A M E R I C A N M AT H E M AT I C A L S O C I E T Y

Markov Chains and Mixing Times

Second Edition

David A. Levin

Yuval Peres

With contributions byElizabeth L. Wilmer

Markov Chains and Mixing Times

Second Edition

David A. LevinUniversity of Oregon

Yuval PeresMicrosoft Research

With contributions by

Elizabeth L. Wilmer

With a chapter on “Coupling from the Past” by

James G. Propp and David B. Wilson

A M E R I C A N M AT H E M AT I C A L S O C I E T Y

Providence, Rhode Island

10.1090/mbk/107

2010 Mathematics Subject Classification. Primary 60J10, 60J27, 60B15, 60C05, 65C05,60K35, 68W20, 68U20, 82C22.

FRONT COVER: The figure on the bottom left of the front cover, courtesy ofDavid B. Wilson, is a uniformly random lozenge tiling of a hexagon (see Section25.2). The figure on the bottom center, also from David B. Wilson, is a randomsample of an Ising model at its critical temperature (see Sections 3.3.5 and 25.2)with mixed boundary conditions. The figure on the bottom right, courtesy of EyalLubetzky, is a portion of an expander graph (see Section 13.6).

For additional information and updates on this book, visitwww.ams.org/bookpages/mbk-107

Library of Congress Cataloging-in-Publication Data

Names: Levin, David Asher, 1971- | Peres, Y. (Yuval) | Wilmer, Elizabeth L. (Elizabeth Lee),1970- | Propp, James, 1960- | Wilson, David B. (David Bruce)

Title: Markov chains and mixing times / David A. Levin, Yuval Peres ; with contributions byElizabeth L. Wilmer.

Description: Second edition. | Providence, Rhode Island : American Mathematical Society, [2017]| “With a chapter on Coupling from the past, by James G. Propp and David B. Wilson.” |Includes bibliographical references and indexes.

Identifiers: LCCN 2017017451 | ISBN 9781470429621 (alk. paper)Subjects: LCSH: Markov processes–Textbooks. | Distribution (Probability theory)–Textbooks.

| AMS: Probability theory and stochastic processes – Markov processes – Markov chains(discrete-time Markov processes on discrete state spaces). msc | Probability theory and sto-chastic processes – Markov processes – Continuous-time Markov processes on discrete statespaces. msc | Probability theory and stochastic processes – Probability theory on algebraicand topological structures – Probability measures on groups or semigroups, Fourier transforms,factorization. msc | Probability theory and stochastic processes – Combinatorial probability– Combinatorial probability. msc | Numerical analysis – Probabilistic methods, simulationand stochastic differential equations – Monte Carlo methods. msc | Probability theory andstochastic processes – Special processes – Interacting random processes; statistical mechan-ics type models; percolation theory. msc | Computer science – Algorithms – Randomizedalgorithms. msc | Computer science – Computing methodologies and applications – Simula-tion. msc | Statistical mechanics, structure of matter – Time-dependent statistical mechanics(dynamic and nonequilibrium) – Interacting particle systems. msc

Classification: LCC QA274.7 .L48 2017 | DDC 519.2/33–dc23 LC record available at https://lccn.loc.gov/2017017451

Copying and reprinting. Individual readers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy select pages foruse in teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Permissions to reuseportions of AMS publication content are handled by Copyright Clearance Center’s RightsLink�service. For more information, please visit: http://www.ams.org/rightslink.

Send requests for translation rights and licensed reprints to reprint-permission@ams.org.Excluded from these provisions is material for which the author holds copyright. In such cases,

requests for permission to reuse or reprint material should be addressed directly to the author(s).Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of thefirst page of each article within proceedings volumes.

Second edition c© 2017 by the authors. All rights reserved.

First edition c© 2009 by the authors. All rights reserved.Printed in the United States of America.

©∞ The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12

Contents

Preface ixPreface to the Second Edition ixPreface to the First Edition ixOverview xiFor the Reader xiiFor the Instructor xiiiFor the Expert xiv

Acknowledgements xvi

Part I: Basic Methods and Examples 1

Chapter 1. Introduction to Finite Markov Chains 21.1. Markov Chains 21.2. Random Mapping Representation 51.3. Irreducibility and Aperiodicity 71.4. Random Walks on Graphs 81.5. Stationary Distributions 91.6. Reversibility and Time Reversals 131.7. Classifying the States of a Markov Chain* 15Exercises 17Notes 19

Chapter 2. Classical (and Useful) Markov Chains 212.1. Gambler’s Ruin 212.2. Coupon Collecting 222.3. The Hypercube and the Ehrenfest Urn Model 232.4. The Polya Urn Model 252.5. Birth-and-Death Chains 262.6. Random Walks on Groups 272.7. Random Walks on Z and Reflection Principles 30Exercises 34Notes 35

