DOCSLIB.ORG

UC Berkeley UC Berkeley Electronic Theses and Dissertations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
UC Berkeley UC Berkeley Electronic Theses and Dissertations UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Mixing time for the Ising model and random walks Permalink https://escholarship.org/uc/item/2b54v885 Author Ding, Jian Publication Date 2011 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Mixing time for the Ising model and random walks by Jian Ding A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Statistics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Yuval Peres, Co-chair Professor Elchanan Mossel, Co-chair Professor David Aldous Professor Fraydoun Rezakhanlou Spring 2011 Mixing time for the Ising model and random walks Copyright 2011 by Jian Ding 1 Abstract Mixing time for the Ising model and random walks by Jian Ding Doctor of Philosophy in Statistics University of California, Berkeley Professor Yuval Peres, Co-chair Professor Elchanan Mossel, Co-chair In this thesis we study the mixing times of Markov chains, e.g., the rate of convergence of Markov chains to stationary measures. We focus on Glauber dynamics for the (classical) Ising model as well as random walks on random graphs. We first provide a complete picture for the evolution of the mixing times and spectral gaps for the mean-field Ising model. In particular, we pin down the scaling window, and prove a cutoff phenomenon at high temperatures, as well as confirm the power law at criticality. We then move to the critical Ising model at Bethe lattice (regular trees), where the criticality corresponds to the reconstruction threshold. We establish that the mixing time and the spectral gap are polynomial in the surface area, which is the height of the tree in this special case. Afterwards, we show that the mixing time of Glauber dynamics for the (ferromagnetic) Ising model on an arbitrary n-vertex graph at any temperature has a lower bound of n log n/4, confirming a folklore theorem in the special case of Ising model. In the second part, we study the random walk on the largest component of the near- supcritical Erd¨os-R´enyi graph. Using a complete characterization of the structure for the near-supercritical random graph, as well as various techniques to bound the mixing times in terms of spectral profile, we obtain the correct order for the mixing time in this regime, which demonstrates a smooth interpolation between the critical and the supercritical regime. i ii Contents 1 Introduction 1 1.1 Glauber dynamics for the Ising model ...................... 2 1.2 Random walks on random graphs ........................ 3 I Mixing time for the Ising model 5 2 Mixing evolution of the mean-field Ising model 6 2.1 Introduction .................................... 6 2.2 Outline of proof .................................. 8 2.2.1 High temperature regime ......................... 8 2.2.2 Low temperature regime ......................... 10 2.3 Preliminaries ................................... 11 2.3.1 The magnetization chain ......................... 11 2.3.2 From magnetization equilibrium to full mixing ............ 13 2.3.3 Contraction and one-dimensional Markov chains ............ 14 2.3.4 Monotone coupling ............................ 16 2.3.5 The spectral gap of the dynamics and its magnetization chain .... 17 2.4 High temperature regime ............................. 19 2.4.1 Cutoff for the magnetization chain ................... 19 2.4.2 Full Mixing of the Glauber dynamics .................. 30 2.4.3 Spectral gap Analysis ........................... 33 2.5 The critical window ................................ 35 2.5.1 Upper bound ............................... 36 2.5.2 Lower bound ............................... 36 2.5.3 Spectral gap analysis ........................... 41 2.6 Low temperature regime ............................. 42 2.6.1 Exponential mixing ............................ 42 2.6.2 Spectral gap analysis ........................... 50 iii 3 Mixing for the Ising-model on regular trees at criticality 51 3.1 Introduction .................................... 51 3.1.1 Background ................................ 53 3.1.2 The critical inverse-gap and mixing-time ................ 54 3.1.3 Techniques and proof ideas ........................ 