Combinatorial Aspects of Spatial Mixing and New Conditions for Pressure Representation
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Combinatorial aspects of spatial mixing and new conditions for pressure representation by Raimundo Jose´ Briceno˜ Dom´ınguez A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate and Postdoctoral Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2016 c Raimundo Jose´ Briceno˜ Dom´ınguez 2016 Abstract Over 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). SSM has connections with decay of correlations, uniqueness of equilibrium states, ap- proximation algorithms for counting problems, and has been particularly useful for proving special representation formulas and the existence of efficient approx- imation algorithms for (topological) pressure. We look into conditions for the existence of Gibbs distributions satisfying SSM, with special emphasis in hard constrained models, and apply this for pressure representation and approximation d techniques in Z lattice models. Given a locally finite countable graph G and a finite graph H, we consider Hom(G ;H) the set of graph homomorphisms from G to H, and we study Gibbs measures supported on Hom(G ;H). We develop some sufficient and other neces- sary conditions on Hom(G ;H) for the existence of Gibbs specifications satisfying SSM (with exponential decay). In particular, we introduce a new combinatorial condition on the support of Gibbs distributions called topological strong spatial mixing (TSSM). We establish many useful properties of TSSM for studying SSM on systems with hard constraints, and we prove that TSSM combined with SSM is sufficient for having an efficient approximation algorithm for pressure. We also show that TSSM is, in fact, necessary for SSM to hold at high decay rate. Later, we prove a new pressure representation theorem for nearest-neighbour d Gibbs interactions on Z 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-core lattice gas, they apply to certain subsets of both the subcritical and supercritical regions. The main novelty of this work is in the latter, where SSM cannot hold. ii Preface The thesis is split in two main parts (Part I and Part II), based on the following three 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 Ronnie Pavlov. Submitted. (See [16].) Part I (Chapter2, Chapter3, and Chapter4) is mainly based on paper 2 and paper 3. Part II (Chapter5, Chapter6, and Chapter7) is mainly based on paper 1 and paper 2. Some of the writing of this thesis was done while the author was visiting the Simons Institute for the Theory of Computing at University of California, Berkeley. iii Table of Contents Abstract ................................... ii Preface .................................... iii Table of Contents .............................. iv List of Figures ................................ viii Nomenclature ................................x Acknowledgements ............................. xv Dedication .................................. xviii 1 Introduction ...............................1 1.1 Preliminaries . .1 1.2 Results overview . .9 1.3 Thesis structure . 15 I Combinatorial aspects of the strong spatial mixing property . 17 2 Basic definitions and notation ..................... 18 2.1 Graphs . 18 2.1.1 Boards . 19 2.1.2 The d-dimensional integer lattice . 20 2.1.3 Constraint graphs . 21 2.2 Configuration spaces . 23 iv Table of Contents 2.2.1 Homomorphism spaces . 24 d 2.2.2 Shift spaces in Z ..................... 24 2.3 Gibbs measures . 26 2.3.1 Borel probability measures and Markov random fields . 26 2.3.2 Shift-invariant measures . 27 2.3.3 Nearest-neighbour interactions . 28 2.3.4 Hamiltonian and partition function . 29 2.3.5 Gibbs specifications . 31 2.3.6 Nearest-neighbour Gibbs measures . 31 3 Mixing properties ............................ 33 3.1 Spatial mixing properties . 33 3.2 Combinatorial mixing properties . 36 3.2.1 Global properties . 36 3.2.2 Local properties . 37 3.3 Topological strong spatial mixing . 38 d 3.4 TSSM in Z shift spaces . 41 3.4.1 Existence of periodic points . 43 3.4.2 Algorithmic results . 45 d 3.5 Examples: Z n.n. SFTs . 