Chapter 3. Markov Chain Monte Carlo: Metropolis and Glauber Chains 383.1. Introduction 383.2. Metropolis Chains 383.3. Glauber Dynamics 41Exercises 45Notes 45

iii

iv CONTENTS

Chapter 4. Introduction to Markov Chain Mixing 474.1. Total Variation Distance 474.2. Coupling and Total Variation Distance 494.3. The Convergence Theorem 524.4. Standardizing Distance from Stationarity 534.5. Mixing Time 544.6. Mixing and Time Reversal 554.7. �p Distance and Mixing 56Exercises 57Notes 58

Chapter 5. Coupling 605.1. Definition 605.2. Bounding Total Variation Distance 615.3. Examples 625.4. Grand Couplings 69Exercises 73Notes 73

Chapter 6. Strong Stationary Times 756.1. Top-to-Random Shuffle 756.2. Markov Chains with Filtrations 766.3. Stationary Times 776.4. Strong Stationary Times and Bounding Distance 786.5. Examples 816.6. Stationary Times and Cesaro Mixing Time 836.7. Optimal Strong Stationary Times* 84Exercises 85Notes 86

Chapter 7. Lower Bounds on Mixing Times 877.1. Counting and Diameter Bounds 877.2. Bottleneck Ratio 887.3. Distinguishing Statistics 917.4. Examples 95Exercises 97Notes 98

Chapter 8. The Symmetric Group and Shuffling Cards 998.1. The Symmetric Group 998.2. Random Transpositions 1018.3. Riffle Shuffles 106Exercises 109Notes 112

Chapter 9. Random Walks on Networks 1159.1. Networks and Reversible Markov Chains 1159.2. Harmonic Functions 1169.3. Voltages and Current Flows 117

CONTENTS v

9.4. Effective Resistance 1189.5. Escape Probabilities on a Square 123Exercises 125Notes 126

Chapter 10. Hitting Times 12710.1. Definition 12710.2. Random Target Times 12810.3. Commute Time 13010.4. Hitting Times on Trees 13310.5. Hitting Times for Eulerian Graphs 13510.6. Hitting Times for the Torus 13610.7. Bounding Mixing Times via Hitting Times 13910.8. Mixing for the Walk on Two Glued Graphs 143Exercises 145Notes 147

Chapter 11. Cover Times 14911.1. Definitions 14911.2. The Matthews Method 14911.3. Applications of the Matthews Method 15111.4. Spanning Tree Bound for Cover Time 15311.5. Waiting for All Patterns in Coin Tossing 155Exercises 157Notes 157

Chapter 12. Eigenvalues 16012.1. The Spectral Representation of a Reversible Transition Matrix 16012.2. The Relaxation Time 16212.3. Eigenvalues and Eigenfunctions of Some Simple Random Walks 16412.4. Product Chains 16812.5. Spectral Formula for the Target Time 17112.6. An �2 Bound 17112.7. Time Averages 172Exercises 176Notes 177

Part II: The Plot Thickens 179

Chapter 13. Eigenfunctions and Comparison of Chains 18013.1. Bounds on Spectral Gap via Contractions 18013.2. The Dirichlet Form and the Bottleneck Ratio 18113.3. Simple Comparison of Markov Chains 18513.4. The Path Method 18813.5. Wilson’s Method for Lower Bounds 19313.6. Expander Graphs* 196Exercises 198Notes 199

vi CONTENTS

Chapter 14. The Transportation Metric and Path Coupling 20114.1. The Transportation Metric 20114.2. Path Coupling 20314.3. Rapid Mixing for Colorings 20714.4. Approximate Counting 209Exercises 213Notes 214

Chapter 15. The Ising Model 21615.1. Fast Mixing at High Temperature 21615.2. The Complete Graph 21915.3. The Cycle 22015.4. The Tree 22115.5. Block Dynamics 22415.6. Lower Bound for the Ising Model on Square* 227Exercises 229Notes 230

Chapter 16. From Shuffling Cards to Shuffling Genes 23316.1. Random Adjacent Transpositions 23316.2. Shuffling Genes 237Exercises 241Notes 242

Chapter 17. Martingales and Evolving Sets 24417.1. Definition and Examples 24417.2. Optional Stopping Theorem 24517.3. Applications 24717.4. Evolving Sets 25017.5. A General Bound on Return Probabilities 25517.6. Harmonic Functions and the Doob h-Transform 25717.7. Strong Stationary Times from Evolving Sets 258Exercises 260Notes 260

Chapter 18. The Cutoff Phenomenon 26218.1. Definition 26218.2. Examples of Cutoff 26318.3. A Necessary Condition for Cutoff 26818.4. Separation Cutoff 269Exercises 270Notes 270