55 3.1.4 Organization ............................... 56 3.2 Preliminaries ................................... 57 3.2.1 Total-variation mixing .......................... 57 3.2.2 The Ising model on trees ......................... 57 3.2.3 L2-capacity ................................ 58 3.2.4 Spectral gap and log-Sobolev constant ................. 59 3.2.5 From single site dynamics to block dynamics .............. 60 3.2.6 Decomposition of Markov chains ..................... 62 3.3 Spatial mixing of Ising model on trees ...................... 63 3.4 Upper bound on the inverse-gap and mixing time ............... 80 3.4.1 Block dynamics for the tree ....................... 80 3.4.2 Proof of Theorem 6 ............................ 81 3.4.3 Proof of Theorem 3.4.1 .......................... 82 3.5 Lower bounds on the mixing time and inverse-gap ............... 90 3.5.1 Lower bound on the inverse-gap ..................... 90 3.5.2 Lower bound on the mixing-time .................... 92 3.6 Phase transition to polynomial mixing ..................... 100 3.7 Concluding remarks and open problems ..................... 106 4 General lower bound on the mixing for Ising model 107 4.1 Introduction .................................... 107 4.2 Proof of Theorem 9 ................................ 108 II Random walk on random graphs 118 5 Mixing time for the random walk on near-supercritical random graphs 119 5.1 Preliminaries ................................... 119 5.1.1 Cores and kernels ............................. 119 5.1.2 Structure of the supercritical giant component ............. 119 5.1.3 Notions of mixing of the random walk ................. 121 5.1.4 Conductance and mixing ......................... 123 5.1.5 Edge set notations ............................ 123 5.2 Random walk on the 2-core ........................... 124 5.2.1 Mixing time of the 2-core ........................ 124 5.2.2 Local times for the random walk on the 2-core ............. 131 iv 5.3 Mixing on the giant component ......................... 134 5.3.1 Controlling the attached Poisson Galton-Watson trees ......... 135 5.3.2 Proof of Theorem 1: Upper bound on the mixing time ........ 140 5.3.3 Proof of Theorem 1: Lower bound on the mixing time ......... 144 5.4 Mixing in the subcritical regime ......................... 145 v Acknowledgments This dissertation would not have been possible without the help of many people. Greatest thanks goes to my amazing advisor Yuval Peres whose help made this disserta- tion possible. He was a fantastic mentor, who provided a constant stream of problems and offered sharp insights to our questions. He constantly supported me as a GSR such that I could focus on research and visit him at Microsoft research, a truly active place for proba- bility. I enjoyed many times having lunches and dinners with him in various restaurants he brought me to, of which I love Sushi and the Whole foods market are my favorite. Moreover, countless dinners at his place with his family were just so enjoyable and refreshing. I am fortunate to work with Eyal Lubetzky, my main collaborator during my first two years of research. We spent so many hours working together in office and at cafes, and amazingly we could indeed sit down together and produce tex file. I improved my English so much by talking with him. Perhaps too much, such that in a trip to Florida I was told that I had a middle-east accent! Another great experience was working with James Lee, who was a master to challenge me in the just right way. We worked intensively for about half a year on a cover time project joint with Yuval and spent much time thinking of other problems. When we took breaks from the main focus of our potential problems, I enjoyed quite a few introductory lectures by him on discrete geometry and metric embeddings, which initialized my interest in that topic. I am also grateful to the other members of my dissertation committee, Professors Elchanan Mossel, David Aldous and Fraydoun Rezakhanlou for reading my thesis and also for serving on my qualification exam committee. I enjoyed many useful conversations with Elchanan on mathematical research as well as about life in general. The visit to his place at Weizmann institute was a wonderful experience. It was a pleasure to teach as a GSI for Prof. Ani Adhikari, and I learned a lot from her on