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