Chapter 19. Lamplighter Walks 27319.1. Introduction 27319.2. Relaxation Time Bounds 27419.3. Mixing Time Bounds 27619.4. Examples 278Exercises 278Notes 279

CONTENTS vii

Chapter 20. Continuous-Time Chains* 28120.1. Definitions 28120.2. Continuous-Time Mixing 28320.3. Spectral Gap 28520.4. Product Chains 286Exercises 290Notes 291

Chapter 21. Countable State Space Chains* 29221.1. Recurrence and Transience 29221.2. Infinite Networks 29421.3. Positive Recurrence and Convergence 29621.4. Null Recurrence and Convergence 30121.5. Bounds on Return Probabilities 302Exercises 303Notes 305

Chapter 22. Monotone Chains 30622.1. Introduction 30622.2. Stochastic Domination 30722.3. Definition and Examples of Monotone Markov Chains 30922.4. Positive Correlations 31022.5. The Second Eigenfunction 31422.6. Censoring Inequality 31522.7. Lower Bound on d 32022.8. Proof of Strassen’s Theorem 321Exercises 322Notes 323

Chapter 23. The Exclusion Process 32423.1. Introduction 32423.2. Mixing Time of k-Exclusion on the n-Path 32923.3. Biased Exclusion 330Exercises 334Notes 334

Chapter 24. Cesaro Mixing Time, Stationary Times, and HittingLarge Sets 336

24.1. Introduction 33624.2. Equivalence of tstop, tCes, and tG for Reversible Chains 33824.3. Halting States and Mean-Optimal Stopping Times 34024.4. Regularity Properties of Geometric Mixing Times* 34124.5. Equivalence of tG and tH 34224.6. Upward Skip-Free Chains 34424.7. tH(α) Are Comparable for α ≤ 1/2 34524.8. An Upper Bound on trel 34624.9. Application to Robustness of Mixing 346Exercises 347Notes 348

viii CONTENTS

Chapter 25. Coupling from the Past 34925.1. Introduction 34925.2. Monotone CFTP 35025.3. Perfect Sampling via Coupling from the Past 35525.4. The Hardcore Model 35625.5. Random State of an Unknown Markov Chain 358Exercise 359Notes 359

Chapter 26. Open Problems 36026.1. The Ising Model 36026.2. Cutoff 36126.3. Other Problems 36126.4. Update: Previously Open Problems 362

Appendix A. Background Material 365A.1. Probability Spaces and Random Variables 365A.2. Conditional Expectation 371A.3. Strong Markov Property 374A.4. Metric Spaces 375A.5. Linear Algebra 376A.6. Miscellaneous 376Exercise 376

Appendix B. Introduction to Simulation 377B.1. What Is Simulation? 377B.2. Von Neumann Unbiasing* 378B.3. Simulating Discrete Distributions and Sampling 379B.4. Inverse Distribution Function Method 380B.5. Acceptance-Rejection Sampling 380B.6. Simulating Normal Random Variables 383B.7. Sampling from the Simplex 384B.8. About Random Numbers 384B.9. Sampling from Large Sets* 385Exercises 388Notes 391

Appendix C. Ergodic Theorem 392C.1. Ergodic Theorem* 392Exercise 393

Appendix D. Solutions to Selected Exercises 394

Bibliography 425

Notation Index 439

Index 441

Preface

Preface to the Second Edition

Since the publication of the first edition, the field of mixing times has continuedto enjoy rapid expansion. In particular, many of the open problems posed in thefirst edition have been solved. The book has been used in courses at numerousuniversities, motivating us to update it.

In the eight years since the first edition appeared, we have made correctionsand improvements throughout the book. We added three new chapters: Chapter 22on monotone chains, Chapter 23 on the exclusion process, and Chapter 24, whichrelates mixing times and hitting time parameters to stationary stopping times.Chapter 4 now includes an introduction to mixing times in �p, which reappear inChapter 10. The latter chapter has several new topics, including estimates for hit-ting times on trees and Eulerian digraphs. A bound for cover times using spanningtrees has been added to Chapter 11, which also now includes a general bound oncover times for regular graphs. The exposition in Chapter 6 and Chapter 17 nowemploys filtrations rather than relying on the random mapping representation. Toreflect the key developments since the first edition, especially breakthroughs on theIsing model and the cutoff phenomenon, the Notes at the end of chapters and theopen problems have been updated.