how to teach effectively and professionally. I am indebted to Jennifer Chayes and Christian Borgs for giving me the opportunity to working as an intern at MSR New England for the last summer, and for exposing me to the new field of game theory and social networks. Special thanks went to my academic siblings. I enjoyed the interactions with Asaf Nach- mias, Yun Long, Lionel Levine, Ron Peled, Gabor Pete, and Manju Krishnapur, from whom I learned not only mathematics, but also how to handle the ups and downs in research. I am grateful to Yun for all the help she provided when I first relocated to Seattle: life would have been much more difficult otherwise. I also thank Tonci Antunovic, Subhroshekhar Ghosh, Weiyang Ning, Julia Ruscher, Elisa Celis for the great time we spent together. I was so fortunate to meet so many wonderful fellow students as well as colleagues, from both our department and Microsoft Research. It was such a
Load more

Recommended publications
  • Perfect Filtering and Double Disjointness
    ANNALES DE L’I. H. P., SECTION B HILLEL FURSTENBERG YUVAL PERES BENJAMIN WEISS Perfect filtering and double disjointness Annales de l’I. H. P., section B, tome 31, no 3 (1995), p. 453-465 <http://www.numdam.org/item?id=AIHPB_1995__31_3_453_0> © Gauthier-Villars, 1995, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section B » (http://www.elsevier.com/locate/anihpb) implique l’accord avec les condi- tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Henri Poincaré, Vol. 31, n° 3, 1995, p. 453-465. Probabilités et Statistiques Perfect filtering and double disjointness Hillel FURSTENBERG, Yuval PERES (*) and Benjamin WEISS Institute of Mathematics, the Hebrew University, Jerusalem. ABSTRACT. - Suppose a stationary process ~ Un ~ is used to select from several stationary processes, i.e., if Un = i then we observe Yn which is the n’th variable in the i’th process. when can we recover the selecting sequence ~ Un ~ from the output sequence ~Yn ~ ? RÉSUMÉ. - Soit ~ Un ~ un processus stationnaire utilise pour la selection de plusieurs processus stationnaires, c’ est-a-dire si Un = i alors on observe Yn qui est le n-ième variable dans le i-ième processus. Quand peut-on reconstruire ~ Un ~ à partir de ~ Yn ~ ? 1. INTRODUCTION Suppose a discrete-time stationary stochastic signal ~ Un ~, taking integer values, is transmitted over a noisy channel.
    [Show full text]
  • Russell David Lyons
    Russell David Lyons Education Case Western Reserve University, Cleveland, OH B.A. summa cum laude with departmental honors, May 1979, Mathematics University of Michigan, Ann Arbor, MI Ph.D., August 1983, Mathematics Sumner Myers Award for best thesis in mathematics Specialization: Harmonic Analysis Thesis: A Characterization of Measures Whose Fourier-Stieltjes Transforms Vanish at Infinity Thesis Advisers: Hugh L. Montgomery, Allen L. Shields Employment Indiana University, Bloomington, IN: James H. Rudy Professor of Mathematics, 2014{present. Indiana University, Bloomington, IN: Adjunct Professor of Statistics, 2006{present. Indiana University, Bloomington, IN: Professor of Mathematics, 1994{2014. Georgia Institute of Technology, Atlanta, GA: Professor of Mathematics, 2000{2003. Indiana University, Bloomington, IN: Associate Professor of Mathematics, 1990{94. Stanford University, Stanford, CA: Assistant Professor of Mathematics, 1985{90. Universit´ede Paris-Sud, Orsay, France: Assistant Associ´e,half-time, 1984{85. Sperry Research Center, Sudbury, MA: Researcher, summers 1976, 1979. Hampshire College Summer Studies in Mathematics, Amherst, MA: Teaching staff, summers 1977, 1978. Visiting Research Positions University of Calif., Berkeley: Visiting Miller Research Professor, Spring 2001. Microsoft Research: Visiting Researcher, Jan.{Mar. 2000, May{June 2004, July 2006, Jan.{June 2007, July 2008{June 2009, Sep.{Dec. 2010, Aug.{Oct. 2011, July{Oct. 2012, May{July 2013, Jun.{Oct. 2014, Jun.{Aug. 2015, Jun.{Aug. 2016, Jun.{Aug. 2017, Jun.{Aug. 2018. Weizmann Institute of Science, Rehovot, Israel: Rosi and Max Varon Visiting Professor, Fall 1997. Institute for Advanced Studies, Hebrew University of Jerusalem, Israel: Winston Fellow, 1996{97. Universit´ede Lyon, France: Visiting Professor, May 1996. University of Wisconsin, Madison, WI: Visiting Associate Professor, Winter 1994.
