |ω2|∂A)| (6.21)=∣∣∣µ (A∣∣∣[δ˜ ]∩B1)−µ (A∣∣∣[δ˜ ]∩B2)∣∣∣ (6.22)=∣∣∣∣∣µ(A∩B1|δ˜ )µ(B1|δ˜ ) −µ(A|δ˜ )+µ(A|δ˜ )− µ(A∩B2|δ˜ )µ(B2|δ˜ )∣∣∣∣∣ (6.23)1226.4. Additional properties≤∣∣∣∣∣ µ(A∩B1|δ˜ )µ(B1|δ˜ )µ(A|δ˜ ) −1∣∣∣∣∣+∣∣∣∣∣1− µ(A∩B2|δ˜ )µ(B2|δ˜ )µ(A|δ˜ )∣∣∣∣∣ (6.24)≤ 2C ∑x∈B,y∈Θe−γdist(x,y) (6.25)≤ |B|2C ∑y∈Θe−γdist(B,y). (6.26)W.l.o.g., we can assume that |Θ|= 1 (see Lemma 3.1.1). Therefore, by takingC′ = 2C, we conclude that∣∣piω1A (β )−piω2A (β )∣∣≤ |B|2K ∑y∈Θe−γdist(B,y) = |B|C′e−γdist(B,Σ∂A(ω1,ω2)). (6.27)Remark 13. The proof of Proposition 6.4.1 seems to require some assumption onthe support of µ (in particular, for the existence of ω˜ in the enumerated item listabove). Fully supported (i.e. supp(µ) = A Z2) suffices, and is the only case inwhich we will apply this result (see Corollary 11), but the conclusion probablyholds under weaker assumptions.Given y,z≥ 0, we define the past boundary of Qy,z as ∂↓Qy,z := ∂Qy,z∩P , i.e.the portion of the boundary of Qy,z included in the past, and the future boundaryof Qy,z as the complement ∂↑Qy,z := ∂Qy,z \P . Clearly, ∂Qy,z = ∂↓Qy,zunionsq∂↑Qy,z.Proposition 6.4.2. Let pi be a Gibbs specification satisfying exponential SSM withparameters C,γ > 0. Then, for all n ∈ N, y,z≥~1n, and a ∈A ,∣∣∣piω1Qn (a{~0})−piω2Qy,z(a{~0})∣∣∣≤Ce−γn, (6.28)uniformly over ω1,ω2 ∈Ω such that ω1|P = ω2|P .Proof. Fix n∈N, y,z≥~1n, a∈A , and ω1,ω2 ∈Ω such that ω1|P = ω2|P . Then,∣∣∣piω1Qn (a{~0})−piω2Qy,z(a{~0})∣∣∣ (6.29)=∣∣∣∣∣∣∣piω1Qn (a{~0})− ∑υ∈ [ω2|Qcn ]ΩpiυQn(a{~0})piω2Qy,z(υ |Qy,z\Qn)∣∣∣∣∣∣∣ (6.30)1236.4. Additional properties≤ ∑υ∈ [ω2|Qcn ]Ω∣∣∣piω1Qn (a{~0})−piυQn(a{~0})∣∣∣piω2Qy,z(υ |Qy,z\Qn) (6.31)≤ ∑υ∈[ω2|Qcn ]ΩCe−γnpiω2Qy,z(υ |Qy,z\Qn) =Ce−γn, (6.32)since for any υ ∈[ω2|Qcn]Ω, we have that Σ∂Qn(ω1,υ)⊆ ∂↑Qn, anddist(~0,Σ∂Qn(ω1,υ))≥ dist(~0,∂↑Qn)= n. (6.33)6.4.2 Stochastic dominanceSuppose that A is a finite linearly ordered alphabet. Then for any set L (in ourcontext, usually a set of sites or bonds), A L is equipped with a natural partialorder  which is defined coordinate-wise: for θ1,θ2 ∈ A L, we write θ1  θ2 ifθ1(x)  θ2(x) for every x ∈ L. A function f :A L→ R is said to be increasing iff (θ1)≤ f (θ2)whenever θ1 θ2. An eventA is said to be increasing if its indicatorfunction 1A is increasing.Definition 6.4.2. Let ρ1 and ρ2 be two probability measures on A L. We say thatρ1 is stochastically dominated by ρ2, writing ρ1 ≤D ρ2, if for every boundedincreasing function f : A L→ R we have ρ1( f ) ≤ ρ2( f ), where ρ( f ) denotes theexpected value Eρ( f ) of f according to the measure ρ .Recall from Section 6.2 the bond random-cluster model on finite subsets of Z2with boundary conditions i = 0,1, and the bond random-cluster measure ϕp,q onZ2 (see page 116).Theorem 6.4.3 ([36, Equation (29)]). For any p ∈ [0,1], q ∈ N, and ∆⊆ Λb Z2,ϕ(0)p,q,∆ ≤D ϕ(0)p,q,Λ and ϕ(1)p,q,Λ ≤D ϕ(1)p,q,∆. (6.34)In particular, if p < pc(q), we have that, for any Λb Z2,ϕ(0)p,q,Λ ≤D ϕp,q ≤D ϕ(1)p,q,Λ, (6.35)1246.4. Additional propertieswhere ≤D is with respect to the restriction of each measure to events on E 0(Λ).Connectivity decay for the bond random-cluster modelThe following result was a key element of the proof that βc(q) = log(1+√q) is thecritical inverse temperature for the Potts model. We will use this result in a crucialway.Recall that for p< pc(q), ϕp,q is the unique bond random cluster measure withparameters p and q.Theorem 6.4.4 ([9, Theorem 2]). Let q ≥ 1 and p < pc(q) =√q1+√q . Then, thetwo-point connectivity function decays exponentially, i.e. there exist constants0 <C(p,q),c(p,q)< ∞ such that for any x,y ∈ Z2,ϕp,q(x↔ y)≤C(p,q)e−c(p,q)‖x−y‖2 , (6.36)where {x↔ y} is the event that the sites x and y are connected by an open pathand ‖ · ‖2 is the Euclidean norm.Stochastic dominance for the site random-cluster modelLemma 6.4.5. Given Λ b Zd and parameters p ∈ [0,1] and q > 0, we have thatfor any x ∈ Λ and any τ ∈ {0,1}Λ\{x},p1(q)≤ ψ(1)p,q,Λ(θ(x) = 1|τ)≤ p2(q), (6.37)where p1(q) :=pqpq+(1−p)q2d and p2(q) :=pqpq+(1−p) . In consequence,ψp1(q),Λ ≤D ψ(1)p,q,Λ ≤D ψp2(q),Λ. (6.38)(Recall that ψp,Λ denotes Bernoulli site percolation.)Proof. This result is obtained by adapting the discussion on [42, p. 339] to thewired site random-cluster model case. See also [43, Lemma 5.4] for the case q =2.1256.4. Additional propertiesStochastic dominance for the Potts modelAs before, let q∈AFP,q denote a fixed, but arbitrary, choice of a colour. Let ΛbZdand consider g :AFP,qΛ→{+,−}Λ be defined by(g(α))(x) =+ if α(x) = q,− if α(x) 6= q. (6.39)The function g makes the non-q colours indistinguishable and gives a reducedmodel (these models are sometimes called fuzzy Potts models). We say α ' α ′ ifg(α) = g(α ′). This relation defines a partition of AFP,qΛ, and unions of elementsof this partition form a sub-algebra of AFP,qΛ which can be identified with thecollection of all subsets of {+,−}Λ. Let pi+β ,Λ := g∗piωqβ ,Λ be the push-forward mea-sure, which is nothing more than the restriction (or projection) of piωqβ ,Λ to {+,−}Λ.Chayes showed that the FKG property holds on events in this reduced model.Proposition 6.4.6 ([23, Lemma on p. 211]). For all β > 0 and Λ b Z2, pi+β ,Λsatisfies the following properties:1. For increasing subsets A,B⊆ {+,−}Λ, pi+β ,Λ (A|B)≥ pi+β ,Λ(A).2. If A is decreasing and B is increasing, then pi+β ,Λ (A|B)≤ pi+β ,Λ(A).3. If ∆⊆Λ and A is an increasing subset of {+,−}∆, then pi+β ,∆(A)≥ pi+β ,Λ(A).Proof.1. This is contained in [23, Lemma on p. 211].2. This is an immediate consequence