47 3.5.1 A strongly irreducible Z2 n.n. SFT that is not TSSM . 47 3.5.2 A TSSM Z2 n.n. SFT that is not SSF . 52 3.5.3 Arbitrarily large gap, arbitrarily high rate . 54 3.6 Relationship between spatial and combinatorial mixing properties 56 3.6.1 SSM criterion . 56 3.6.2 Uniform bounds of conditional probabilities . 57 3.6.3 SSM with high decay rate . 60 4 SSM in homomorphism spaces ..................... 63 4.1 Dismantlable graphs and homomorphism spaces . 63 4.2 Dismantlable graphs and WSM . 65 4.3 The unique maximal configuration property . 71 4.4 SSM and the UMC property . 72 v Table of Contents 4.5 UMC and chordal/tree decomposable graphs . 75 4.5.1 Chordal/tree decompositions . 75 4.5.2 A natural linear order . 77 4.6 The looped tree case . 82 4.7 Examples: Homomorphism spaces . 87 II Representation and poly-time approximation for pressure .. 94 5 Entropy and pressure .......................... 95 5.1 Topological entropy . 95 5.2 Topological pressure . 99 5.3 Pressure representation . 102 5.3.1 A generalization of previous results . 103 5.3.2 The function pˆ and a new pressure representation theorem 106 6 Classical lattice models and related properties ............ 112 d 6.1 Three Z lattice models . 112 6.1.1 The (ferromagnetic) Potts model . 112 6.1.2 The (multi-type) Widom-Rowlinson model . 113 6.1.3 The hard-core lattice gas model . 113 6.2 The bond random-cluster model . 115 6.3 The site random-cluster model . 118 6.4 Additional properties . 122 6.4.1 Spatial mixing properties . 122 6.4.2 Stochastic dominance . 124 6.5 Exponential convergence of pn ................... 127 6.5.1 Exponential convergence in the Potts model . 127 6.5.2 Exponential convergence in the Widom-Rowlinson model 131 6.5.3 Exponential convergence in the hard-core lattice gas model 135 7 Algorithmic implications ........................ 141 7.1 Previous results . 141 7.2 New results . 142 vi Table of Contents 8 Conclusion ................................ 149 References .................................. 151 vii List of Figures 2 2.1 A sample of the boards Z and T3.................. 19 2.2 The graphs Kn and Kn , for n = 5. 21 2.3 The graph Hj ............................. 22 o 2.4 The graphs S6,S6, and S6 ...................... 22 3.1 The weak and strong spatial mixing properties. 34 3.2 A topologically mixing Z2 shift space. 37 3.3 The topological strong spatial mixing property. 39 3.4 Construction of a periodic point using TSSM. 44 2 3.5 Proof that W = Hom(Z ;K4) does not satisfy TSSM nor SSM. 51 3.6 Representation of the proof of Theorem 3.6.5............ 61 4.1 Decomposition in the proofs of Lemma 4.2.3 and Proposition 4.2.5. 69 4.2 A chordal/tree decomposition. 76 4.3 The n-barbell graph Bn........................ 82 4.4 A “channel” in Hom(Z2;T) with two incompatible extremes. 85 4.5 A dismantlable graph H such that Hom(Z2;H) is not TSSM. 88 4.6 Two incompatible configurations a and b (both in red). 89 4.7 A scenario where SSM fails for any Gibbs (Z2;H;f)-specification. 89 4.8 The graph Hq, for q = 5. ...................... 91 4.9 A Markov chain embedded in a Z2 Markov random field. 92 6.1 A ?-connected set Q (in black), D = ¶ ?Q \ L (in dark grey), and c L (in light grey) for L = Qy;z.................... 121 6.2 A configuration q 2 EC? . On the left, the associated sets S~0(q) and K(q). On the right, the sets I(C?) and O(C?) for G(q) = C?.... 138 viii List of Figures 7.1 Decomposition in the proof of Proposition 7.2.1.......... 143 ix Nomenclature x;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. D(G) Maximum degree, p. 19. G Board graph (simple, connected, locally finite), p. 19. d Z 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. d 4 Lexicographic order in Z , p. 20. d P Lexicographic past of Z , p. 20. Bn n-block, p. 20. x Nomenclature Qy;z Broken [y;z]-block, p. 20. Rn n-rhomboid, p. 21. d ? Superindex for concepts of Z with diagonal edges, p. 21. H Constraint graph (finite, possibly with loops), p. 21. H Constraint graph with loops everywhere, p. 21. Kn Complete graph with n vertices, p. 22. Hj Constraint graph of hard-core model, p. 22. Sn n-star graph, p. 22. A Alphabet, p. 23. W Configuration space (shift space, homomorphism space, etc.), p.