    [Show full text]
  • Affine Embeddings of Cantor Sets on the Line Arxiv:1607.02849V3
    Affine Embeddings of Cantor Sets on the Line Amir Algom August 5, 2021 Abstract Let s 2 (0; 1), and let F ⊂ R be a self similar set such that 0 < dimH F ≤ s . We prove that there exists δ = δ(s) > 0 such that if F admits an affine embedding into a homogeneous self similar set E and 0 ≤ dimH E − dimH F < δ then (under some mild conditions on E and F ) the contraction ratios of E and F are logarithmically commensurable. This provides more evidence for Conjecture 1.2 of Feng et al.(2014), that states that these contraction ratios are logarithmically commensurable whenever F admits an affine embedding into E (under some mild conditions). Our method is a combination of an argument based on the approach of Feng et al.(2014) with a new result by Hochman(2016), which is related to the increase of entropy of measures under convolutions. 1 Introduction Let A; B ⊂ R. We say that A can be affinely embedded into B if there exists an affine map g : R ! R; g(x) = γ · x + t for γ 6= 0; t 2 R, such that g(A) ⊆ B. The motivating problem of this paper to show that if A and B are two Cantor sets and A can be affinely embedded into B, then the there should be some arithmetic dependence between the scales appearing in the multiscale structures of A and B. This phenomenon is known to occur e.g. when A and B are two self similar sets satisfying the strong separation condition that are Lipschitz equivalent, see Falconer and Marsh(1992).
    [Show full text]
  • Characterization of Cutoff for Reversible Markov Chains
    Characterization of cutoff for reversible Markov chains Riddhipratim Basu ∗ Jonathan Hermon y Yuval Peres z Abstract A sequence of Markov chains is said to exhibit (total variation) cutoff if the conver- gence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of \worst" (in some sense) sets of stationary measure at least α, for some α (0; 1). 2 We also give general bounds on the total variation distance of a reversible chain at time t in terms of the probability that some \worst" set of stationary measure at least α was not hit by time t. As an application of our techniques we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to . 1 Keywords: Cutoff, mixing-time, finite reversible Markov chains, hitting times, trees, maximal inequality. arXiv:1409.3250v4 [math.PR] 18 Feb 2015 ∗Department of Statistics, UC Berkeley, California, USA. E-mail: [email protected]. Supported by UC Berkeley Graduate Fellowship. yDepartment of Statistics, UC Berkeley, USA. E-mail: [email protected]. zMicrosoft Research, Redmond, Washington, USA. E-mail: [email protected]. 1 1 Introduction In many randomized algorithms, the mixing-time of an underlying Markov chain is the main component of the running-time (see [21]). We obtain a tight bound on tmix() (up to an absolute constant independent of ) for lazy reversible Markov chains in terms of hitting times of large sets (Proposition 1.7,(1.6)).
    [Show full text]
  • Amber L. Puha, Phd June 2021 Department of Mathematics Craven Hall 6130 California State University San Marcos Office Phone: 1-760-750-4201 333 S
    Amber L. Puha, PhD June 2021 Department of Mathematics Craven Hall 6130 California State University San Marcos Office Phone: 1-760-750-4201 333 S. Twin Oaks Valley Road Department Phone: 1-760-750-8059 San Marcos, CA 92096-0001 https://faculty.csusm.edu/apuha/ [email protected] Appointments 2010{present, Professor, Department of Mathematics, CSU San Marcos 2020-2021, Sabbatical Leave of Absence, Research Associate and Teaching Visitor Department of Mathematics, UCSD 2013-2014, Sabbatical Leave of Absence, Research Associate and Teaching Visitor Department of Mathematics, UCSD 2010-2011, Professional Leave of Absence 2009{2011, Associate Director, Institute for Pure and Applied Mathematics (IPAM), UCLA 2004-2010, Associate Professor, Department of Mathematics, CSU San Marcos 2009-2010, Professional Leave of Absence 2005-2006, Sabbatical Leave of Absence, Research Associate and Teaching Visitor, Department of Mathematics, UCSD 1999-2004, Assistant Professor, Department of Mathematics, CSU San Marcos 2000-2001 & Spring 2002, Professional Leave of Absence, National Science Foundation Mathematical Sciences Postdoctoral Fellow Research Interests Probability Theory and Stochastic Processes, with emphasis on Stochastic Networks Professional Awards 2021-2024, PI, National Science Foundation Single-Investigator Award, DMS-2054505, $232,433 2020{2021, CSUSM Research, Scholarship and Creative Activity (RSCA) Grant, $7,200 2019{2020, CSUSM Research, Scholarship and Creative Activity (RSCA) Grant, $2,000 2019, National Scholastic Surfing Association,