of (1).3. This is a standard consequence of (1): let B = [+∂∆]ΩFP . Since g−1(B) isa single configuration, namely q∂∆, we obtain from the Markov property ofpiωqβ ,Λ that pi+β ,∆(A) = pi+β ,Λ (A|B). From (1), we have pi+β ,Λ (A|B) ≥ pi+β ,Λ(A).Now, combine the previous two statements.Remark 14. The preceding result immediately applies to piωqβ ,Λ for events inAFP,qΛthat are measurable with respect to {+,−}Λ, viewed as a sub-algebra of AFP,qΛ.1266.5. Exponential convergence of pinVolume monotonicity for the Widom-Rowlinson model with 2 typesFor the classical Widom-Rowlinson model (q= 2), Higuchi and Takei showed thatthe FKG property holds. In particular,Proposition 6.4.7 ([43, Lemma 2.3]). Fix q = 2 and let ∆ ⊆ Λ b Zd and ζ > 0.Then,piζΛ(ωq)≤ piζ∆ (ωq). (6.40)However, this kind of stochastic monotonicity can fail for general q (see [36,p. 60]).6.5 Exponential convergence of pinIn this section, we consider the Potts, Widom-Rowlinson, and hard-core latticegas models and establish exponential convergence results that will lead to pressurerepresentation and approximation algorithms for these lattice models.Recall that for the Potts model, piβy,z(ω) = piωβ ,Qy,z(ω|{~0}) and, in particular,piβn (ω) = piωβ ,Qn(ω|{~0}), with similar notation for the Widom-Rowlinson and hard-core lattice gas models.6.5.1 Exponential convergence in the Potts modelTheorem 6.5.1. For the (ferromagnetic) Potts model with q types and inverse tem-perature β > 0, there exists a critical parameter βc(q)> 0 such that for β 6= βc(q),there exists C,γ > 0 such that, for every y,z≥~1n,∣∣∣piβn (ωq)−piβy,z(ωq)∣∣∣≤Ce−γn. (6.41)Proof. In the supercritical region β > βc(q), our proof very closely follows [22,Theorem 3], which treated the Ising model case. We fill in some details of theirproof, adapting that proof in two ways: to a half-plane version of their result (thequantities in Equation (6.41) are effectively half-plane quantities) and to the gen-eral Potts case. For the subcritical region β < βc(q), the proposition will followeasily from [4, Theorem 1.8 (ii)].1276.5. Exponential convergence of pinPart 1: β > βc(q). Let P−?∂Qn denote the event that there is a ?-path of − from~0to ∂Qn, i.e. a path that runs along ordinary Z2 bonds and diagonal bonds wherethe colour at each site is not q (in our context below, the configuration on the pastboundary of Qn will be all q and thus a ?-path of− from~0 to ∂Qn cannot terminateon ∂↓Qn). Note that P−?∂Qn is an event that is measurable with respect to the sub-algebra {+,−}Λ, for any finite set Λ containing Qn, introduced in Section 6.4.2(recall that this sub-algebra corresponds to the reduced Potts model).By decomposing piβy,z(ωq) into probabilities conditional on P−?∂Qn and (P−?∂Qn)c ,we obtainpiβn (ωq)−piβy,z(ωq) (6.42)= piωqβ ,Qn(q{~0})−piωqβ ,Qy,z(q{~0}) (6.43)= piωqβ ,Qy,z(P−?∂Qn)(piωqβ ,Qn(q{~0})−piωqβ ,Qy,z(q{~0}|P−?∂Qn))+(1−piωqβ ,Qy,z(P−?∂Qn))(piωqβ ,Qn(q{~0})−piωqβ ,Qy,z(q{~0}|(P−?∂Qn)c)). (6.44)We claim that the expression in Equation (6.42) is nonnegative. To see this,observe that the events [q{~0}], [q∂Qn ], and [q∂Qy,z ] may be viewed as the events[+{~0}], [+∂Qn ], and [+∂Qy,z ] in the sub-algebra {+,−}Qy,z of the reduced model, asdiscussed in Section 6.4.2. Now, apply Proposition 6.4.6 (part 3) and Remark 14.We next claim thatpiωqβ ,Qy,z(q{~0}|(P−?∂Qn)c)≥ piωqβ ,Qn(q{~0}). (6.45)To be precise, first observe that ω ∈ (P−?∂Qn)c iff ω contains an all-q path inQn from ∂P ∩{x1 < 0} to ∂P ∩{x1 > 0}. So, (P−?∂Qn)c can be decomposed intoa disjoint collection of events determined by the unique furthest such path from~0. Using the MRF property of Gibbs specifications, it follows that we can regardeach of these events as an increasing event in {+,−}Qm . Now, apply Proposition6.4.6 and Remark 14. The reader may notice that here we have essentially used thestrong Markov property (see [37, p. 1154]).Thus, the expression in Equation (6.44) is nonpositive. This, together with the1286.5. Exponential convergence of pinfact that piωqβ ,Qn(q{~0}|P−?∂Qn) = 0, yields0≤ piβn (ωq)−piβy,z(ωq)≤ piωqβ ,Qy,z(P−?∂Qn)piωqβ ,Qn(q{~0})≤ piωqβ ,Qy,z(P−?∂Qn). (6.46)So, it suffices to show that supy,z≥~1npiωqβ ,Qy,z(P−?∂Qn) decays exponentially in n.Fix y,z ≥~1n and let m > n such that~1m ≥ y,z. By Proposition 6.4.6 (part 2 andpart 3) and Remark 14,piωqβ ,Qy,z(P−?∂Qn)≤ piωqβ ,Qm(P−?∂Qn) (6.47)= piωqβ ,Bm(P−?∂Qn |qP)≤ piωqβ ,Bm(P−?∂Qn)≤ piωqβ ,Bm(P−?∂Bn). (6.48)So, it suffices to show that supm>npiωqβ ,Bm(P−?∂Bn) decays exponentially in n. Re-call the Edwards-Sokal coupling P(1)p,q,Bm for the Gibbs distribution and the corre-sponding bond random-cluster measure with wired boundary condition ϕ(1)p,q,Bm (seeSection 6.2).W.l.o.g., let’s suppose that n is even, i.e. n = 2k < m, for some k ∈ N. Weconsider the following two events in the bond random-cluster model, as in [23,Theorem 3]. Let Rn be the event of an open cycle in B2k \Bk that surrounds Bk.Let Mn,m be the event in which there is an open path from some site in Bk to∂Bm. The joint occurrence of these two events forces the Potts event (P−?∂Bn)c in thecoupling: Rn∩Mn,m ⊆ (P−?∂Bn)c (here, technically, we are identifying these eventswith their inverse images of the projections in the coupling).Then, by the coupling property,piωqβ ,Bm((P−?∂Bn)c)= P(1)p,q,Bm((P−?∂Bn)c)(6.49)≥ P(1)p,q,Bm((P−?∂Bn)c∣∣∣Rn∩Mn,m)P(1)p,q,Bm (Rn∩Mn,m) (6.50)= ϕ(1)p,q,Bm (Rn∩Mn,m) , (6.51)sopiωqβ ,Bm(P−?∂Bn)≤ 1−ϕ(1)p,q,Bm(Rn∩Mn,m)≤ ϕ(1)p,q,Bm(Rcn)+ϕ(1)p,q,Bm(Mcn,m). (6.52)1296.5. Exponential convergence of pinTherefore,supm>npiωqβ ,Bm(P−?