    [Show full text]
  • Front Matter
    Cambridge University Press 978-1-107-13411-9 — Fractals in Probability and Analysis Christopher J. Bishop , Yuval Peres Frontmatter More Information CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 162 Editorial Board B. BOLLOBAS,´ W. FULTON, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO FRACTALS IN PROBABILITY AND ANALYSIS A mathematically rigorous introduction to fractals which emphasizes examples and fundamental ideas. Building up from basic techniques of geometric measure theory and probability, central topics such as Hausdorff dimension, self-similar sets and Brownian motion are introduced, as are more specialized topics, including Kakeya sets, capacity, percolation on trees and the Traveling Salesman Theorem. The broad range of techniques presented enables key ideas to be highlighted, without the distraction of excessive technicalities. The authors incorporate some novel proofs which are simpler than those available elsewhere. Where possible, chapters are designed to be read independently so the book can be used to teach a variety of courses, with the clear structure offering students an accessible route into the topic. Christopher J. Bishop is a professor in the Department of Mathematics at Stony Brook University. He has made contributions to the theory of function algebras, Kleinian groups, harmonic measure, conformal and quasiconformal mapping, holomorphic dynamics and computational geometry. Yuval Peres is a Principal Researcher at Microsoft Research in Redmond, WA. He is particularly known for his research in topics such as fractals and Hausdorff measure, random walks, Brownian motion, percolation and Markov chain mixing times. In 2016 he was elected to the National Academy of Science. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-13411-9 — Fractals in Probability and Analysis Christopher J.
    [Show full text]
  • CHRISTOPHER J. BISHOP Department of Mathematics Stony Brook University Stony Brook, NY 11794-3651
    CHRISTOPHER J. BISHOP Department of Mathematics Stony Brook University Stony Brook, NY 11794-3651 RESEARCH INTERESTS Real and complex analysis, geometric function theory, conformal dynamics, probability theory, numerical analysis, analysis on fractals, quasiconformal geometry, computational geometry. Some of my more particular interests have included: potential theory, fractal properties of harmonic measure, geometric properties of Brownian motion and other random processes, al- gebras generated by harmonic and holomorphic functions, geometry of hyperbolic manifolds and their covering groups, numerical computation of conformal mappings, multipole methods, optimal meshing algorithms, nonobtuse triangulation, quadrilateral meshing, iteration of entire functions and dimension distorting properties of quasiconformal maps. I program in C, Mathe- matica and Matlab as part of my investigations in these areas, and teach a class on experimental aspects of mathematics. PROFESSIONAL EXPERIENCE Sept. 1997 to present: Professor at Stony Brook University. Sept. 1992 to Aug. 1997: Assoc. professor at Stony Brook. Sept. 1991 to Aug. 1992: Asst. professor at Stony Brook. Sept. 1988 to Aug. 1991: Hedrick Asst. professor at UCLA. Sept. 1987 to Aug. 1988: NSF postdoc at MSRI, Berkeley. EDUCATION Cambridge University, Master of Advanced Study (MASt), 2011. University of Chicago, Mathematics, Ph. D., 1987, Advisor Peter W. Jones. Visiting graduate student and programmer, Dept. of Mathematics, Yale University, 1985-1987. University of Chicago, Mathematics, Master of Science, 1984. Cambridge University, Certificate of Advanced Study (Part III of Math. Tripos), 1983. Michigan State University, Mathematics, Bachelor of Science, 1982. PH.D. STUDENTS Zsuzsanna Gonye, Ph. D. 2001, Geodesics in hyperbolic manifolds, Assoc. Professor, University of West Hungary Karyn Lundberg, Ph. D. 2005, Boundary behavior of conformal mappings, researcher, Lincoln Labs Hrant Hakobyan, Ph.D.
    [Show full text]
  • Diffusive Estimates for Random Walks on Stationary Random Graphs Of
    Diffusive estimates for random walks on stationary random graphs of polynomial growth Shirshendu Ganguly ∗ James R. Lee y Yuval Peres z Abstract G Let (G; ρ) be a stationary random graph, and use Bρ (r) to denote the ball of radius r about ρ G k in G. Suppose that (G; ρ) has annealed polynomial growth, in the sense that E[ Bρ (r) ] 6 O(r ) for some k > 0 and every r > 1. j j Then there is an infinite sequence of times tn at which the random walk Xt on (G; ρ) is at most diffusive: Almost surely (over the choicef ofg (G; ρ)), there is a number C f> 0g such that h 2 i E distG(X ; Xt ) X = ρ, (G; ρ) 6 Ctn n > 1 : 0 n j 0 8 This result is new even in the case when G is a stationary random subgraph of Zd. Combined with the work of Benjamini, Duminil-Copin, Kozma, and Yadin (2015), it implies that G almost surely does not admit a non-constant harmonic function of sublinear growth. To complement this, we argue that passing to a subsequence of times tn is necessary, as there are stationary random graphs of (almost sure) polynomial growth wheref theg random walk is almost surely superdiffusive at an infinite subset of times. Contents 1 Introduction 2 1.1 The absence of non-constant sublinear growth harmonic functions..........3 1.2 Speed via Euclidean embeddings..............................4 1.3 A deterministic example: Planar graphs..........................6 2 Martingales, embeddings, and growth rates8 2.1 Control by martingales....................................9 2.2 Embeddings and growth rates...............................
    [Show full text]
  • PACKING DIMENSION and CARTESIAN PRODUCTS 1. Introduction Marstrand's Product Theorem
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 11, November 1996 PACKING DIMENSION AND CARTESIAN PRODUCTS CHRISTOPHER J. BISHOP AND YUVAL PERES Abstract. We show that for any analytic set A in Rd, its packing dimension dimP (A) can be represented as supB dimH (A B) dimH (B) , where the { d × − } supremum is over all compact sets B in R , and dimH denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if dimP (A) <d.In contrast, we show that the dual quantity infB dimP (A B) dimP (B) , is at least the “lower packing dimension” of A, but{ can be× strictly− greater.} (The lower packing dimension is greater than or equal to the Hausdorff dimension.) 1. Introduction Marstrand’s product theorem ([9]) asserts that if A, B Rd then ⊂ dim (A B) dim (A)+dim (B), H × ≥ H H where “dimH ” denotes Hausdorff dimension. Refining earlier results of Besicovitch and Moran [3], Tricot [14] showed that (1) dim (A B) dim (B) dim (A) , H × − H ≤ P where “dimP ” denotes packing dimension (see the next section for background). Tricot [15] and also Hu and Taylor [6] asked if this is sharp, i.e., given A,canthe right-hand side of (1) be approximated arbitrarily well by appropriate choices of B on the left-hand side? Our first result, proved in Section 3, states that this is possible. Theorem 1.1. For any analytic set A in Rd (2) sup dimH (A B) dimH (B) =dimP(A), B { × − } where the supremum is over all compact sets B Rd.
    [Show full text]
  • Critical Percolation on Certain Nonunimodular Graphs
    New York Journal of Mathematics New York J. Math. 12 (2006) 1–18. Critical percolation on certain nonunimodular graphs Yuval Peres, G´abor Pete and Ariel Scolnicov Abstract. An important conjecture in percolation theory is that almost sure- ly no infinite cluster exists in critical percolation on any transitive graph for which the critical probability is less than 1. Earlier work has established this for the amenable cases Z2 and Zd for large d, as well as for all nonamenable graphs with unimodular automorphism groups. We show that the conjecture holds for the basic classes of nonamenable graphs with nonunimodular automorphism groups: for decorated trees and the nonunimodular Diestel–Leader graphs. We also show that the connection probability between two vertices decays exponentially in their distance. Finally, we prove that critical percolation on the positive part of the lamplighter group has no infinite clusters. Contents 1. Introduction and preliminaries 1 2. Decorated trees 4 3. Diestel–Leader graphs 8 4. The lamplighter group 14 References 17 1. Introduction and preliminaries 1.1. Introduction. We will focus on the following general conjecture of Benjamini and Schramm [BS96] on critical percolation (see Subsection 1.2 for definitions): Conjecture 1.1. Let G be a transitive graph. If pc < 1, then almost surely critical percolation on G has no infinite clusters. Received February 14, 2005. Mathematics Subject Classification. 60B, 60K, 82B, 20F. Key words and phrases. Critical percolation, nonunimodular, nonamenable, Diestel–Leader graphs, grandmother graph, lamplighter group, decay of connection probability. Our work was partially supported by the NSF grant DMS-0244479 (Peres, Pete), and OTKA (Hungarian National Foundation for Scientific Research) grants T30074 and T049398 (Pete).