∂Bn)≤ supm>nϕ(1)p,q,Bm(Rcn)+ supm>nϕ(1)p,q,Bm(Mcn,m). (6.53)The first term on the right hand side of Equation (6.53) is bounded from aboveas follows:ϕ(1)p,q,Bm(Rcn)≤ ϕ(1)p,q,B˜m+1(Rcn) (6.54)≤ ∑x∈∂Bk,y∈∂B2kϕ(0)p∗,q,Bm+1(x↔ y) (6.55)≤ ∑x∈∂Bk,y∈∂B2kϕp∗,q(x↔ y), (6.56)where B˜m = [−m+ 1,m]2 ∩Z2 and p∗ denotes the dual of p and the inequalitiesfollow from Proposition 6.2.3 and Theorem 6.4.3.If p> pc(q), then p∗ < pc(q), and by Theorem 6.4.4, the first term on the rightside of Equation (6.53) is upper bounded by 64C(p∗,q)n2 exp(−c(p∗,q)n/4), since|∂Bk||∂B2k| ≤ 64n2 and ‖x− y‖2 ≥ k− 1 ≥ n4 , for all x ∈ ∂Bk and y ∈ ∂B2k. So,the first term on the right side of Equation (6.53) decays exponentially.As for the second term, in order forMn,m to fail to occur, there must be a closedcycle in Bm\Bk and in particular a closed path from Lm,n :=Bm\Bk∩{x1 < 0,x2 =0} to Rm,n := Bm \Bk∩{x1 > 0,x2 = 0} in Bm. Thus,ϕ(1)p,q,Bm(Mcn,m)≤ ϕ(1)p,q,B˜m+1(Mcn,m) (6.57)≤ ∑x∈Lm,n,y∈Rm,nϕ(0)p∗,q,Bm+1(x↔ y) (6.58)≤ ∑x∈Lm,n,y∈Rm,nϕp∗,q(x↔ y), (6.59)where the last inequality follows by Proposition 6.2.3 and Proposition 6.4.3. ByTheorem 6.4.4, this is less than∑i=n, j=nC(p∗,q)e−c(p∗,q)(i+ j) ≤C(p∗,q)(e−c(p∗,q)n 11− e−c(p∗,q))2(6.60)1306.5. Exponential convergence of pin=C(p∗,q)(1− e−c(p∗,q))2 e−2c(p∗,q)n. (6.61)Thus, the 2nd term on the right side of Equation (6.53) also decays exponen-tially. Thus, supm>npiωqβ ,m(P−?∂Bn) decays exponentially in n. Thus, by Equation(6.47), supm>npiωqβ ,m(P−?∂Qn) also decays exponentially in n, as desired.Part 2: β < βc(q). Recall from Section 6.4 the notions of strong spatial mixingand ratio strong mixing property. We will use the following result.Theorem 6.5.2 ([4, Theorem 1.8 (ii)]). For the Z2 (ferromagnetic) Potts modelwith q types and inverse temperature β , if 0 < β < βc(q) and exponential decayof the two-point connectivity function holds for the corresponding bond random-cluster model, then the unique µ ∈G(piFPβ ) satisfies the ratio strong mixing propertyfor the class of finite simply lattice-connected sets.Corollary 11. For the Z2 Potts model with q types and inverse temperature 0 <β < βc(q), the Gibbs specification piFPβ satisfies exponential SSM for the family ofsets {Qy,z}y,z≥0.Proof. This follows immediately from Theorem 6.4.4, Theorem 6.5.2, and Propo-sition 6.4.1.Then, since exponential SSM holds for the class of finite simply lattice-connec-ted sets when β < βc(q), the desired result follows directly from Proposition 6.4.2.This completes the proof of Theorem Exponential convergence in the Widom-Rowlinson modelRecall that for Bernoulli site percolation in Z2 there exists a critical value pc(Z2)(or just pc), known as the percolation threshold, such that for p < pc, there is noinfinite cluster of 1’s ψp,Z2-almost surely and, for p > pc, there is such a (unique)cluster ψp,Z2-a.s. Similarly, one can define an analogous parameter p?c for the Z2,?lattice, which satisfies pc+ p?c = 1 (see [72]).Theorem 6.5.3. For the Widom-Rowlinson model with q types and activity ζ , thereexist two critical parameters 0 < ζ1(q) < ζ2(q) such that for ζ < ζ1(q) or ζ >1316.5. Exponential convergence of pinζ2(q), there exists C,γ > 0 such that, for every y,z≥~1n,∣∣∣piζn (ωq)−piζy,z(ωq)∣∣∣≤Ce−γn. (6.62)Proof. As in Theorem 6.5.1, we split the proof in two parts.Part 1: ζ > ζ2(q) := q3(pc1−pc). Fix n ∈ N and y,z ≥~1n. Notice that, due to thehard constraints of the Widom-Rowlinson model, and recalling Proposition 6.3.1,piζy,z(ωq) = piωqζ ,Qy,z(q{~0}) = ψ(1)p,q,Qy,z(1{~0}), (6.63)where p = ζ1+ζ , and the same holds for piζn (ωq). Then, it suffices to prove that∣∣∣ψ(1)p,q,Qn(1{~0})−ψ(1)p,q,Qy,z(1{~0})∣∣∣≤Ce−γn, (6.64)for some C,γ > 0.Notice that ~0 ∈ Qn ⊆ Qy,z =: Λ. Fix any ordering on the set Λ. From nowon, when we talk about comparing sites in Λ, it is assumed we are speaking ofthis ordering. For convenience, we will extend configurations on Qn and Λ toconfigurations on Λ by appending 1Λ\Qn and 1∂Λ, respectively.We proceed to define a coupling Pn,y,z of ψ(1)p,q,Qn and ψ(1)p,q,Λ, defined on pairsof configurations (θ1,θ2) ∈ {0,1}Λ×{0,1}Λ. The coupling is defined one site ata time, using values from previously defined sites.We use (τ t1,τt2) to denote the (incomplete) configurations on Λ×Λ at step t =0,1, . . . , |Qn|. We therefore begin with τ01 = 1Λ\Qn and τ02 = 1∂Λ. Next, we setτ11 = τ01 and form τ12 by extending τ02 to Λ \Qn, choosing randomly according tothe distribution ψ(1)p,q,Λ(·∣∣1∂Λ). At this point of the construction, both τ11 and τ12have shape Λ\Qn. In the end, (τ |Qn|1 ,τ |Qn|2 ) will give as a result a pair (θ1,θ2), bothwith shape Λ.At any step t, we use W t to denote the set of sites in Λ on which τ t1 and τt2have already received values in previous steps. In particular, W 1 = Λ \Qn. At anarbitrary step t of the construction, we choose the next site xt+1 on which to assignvalues in τ t+11 and τt+12 as follows:(i) If possible, take xt+1 to be the first site in ∂ ?W t that is ?-adjacent to a site1326.5. Exponential convergence of piny ∈W t for which (τ t1(y),τ t2(y)) 6= (1,1).(ii) Otherwise, just take xt+1 to be the smallest site in ∂ ?W t .Notice that at any step t, W t is a ?-connected set, and that it it always possibleto find the next site xt+1 for any t < |Qn| (i.e. the two rules above give a welldefined procedure).Now we are ready to augment the coupling from W t to W t∪{xt+1} by assigningτ t+11∣∣{xt+1} and τt+12∣∣{xt+1} according to an optimal coupling of ψ(1)p,q,Qn (·|τ t1)∣∣∣{xt+1}and ψ(1)p,q,Qy,z(· | τ t2)∣∣∣{xt+1}, i.e. a coupling which minimizes the probability that,given (τ t1,τt2), θ1(xt+1) 6= θ2(xt+1). Since Pn,y,z is defined site-wise, and at eachstep is assigned according to ψ(1)p,q,Qn (·|τ t1) in the first coordinate and ψ(1)p,q,Qy,z(· | τ t2)in the second, the reader may check that it is indeed a coupling of ψ(1)p,q,Qn andψ(1)p,q,Qy,z . The key property of Pn,y,z is the following.Lemma 6.5.4. θ1(~0) 6= θ2(~0) Pn,y,z-a.s. iff there exists a path P of ?-adjacent sitesfrom~0 to ∂Qn, such that for each site y ∈ P, (θ1(y),θ2(y)) 6= (1,1).Proof. Suppose, for a contradiction, that θ1(~0) 6= θ2(~0) and that there exists nosuch path. This implies that there exists a cycle C surrounding ~0 (when we in-clude the past boundary as part of C) and contained in Qn such that for all y ∈ C,(θ1(y),θ2(y)) = (1,1). Define by I the simply lattice-?-connected set of sites in theinterior of C and, let’s say that at time t0, xt0 was the first site within I defined ac-cording to the site-by-site evolution of Pn,y,z. Then, (τ t01 (xt0),τt02 (xt0)) cannot havebeen defined according to rule (i) since all sites ?-adjacent to xt0 are either in I (andtherefore not yet defined by definition of xt0), or in C (and therefore either not yetdefined or sites at which θ1 and θ2 are both 1).Therefore, (θ1(xt0),θ2(xt0)) was defined according to rule (ii). We thereforedefine the set D := Λ \W t0−1 ⊇ I, and note that ~0 and xt0 belong to the same?-connected component Θ of D. We also know that τ t0−11∣∣∣∂ ?D= τ t0−12∣∣∣∂ ?D=1∂?D, otherwise some unassigned site in D would be ?-adjacent to a 0 in eitherτ t0−11∣∣∣∂ ?Dor τ t0−12∣∣∣∂ ?D, and so rule (i) would be applied instead. We may nowapply Lemma 6.3.2 (combined with Remark 12) to Θ and Λ in order to see that1336.5. Exponential convergence of pinψ(1)p,q,Qn(θ1|Θ |τt0−11 ) and ψ(1)p,q,Qy,z(θ2|Θ |τt0−12 ) are identical. This means that the op-timal coupling according to which τ t01 (xt0) and τ t02 (xt0) are assigned is supported onthe diagonal, and so τ t01 (xt0) = τ t02 (xt0), Pn,y,z-almost surely. This will not changethe conditions under which we applied Lemma 6.3.2, and so inductively, the samewill be true for each site in I as it is assigned, including~0. We have shown thatθ1(~0) = θ2(~0), Pn,y,z-almost surely, regardless of when~0 is assigned in the site-by-site evolution of Pn,y,z. This is a contradiction, and so our original assumption wasincorrect, implying that the desired path P exists.Given an arbitrary time t, letρ t1(·) := ψ(1)p,q,Qn(·|τ t−1i )∣∣∣{xt}and ρ t2(·) := ψ(1)p,q,Λ(·|τ t−1i )∣∣∣{xt}(6.65)be the two corresponding probability measures defined on the set {0,1}{xt}. Notethat at any step within the site-by-site definition of Pn,y,z, Lemma 6.4.5 implies thatζζ+q3 ≤ ρ ti (1), where ζ = p1−p and i = 1,2. Now, w.l.o.g., suppose that ρ t2(0) ≥ρ t1(0). Then, an optimal coupling Qt of ρ t1 and ρ t2 will assign Qt({(0,0)}) =ρ t1(0), Qt({(0,1)}) = 0, Qt({(1,0)}) = ρ t2(0)− ρ t1(0), and Qt({(1,1)}) = 1−ρ t2(0). Therefore,Qt({(1,1)}c) = ρ t2(0)≤q3ζ +q3. (6.66)Next, define the map h : {0,1}Qn×{0,1}Qn →{0,1}Qn given by(h(θ1,θ2))(x) =1 if (θ1(x),θ2(x)) 6= (1,1),0 if (θ1(x),θ2(x)) = (1,1). (6.67)By Equation (6.66), h∗Pn,y,z (the push-forward measure) can be coupled againstan i.i.d. measure on {0,1}Qn which assigns 1 with probability q3ζ+q3 and 0 withprobability ζζ+q3 , and that the former is stochastically dominated by the latter. This,together with Lemma 6.5.4, yields∣∣∣ψ(1)p,q,Qn(1{~0})−ψ(1)p,q,Qy,z(1{~0})∣∣∣≤ Pn,y,z(θ1(~0) 6= θ2(~0)) (6.68)≤ ψ q3ζ+q3,Qn(~0 ?←→ ∂Qn), (6.69)1346.5. Exponential convergence of pinwhere {~0 ?←→ ∂Qn} denotes the event of an open ?-path from ~0 to ∂Qn. Sincewe have assumed ζ > q3(pc1−pc)and pc + p?c = 1, we haveq3ζ+q3 < p?c . It followsby [2, 62] that the expression in Equation (6.69) decays exponentially in n. Thiscompletes the proof.Part 2: ζ < ζ1(q) := 1q(pc1−pc). Observe that, by virtue of Proposition 6.4.2, itsuffices to prove that piWRζ satisfies exponential SSM. By considering all cases ofnearest-neighbour configurations at the origin, one can computeQ(piWRζ ) = maxω1,ω2∈ΩWRdTV(piω1ζ ,{~0},piω2ζ ,{~0}) =qζ1+qζ. (6.70)By Theorem 3.6.1, we obtain exponential SSM whenζ <1q(pc1− pc)= ζ1(q). (6.71)Uniqueness of Gibbs states in this same region was mentioned in [38, p. 40],by appealing to [11, Theorem 1] (which is the crux of Theorem 3.6.1).Remark 15. In the case q= 2, it is possible to give an alternative proof of Theorem6.5.3 (Part 1) using the framework of the proof of Theorem 6.5.1 (Part 1). Thearguments through Equation (6.47) go through, with an appropriate redefinition ofevents and use of Proposition 6.4.7 for stochastic dominance. One can then applyLemma 6.4.5 to give estimates based on the site random-cluster model. (In contrastto Theorem 6.5.1 (Part 1), this does not require the use of planar duality). So far,this approach is limited to q = 2 because we do not know appropriate versions ofProposition 6.4.7 for q > Exponential convergence in the hard-core lattice gas modelOur argument again relies on proving exponential convergence for conditionalmeasures with respect to certain “extremal” boundaries on Qn, but these now willconsist of alternating 0 and 1 symbols rather than a single symbol (recall from Sec-tion 6.1 that ω(o) is defined as the configuration of 1’s in all even sites and 0 in allodd sites).1356.5. Exponential convergence of pinTheorem 