    [Show full text]
  • Modern Discrete Probability an Essential Toolkit
    Modern Discrete Probability An Essential Toolkit Sebastien´ Roch November 20, 2014 Modern Discrete Probability: An Essential Toolkit Manuscript version of November 20, 2014 Copyright c 2014 Sebastien´ Roch Contents Preface vi I Introduction 1 1 Models 2 1.1 Preliminaries ............................ 2 1.1.1 Review of graph theory ................... 2 1.1.2 Review of Markov chain theory .............. 2 1.2 Percolation ............................. 2 1.3 Random graphs ........................... 2 1.4 Markov random fields ........................ 2 1.5 Random walks on networks ..................... 3 1.6 Interacting particles on finite graphs ................ 3 1.7 Other random structures ...................... 3 II Basic tools: moments, martingales, couplings 4 2 Moments and tails 5 2.1 First moment method ........................ 6 2.1.1 The probabilistic method .................. 6 2.1.2 Markov’s inequality .................... 8 2.1.3 . Random permutations: longest increasing subsequence . 9 2.1.4 . Constraint satisfaction: bound on random k-SAT threshold 11 2.1.5 . Percolation on Zd: existence of a phase transition .... 12 2.2 Second moment method ...................... 16 2.2.1 Chebyshev’s and Paley-Zygmund inequalities ....... 17 2.2.2 . Erdos-R¨ enyi´ graphs: small subgraph containment .... 21 2.2.3 . Erdos-R¨ enyi´ graphs: connectivity threshold ....... 25 i 2.2.4 . Percolation on trees: branching number, weighted sec- ond moment method, and application to the critical value . 28 2.3 Chernoff-Cramer´ method ...................... 32 2.3.1 Tail bounds via the moment-generating function ..... 33 2.3.2 . Randomized algorithms: Johnson-Lindenstrauss, "-nets, and application to compressed sensing .......... 37 2.3.3 . Information theory: Shannon’s theorem ......... 44 2.3.4 .
    [Show full text]
  • Smoothness of Projections, Bernoulli Convolutions, and Dimensions Of
    SMOOTHNESS OF PROJECTIONS, BERNOULLI CONVOLUTIONS, AND THE DIMENSION OF EXCEPTIONS YUVAL PERES AND WILHELM SCHLAG n Abstract. Erd˝os (1939, 1940) studied the distribution ν of the random series ∞ λ ,andshowedthat λ 0 ± ν is singular for infinitely many λ (1/2, 1), and absolutely continuous for a.e. λ in a small interval λ ∈ P (1 δ, 1). Solomyak (1995) proved a conjecture made by Garsia (1962) that ν is absolutely continuous − λ for a.e. λ (1/2, 1). In order to sharpen this result, we have developed a general method that can be used ∈ to estimate the Hausdorffdimension of exceptional parameters in several contexts. In particular, we prove: For any λ0 > 1/2, the set of λ [λ0, 1) such that ν is singular has Hausdorffdimension strictly less than 1. • ∈ λ For any Borel set A d with Hausdorffdimension dim A>(d +1)/2, there are points x A such that • ⊂ R ∈ the pinned distance set x y : y A has positive Lebesgue measure. Moreover, the set of x where this {| − | ∈ } fails has Hausdorffdimension at most d +1 dim A. − Let denote the middle-α Cantor set for α =1 2λ and let K be any compact set. Peres and • Kλ − ⊂ R Solomyak (1998) showed that for a.e. λ (λ0, 1/2) such that dim K +dim > 1, the sum K + has ∈ Kλ Kλ positive length; we show that the set of exceptional λ in this statement has Hausdorffdimension at most 2 dim K dim . − − Kλ0 For any Borel set E d with dim E>2, almost all orthogonal projections of E onto lines through • ⊂ R the origin have nonempty interior, and the exceptional set of lines where this fails has dimension at most d +1 dim E.
    [Show full text]