6.5.5. For the Z2 hard-core lattice gas model with activity λ , there existtwo critical parameters 0 < λ1 < λ2 such that for any 0 < λ < λ1 or λ > λ2, thereexist C,γ > 0 such that, for every y,z≥~1n,∣∣∣piλn (ω(o))−piλy,z(ω(o))∣∣∣≤Ce−γn. (6.72)Proof. As in the previous two theorems, we consider two cases.Part 1: λ > λ2 := 468. Our proof essentially combines the disagreement percola-tion techniques of [11] and the proof of non-uniqueness of equilibrium states for thehard-core lattice gas model due to Dobrushin (see [26]). We need enough detailsnot technically contained in either proof that we present a mostly self-containedargument here. From [11, Theorem 1] and an averaging argument (as in the proofof Proposition 6.4.2) on ∂↑Qn induced by a boundary condition on Qy,z, we knowthat for any y,z≥~1n,∣∣∣piλn (ω(o))−piλy,z(ω(o))∣∣∣≤ Pn,y,z(~0 6=←→ ∂↑Qn) (6.73)for a certain coupling Pn,y,z of piω(o)λ ,Qn and piω(o)λ ,Qy,z∣∣∣Qn. We do not need the structure ofPn,y,z here, but instead note the following: a path of disagreement for the boundariesω(o)∣∣∂Qnand ω(o)∣∣∂Qy,zimplies that in one of the configurations, all entries on thepath will be “out of phase” with respect to ω(o), i.e. that all entries along thepath will have 1 at every odd site and 0 at every even site rather than the oppositealternating pattern of ω(o). Then, if we denote by Tn the event that there is a pathP from~0 to ∂↑Qn with 1 at every odd site and 0 at every even site, it is clear thatPn,y,z(~06=←→ ∂↑Qn)≤ piω(o)λ ,Qn(Tn)+piω(o)λ ,Qy,z(Tn). (6.74)Since y,z ≥~1n are arbitrary (in particular, y and z can be chosen to be~1n), itsuffices to prove that supy,z≥~1npiω(o)λ ,Qy,z(Tn) decays exponentially with n. Define thesetΘy,z = {θ ∈ {0,1}S?y,z : θ is feasible and θ |∂ ?Qy,z = ω(o)|∂ ?Qy,z}. (6.75)1366.5. Exponential convergence of pinFor any θ ∈Θy,z, we define Σ~0(θ) to be the connected component ofΣQy,z(θ ,ω(o)) = {x ∈ Qy,z : θ(x) 6= ω(o)(x)} (6.76)containing the origin~0. Since Tn ⊆ {Σ~0(θ)∩ ∂↑Qn 6= /0}, our proof will then becomplete if we can show that there exist C,γ > 0 so that, for any n and y,z≥~1n,piω(o)λ ,Qy,z(Σ~0(θ)∩∂↑Qn 6= /0)≤Ce−γn. (6.77)To prove this, we use a Peierls argument, similar to [26].Fix any y,z≥~1n and for any θ ∈Θy,z, define Σ~0(θ) as above, and let K(θ) to bethe connected component of {x ∈ S?y,z : θ(x) =ω(o)(x)} containing ∂ ?Qy,z. Clearly,Σ~0(θ) and K(θ) are disjoint, K(θ) 6= /0 and, provided θ(~0) = 0, Σ~0(θ) 6= /0. Then,define Γ(θ) := Σ~0(θ)∩∂K(θ)⊆Qy,z. We note that for any θ ∈Θy,z with θ(~0) = 0,we have that θ |Γ(θ) = 0Γ(θ), since adjacent sites in Σ~0(θ) and K(θ) must have thesame symbol by definition of Σ~0(θ), and adjacent 1 symbols are forbidden in thehard-core lattice gas model. Therefore, every x ∈ Γ(θ) is even.We need the concept of inner external boundary for a connected set Σ b Z2.The inner external boundary of Σ is defined to be the inner boundary of thesimply lattice-connected set consisting of the union of Σ and the union of all thefinite components ofZ2\Σ. Intuitively, the inner external boundary of Σ is the innerboundary of the set Σ obtained after “filling in the holes” of Σ. Notice that the setΓ(θ) corresponds exactly to the inner external boundary of Σ~0(θ). In addition, by[25, Lemma 2.1 (i)], we know that the inner external boundary of a finite connectedset (more generally a finite ?-connected set) is ?-connected. Thus, Γ(θ) ⊆ Qy,z isa ?-connected set C? that consists only of even sites and contains the origin~0, forany θ ∈Θy,z with θ(~0) = 0.Then, for C?⊆Qy,z, we define the event EC? := {θ ∈Θy,z :Γ(θ) =C?}, and willbound from above piω(o)λ ,Qy,z(EC?), for every C? such that EC? is nonempty. We makesome more notation: for every such a set C?, define O(C?) (for ‘outside’) as theconnected component of (C?)c containing ∂ ?Qy,z, and define I(C?) (for ‘inside’) asQy,z \ (C? ∪O(C?)). Then C?, I(C?), and O(C?) form a partition of Q?y,z. We notethat there cannot be a pair of adjacent sites from I(C?) and O(C?) respectively,1376.5. Exponential convergence of pinsince they would then be in the same connected component of (C?)c. We also notethat for every θ ∈ EC? , C?⊆ Σ~0(θ)⊆ C?∪ I(C?) and K(θ)⊆O(C?), though the setsneed not be equal, since Σ~0(θ) or K(θ) could contain “holes” which are “filled in”in I(C?) and O(C?), respectively.origin1100111100000000000000100 0 01111011011010011100001110001 011001100010011100010110010000010010000010010100100001100100001000110000001011100001110001101100100000111000Figure 6.2: A configuration θ ∈ EC? . On the left, the associated sets Σ~0(θ) andK(θ). On the right, the sets I(C?) and O(C?) for Γ(θ) = C?.Choose any set C? such that EC? 6= /0. For each θ ∈ EC? and x ∈ C?, using thedefinition of C? and the fact that K(θ)⊆O(C?), there exists x0 ∈ {~e1,−~e1,~e2,−~e2}for which x− x0 ∈ O(C?). Fix an x0 which is associated to at least |C?|/4 of thesites in C? in this way. Then, we define a function s : EC? → {0,1}Q?y,z that, givenθ ∈ EC? , defines a new configuration s(θ) such that(s(θ))(x) =θ(x− x0) if x ∈ I(C?),θ(x) if x ∈ O(C?),1 if x ∈ C? and x− x0 ∈ O(C?),0 if x ∈ C? and x− x0 ∈ I(C?).(6.78)Informally, we move all 1 symbols inside I(C?) in the x0-direction by 1 unit(even if those symbols were not part of Σ~0(θ)), add new 1 symbols at some sites inC?, and leave everything in O(C?) unchanged.It should be clear that s(θ) has at least |C?|/4 more 1 symbols than θ did. We1386.5. Exponential convergence of pinmake the following two claims: s is injective on EC? , and for every θ ∈ EC? , s(θ) ∈Θy,z. If these claims are true, then clearly piω(o)λ ,Qy,z(s(EC?)) ≥ λ |C?|/4piω(o)λ ,Qy,z(EC?),implying thatpiω(o)λ ,Qy,z(EC?)≤ λ−|C?|/4. (6.79)Firstly, we show that s is injective. Suppose that θ1 6= θ2, for θ1,θ2 ∈ EC? .Then there is a site x at which θ1(x) 6= θ2(x). If x ∈ O(C?), then (s(θ1))(x) =θ1(x) 6= θ2(x) = (s(θ2))(x) and so s(θ1) 6= s(θ2). If x ∈ I(C?), then (s(θ1))(x+x0) = θ1(x) 6= θ2(x) = (s(θ2))(x+ x0), and again s(θ1) 6= s(θ2). Finally, we notethat x cannot be in C?, since at all sites in C?, both θ1 and θ2 must have 0 symbols.Secondly, we show that for any θ ∈ EC? , s(θ) is feasible. All that must beshown is that s(θ) does not contain adjacent 1 symbols. We break 1 symbols ins(θ) into three categories:1. shifted, meaning that the 1 symbol came from shifting a 1 symbol at a site inI(C?) in the x0-direction,2. new, meaning that the 1 symbol was placed at a site x ∈ C? such that x−x0 ∈O(C?), or3. untouched, meaning that the 1 symbol was at a site in O(C?) (⊇ ∂ ?Qy,z).Note that untouched 1 symbols cannot be adjacent to C?: θ contains all 0symbols on C?, and so since C? ⊆ Σ~0(θ), a 1 symbol adjacent to a symbol in C?would be in Σ~0(θ) as well, a contradiction since Σ~0(θ)⊆ C?∪ I(C?), and so Σ~0(θ)and O(C?) are disjoint.Clearly, shifted 1 symbols cannot be adjacent to each other, since there wereno adjacent 1 symbols in θ in the first place. All new 1’s were placed at sites inC?, and all sites in C? are even, so new 1 symbols cannot be adjacent to each other.Untouched 1’s cannot be adjacent for the same reason as shifted 1’s. We nowaddress the possibility of adjacent 1 symbols in s(θ) from different categories. Ashifted or new 1 in s(θ) is on a site in C? ∪ I(C?), and an untouched 1 cannot beadjacent to a site in C? as explained above, and also cannot be adjacent to a site inI(C?), since I(C?) and O(C?) do not contain adjacent sites. Therefore, shifted ornew 1’s cannot be adjacent to untouched 1’s. The only remaining case which we1396.5. Exponential convergence of pinneed to rule out is a new 1 adjacent to a shifted 1. Suppose that (s(θ))(x) is a new 1and (s(θ))(x′) is a shifted 1. Then by definition, x′−x0 ∈ I(C?) and x−x0 ∈O(C?).We know that I(C?) and O(C?) do not contain adjacent sites, so x− x0 and x′− x0are not adjacent, implying that x and x′ are not adjacent. We’ve then shown thats(θ) is feasible and then, since ∂ ?Qy,z ⊆ O(C?), s(θ) ∈ Θy,z, completing the proofof Equation (6.79).Recall that every set C? which we are considering is ?-connected, occupiesonly even sites, and contains the origin ~0. Then, given k ∈ N, it is direct to seethat the number of such C? with |C?|= k is less than or equal to k · t(k), where t(k)denotes the number of site animals (see [49] for the definition) of size k (the first kfactor comes from the fact that site animals are defined up to translation, and heregiven a site animal of size k, exactly k translations of it will contain the origin~0).We know that for every ε > 0, there exists Cε > 0 such that t(k) ≤Cε(δ + ε)k forevery k, where δ := limk→∞ (t(k))1/k ≤ 4.649551 (see [49]).If Σ~0(θ)∩∂↑Qn 6= /0, then Σ~0(θ) has to intersect the top, right, or bottom bound-ary of Qn. W.l.o.g., we may assume that Σ~0(θ) intersects the bottom boundary ofQn. Then, every horizontal segment in the bottom half of Qn must intersect Σ~0(θ)and, therefore, at least one element of its inner external boundary, namely Γ(θ).Then,Σ~0(θ)∩∂↑Qn 6= /0 =⇒ |Γ(θ)| ≥ n. (6.80)Therefore, taking an arbitrary ε > 0, we may bound piω(o)λ ,Qy,z(Σ~0(θ)∩∂↑Qn 6= /0)from above:piω(o)λ ,Qy,z(Σ~0(θ)∩∂↑Qn 6= /0)≤ ∑C?:|C?|≥nλ−|C?|/4 ≤∞∑k=nkCε(δ + ε)k ·λ−k/4, (6.81)which decays exponentially in n as long as λ > (δ + ε)4, independently of y andz. Since ε was arbitrary, λ > 468 > δ 4 suffices for justifying Equation (6.77),completing the proof.Part 2: λ < λ1 := 2.48. It is known (see [75]) that when d = 2 and λ < 2.48, piHCλsatisfies exponential SSM. Then, by applying Proposition 6.4.2, we conclude.140Chapter 7Algorithmic implicationsIn this chapter we combine previous results (in particular, pressure representationtheorems) in order to compute pressure efficiently.By a poly-time approximation algorithm to compute a number r, we meanan algorithm that, given N ∈N, produces an estimate rN such that |r− rN |< 1N andthe time to compute rN is polynomial in N. In that case, we regard this algorithmas an efficient way to approximate r and we say that r is poly-time computable.One of our goals is to prove that, under certain assumptions on Φ and thesupport Ω, PΩ(Φ) is poly-time computable.7.1 Previous resultsFirst, we give some previously known algorithmic results related with pressureapproximation.Proposition 7.1.1 ([32]). Consider the Zd hard-core lattice gas model with activityλ . Ifλ < λc(T2d) :=(2d−1)2d−1(2d−2)2d , (7.1)then, there is an algorithm to compute P(Φλ ) to within 1N in time poly(N).Note 10. The value λc(T2d) corresponds to the critical activity of the hard-coremodel in the 2d-regular tree T2d . This model satisfies exponential SSM if λ <λc(T2d). It is also known that the partition function of the hard-core model withλ < λc(T2d) in any finite board G of maximum degree ∆(G )≤ 2d can be efficientlyapproximated (for these and more results, see the fundamental work of D. Weitz in[79]).1417.2. New resultsProposition 7.1.2 ([58, Proposition 4.1]). Let Ω be a Zd n.n. SFT, Φ a shift-invariant n.n. interaction on Ω, and pi the corresponding n.n. Gibbs (Ω,Φ)-specification such that• Ω satisfies SSF, and• pi satisfies exponential SSM.Then, there is an algorithm to compute PΩ(Φ) to within 1N in time eO((logN)d−1).In particular, if d = 2, PΩ(Φ) is poly-time computable.7.2 New resultsThe following result is based on a slight modification of the approach used to proveProposition 7.1.2, but we include here the whole proof for completeness.Proposition 7.2.1. Let Ω be a Zd n.n. SFT, Φ a shift-invariant n.n. interaction onΩ, and pi the corresponding n.n. Gibbs (Ω,Φ)-specification such that• Ω satisfies TSSM, and• pi satisfies exponential SSM.Then, there is an algorithm to compute PΩ(Φ) to within 1N in time eO((logN)d−1).In particular, if d = 2, PΩ(Φ) is poly-time computable.Proof. Given ε > 0 and the values of the n.n. interaction Φ, the algorithm wouldbe the following:1. Look for a periodic point ω ∈Ω, provided by Proposition 3.4.3. W.l.o.g., ωhas period 2g in every coordinate direction, for some g ∈ N. This step doesnot need the gap of TSSM explicitly, and it does not depend on ε .2. Take νω the shift-invariant atomic measure supported on the orbit of ω .From Corollary 9, we have thatP(Φ) =∫ (Iµ +AΦ)dν (7.2)1427.2. New results=1(2g)d ∑x∈[1,2g]d∩Zd(− log pµ(σx(ω))+AΦ(σx(ω))) , (7.3)for any µ an equilibrium state for Φ. We need to compute the desired ap-proximations of pµ(ω), for all ω = σx(ω) and x ∈ [1,2g]d ∩Zd . We mayassume x =~0 (the proof is the same for all x).3. For n = 1,2, . . . , consider the sets Wn = Rn \Pn and ∂Wn = SnunionsqVn, whereSn = ∂Wn∩P and Vn = ∂Wn \P .4. Represent pµ(ω) as a weighted average, using the MRF property,pµ(ω) = ∑δ∈A Vn : µ(ω|Snδ )>0µ(ω|{~0}∣∣∣ω|Sn δ)µ(δ ). (7.4)Figure 7.1: Decomposition in the proof of Proposition Take δ ∈ argmaxδ µ(ω|{~0}∣∣∣ω|Sn δ) and δ ∈ argminδ µ (ω|{~0}∣∣∣ω|Sn δ),over all δ ∈ A Vn such that µ(ω|Sn δ ) > 0 (or, since TSSM implies the D-condition, such that ω|Sn δ ∈L (Ω)). Then,µ(ω|{~0}∣∣∣ω|Sn δ)≤ pµ(ω)≤ µ (ω|{~0}∣∣∣ω|Sn δ) . (7.5)1437.2. New results6. By exponential SSM, there are constants C,γ > 0 such that these upper andlower bounds on pµ(ω) differ by at most Ce−γn. Taking logarithms and con-sidering that µ(ω|{~0}∣∣∣ω|Sn δ) ≥ cµ > 0, a direct application of the MeanValue Theorem gives sequences of upper and lower bounds on log pµ(ω)with accuracy e−Ω(n), that is less than ε for sufficiently large n.For δ ∈ A Vn , the time to compute µ(ω|{~0}∣∣∣ω|Sn δ) is eO(nd−1), because thisis the ratio of two probabilities of configurations of size O(nd−1), each of whichcan be computed using the transfer matrix method from [60, Lemma 4.8] in timeeO(nd−1). Thanks to Corollary 2, the necessary time to check if ω|Sn δ ∈L (Ω) ornot is eO(nd−1). Since∣∣A Vn∣∣ = eO(nd−1), the total time to compute the upper andlower bounds is eO(nd−1)eO(nd−1) = eO(nd−1).Remark 16. In the previous algorithm it is not necessary to know explicitly thegap g of TSSM or the constants C,γ > 0 of the decay function f (n) =Ce−γn fromexponential SSM.Corollary 12. Let Ω⊆A Z2 be a Z2 n.n. SFT, Φ a shift-invariant n.n. interactionon Ω, and pi the corresponding n.n. Gibbs (Ω,Φ)-specification. Suppose that pisatisfies exponential SSM with decay rate γ > 4log |A |. Then, PΩ(Φ) is poly-timecomputable.Proof. This follows from Theorem 3.6.5 and Proposition 7.2.1.Notice that, in contrast to preceding results, no mixing condition on the supportis explicitly needed in Corollary 12.Now we present a new algorithmic result for pressure representation, speciallyuseful when SSM fails. We make heavy use of the representation and convergenceresults from the previous chapters.Theorem 7.2.2. LetΩ be a Zd n.n. SFT that satisfies the square block D-condition,Φ a shift-invariant n.n. interaction on Ω, and pi the corresponding n.n. Gibbs(Ω,Φ)-specification. Let ω ∈ Ω be a periodic point such that cpi(νω) > 0. Inaddition, suppose that there exists C,γ > 0 such that, for every y,z≥~1n,|pin(ω)−piy,z(ω)| ≤Ce−γn over ω ∈ O(ω). (7.6)1447.2. New resultsThen,PΩ(Φ) =1|O(ω)| ∑ω∈O(ω)Ipi(ω)+AΦ(ω), (7.7)and, when d = 2, PΩ(Φ) is poly-time computable.Proof. Notice that supp(νω) = O(ω) ⊆ Ω, since Ω is shift-invariant and ω ∈ Ω.Now, since |pin(ω)−piy,z(ω)| ≤Ce−γn over ω ∈ supp(νω), we can easily concludethat limy,z→∞piy,z(ω) = pˆi(ω) uniformly over ω ∈ supp(νω). This, combined withΩ satisfying the square block D-condition and cpi(νω)> 0, gives usPΩ(Φ) =∫(Ipi +AΦ)dνω =1|supp(νω)| ∑ω∈supp(νω )Ipi(ω)+AΦ(ω), (7.8)thanks to Theorem 5.3.2.For the algorithm, it suffices to show that there is a poly-time algorithm tocompute pˆi(ω), for any ω ∈ O(ω).By Equation (7.6), there exist C,γ > 0 such that |pin(ω)− pˆi(ω)| < Ce−γn.Since |∂Qn| is linear in n when d = 2, by the modified transfer matrix methodapproach from [60, Lemma 4.8], we can compute pin(ω) in exponential time Keρnfor some K,ρ > 0. Combining the exponential time to compute pin(ω) for the ex-ponential decay of |pin(ω)− pˆi(ω)|, we get a poly-time approximation algorithmto compute PΩ(Φ): namely, given N ∈ N, let n be the smallest integer such thatCe−γ(n+1) < 1N . Then, pin+1(ω) is within1N of pˆi(ω) and since1N ≤Ce−γn, the timeto compute pin+1(ω) is at mostKeρ(n+1) = (KeρCρ/γ)1(Ce−γn)ρ/γ≤ (KeρCρ/γ)Nρ/γ , (7.9)which is a polynomial in N.Corollary 13. The following holds:1. For the Z2 (ferromagnetic) Potts model with q types and inverse temperatureβ > 0,PΩFP(Φβ ) = Iβpi (ωq)+2β . (7.10)1457.2. New results2. For the Z2 (multi-type) Widom-Rowlinson model with q types and activityζ ∈ (0,ζ1(q))∪ (ζ2(q),∞),PΩWR(Φζ ) = Iζpi(ωq)+ logζ , (7.11)where ζ1(q) := 1q(pc1−pc)and ζ2(q) := q3(pc1−pc).3. For the Z2 hard-core lattice gas model with activity λ ∈ (0,λ1)∪ (λ2,∞),PΩHC(Φλ ) =12Iλpi (ω(o))+12logλ , (7.12)where λ1 = 2.48 and λ2 = 468.Moreover, for the three models in the corresponding regions (except in the casewhen β = βc(q) in the Potts model), there is a poly-time approximation algorithmfor pressure, where the polynomial involved depends on the parameters of the mod-els.Proof. The representation of the pressure given in the previous statement for theZ2 Potts model with q types and inverse temperature β 6= βc(q), the Z2 Widom-Rowlinson model with q types and activity ζ ∈ (0,ζ1(q))∪ (ζ2(q),∞), and the Z2hard-core lattice gas model with activity λ ∈ (0,λ1)∪ (λ2,∞), is a direct conse-quence of Theorem 7.2.2, by virtue of the following facts:• Recall that the corresponding Z2 n.n. SFT Ω for the Potts, Widom-Rowlin-son, and hard-core lattice gas model has a safe symbol, respectively, so Ωsatisfies the square block D-condition and cpi(ν)> 0, for any shift-invariantν with supp(ν)⊆Ω, in each case.• If we consider the δ -measure νωq = δωq , both in the Potts and Widom-Rowlinson cases (in a slight abuse of notation, since the Potts and Widom-Rowlinson σ -algebras are defined in different alphabets), and the measureν = νω(o) = 12δω(e) +12δω(o) in the hard-core lattice gas case, we have thatin all three models, for the range of parameters specified, except for whenβ = βc(q) in the Potts model, there exists C,γ > 0 such that, for every1467.2. New resultsy,z≥~1n,|pin(ω)−piy,z(ω)| ≤Ce−γn, over ω ∈ supp(ν), (7.13)thanks to Theorem 6.5.1, Theorem 6.5.3, and Theorem 6.5.5, respectively.(Notice that Iλpi (ω(e)) = AΦ(ω(e)) = 0.)This proves Equation (7.10), Equation (7.11), and Equation (7.12), except inthe Potts case when β = βc(q). To establish this case, first note that it is easyto prove that P(Φβ ) is continuous with respect to β . Second, if β1 ≤ β2, thenpiβ1n (ωq) ≤ piβ2n (ωq). This follows by the Edwards-Sokal coupling (see Theorem6.2.2) and the comparison inequalities for the bond random-cluster model (see [3,Theorem 4.1]).As an exercise in analysis, it is not difficult to prove that if am,n ≥ 0, and eacham+1,n ≤ am,n and am,n+1 ≤ am,n, then limm limn am,n = limn limm am,n = a, for somea≥ 0.Now, consider the sequence am,n := piβc(q)+ 1mn (ωq). By stochastic dominance(see Proposition 6.4.6), am,n is decreasing in n. By the previous discussion (i.e.Edwards-Sokal coupling and comparison inequalities), it is also decreasing in m.Therefore, and since am,n≥ 0, we conclude that limm limn am,n = limn limm am,n = a,for some a. Then, we have thatP(Φβc(q)) = limm P(Φβc(q)+ 1m ) (7.14)= limm− log limnpiβc(q)+1mn (ωq)+2(βc(q)+1m)(7.15)=− log limmlimnpiβc(q)+1mn (ωq)+2βc(q) (7.16)=− log limnlimmpiβc(q)+1mn (ωq)+2βc(q) (7.17)=− log limnpiβc(q)n (ωq)+2βc(q) (7.18)= Iˆβc(q)pi (ωq)+2βc(q). (7.19)(To prove that limmpiβc(q)+ 1mn (ωq) = piβc(q)n (ωq) is straightforward.) Finally, thealgorithmic implications are also a direct application of Theorem New resultsRemark 17. The algorithm in Theorem 7.2.2 seems to require explicit bounds onthe constants C and γ , so that given N ∈ N, we can find an explicit n such thatCe−γ(n+1) < 1N . Without such bounds, while there exists a poly-time approximationalgorithm, we do not always know how to exhibit an explicit algorithm. However,for all three models, for regions sufficiently deep within the supercritical region (i.e.β , ζ , or λ sufficiently large), one can find crude, but adequate, estimates on C andγ and thus can exhibit a poly-time approximation algorithm. This is the case for thehard-core lattice gas model, where our proof does allow an explicit estimate of theconstants for any λ > 468. On the other hand, in the regions specified in Corollary13 within the subcritical region, all three models satisfy exponential SSM and thenusing [60, Corollary 4.7], one can, in principle, exhibit a poly-time approximationalgorithm (even without estimates on C and γ).148Chapter 8ConclusionThe current plan is to extend our research in the following directions:1. We would like to develop a characterization of constraint graphs H for whichHom(G ,H) satisfies TSSM, for every board G . In addition, it would beinteresting to understand the constraint graphs H for which, for every boardof bounded degree G , there always exist a constrained energy function φsuch that the Gibbs (G ,H,φ)-specification satisfies (exponential) SSM.2. We would like to make progress in determining regimes where SSM holdsin classical models. There is already some progress in this direction (see[79, 39]), but not everything is completely understood. For example, it is notknown whether the uniform 5-colourings model in Z2 satisfies SSM or not.3. In [32], Gamarnik and Katz gave a representation theorem and an approx-imation algorithm for surface pressure, which is a first order correction ofpressure. Their result relied on the existence of a (unique) Gibbs measuresatisfying SSM and a safe symbol. We plan to generalize their result, byrelaxing the combinatorial hypothesis and also extending both representa-tion and approximation of surface pressure to the supercritical regime in Zdlattice models.4. We would like to develop new pressure approximation techniques, by findingmore connections between pressure representation/approximation and recentalgorithmic developments for approximate counting. 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