We thank the many careful readers who sent us comments and corrections:Anselm Adelmann, Amitabha Bagchi, Nathanael Berestycki, Olena Bormashenko,Krzysztof Burdzy, Gerandy Brito, Darcy Camargo, Varsha Dani, Sukhada Fad-navis, Tertuliano Franco, Alan Frieze, Reza Gheissari, Jonathan Hermon, AnderHolroyd, Kenneth Hu, John Jiang, Svante Janson, Melvin Kianmanesh Rad, YinTat Lee, Zhongyang Li, Eyal Lubetzky, Abbas Mehrabian, R. Misturini, L. Mor-gado, Asaf Nachmias, Fedja Nazarov, Joe Neeman, Ross Pinsky, Anthony Quas,Miklos Racz, Dinah Shender, N. J. A. Sloane, Jeff Steif, Izabella Stuhl, Jan Swart,Ryokichi Tanaka, Daniel Wu, and Zhen Zhu. We are particularly grateful to DanielJerison, Pawel Pralat, and Perla Sousi, who sent us long lists of insightful comments.

Preface to the First Edition

Markov first studied the stochastic processes that came to be named after himin 1906. Approximately a century later, there is an active and diverse interdisci-plinary community of researchers using Markov chains in computer science, physics,statistics, bioinformatics, engineering, and many other areas.

The classical theory of Markov chains studied fixed chains, and the goal wasto estimate the rate of convergence to stationarity of the distribution at time t, ast → ∞. In the past two decades, as interest in chains with large state spaces hasincreased, a different asymptotic analysis has emerged. Some target distance to

ix

x PREFACE

the stationary distribution is prescribed; the number of steps required to reach thistarget is called the mixing time of the chain. Now, the goal is to understand howthe mixing time grows as the size of the state space increases.

The modern theory of Markov chain mixing is the result of the convergence, inthe 1980s and 1990s, of several threads. (We mention only a few names here; seethe chapter Notes for references.)

For statistical physicists Markov chains become useful in Monte Carlo simu-lation, especially for models on finite grids. The mixing time can determine therunning time for simulation. However, Markov chains are used not only for sim-ulation and sampling purposes, but also as models of dynamical processes. Deepconnections were found between rapid mixing and spatial properties of spin systems,e.g., by Dobrushin, Shlosman, Stroock, Zegarlinski, Martinelli, and Olivieri.

In theoretical computer science, Markov chains play a key role in sampling andapproximate counting algorithms. Often the goal was to prove that the mixingtime is polynomial in the logarithm of the state space size. (In this book, we aregenerally interested in more precise asymptotics.)

At the same time, mathematicians including Aldous and Diaconis were inten-sively studying card shuffling and other random walks on groups. Both spectralmethods and probabilistic techniques, such as coupling, played important roles.Alon and Milman, Jerrum and Sinclair, and Lawler and Sokal elucidated the con-nection between eigenvalues and expansion properties. Ingenious constructions of“expander” graphs (on which random walks mix especially fast) were found usingprobability, representation theory, and number theory.

In the 1990s there was substantial interaction between these communities, ascomputer scientists studied spin systems and as ideas from physics were used forsampling combinatorial structures. Using the geometry of the underlying graph tofind (or exclude) bottlenecks played a key role in many results.

There are many methods for determining the asymptotics of convergence tostationarity as a function of the state space size and geometry. We hope to presentthese exciting developments in an accessible way.

We will only give a taste of the applications to computer science and statisticalphysics; our focus will be on the common underlying mathematics. The prerequi-sites are all at the undergraduate level. We will draw primarily on probability andlinear algebra, but we will also use the theory of groups and tools from analysiswhen appropriate.

Why should mathematicians study Markov chain convergence? First of all, it isa lively and central part of modern probability theory. But there are ties to severalother mathematical areas as well. The behavior of the random walk on a graphreveals features of the graph’s geometry. Many phenomena that can be observed inthe setting of finite graphs also occur in differential geometry. Indeed, the two fieldsenjoy active cross-fertilization, with ideas in each playing useful roles in the other.Reversible finite Markov chains can be viewed as resistor networks; the resultingdiscrete potential theory has strong connections with classical potential theory. Itis amusing to interpret random walks on the symmetric group as card shuffles—andreal shuffles have inspired some extremely serious mathematics—but these chainsare closely tied to core areas in algebraic combinatorics and representation theory.

OVERVIEW xi

In the spring of 2005, mixing times of finite Markov chains were a major themeof the multidisciplinary research program Probability, Algorithms, and StatisticalPhysics, held at the Mathematical Sciences Research Institute. We began work onthis book there.

Overview

We have divided the book into two parts.In Part I, the focus is on techniques, and the examples are illustrative and

accessible. Chapter 1 defines Markov chains and develops the conditions necessaryfor the existence of a unique stationary distribution. Chapters 2 and 3 both coverexamples. In Chapter 2, they are either classical or useful—and generally both;we include accounts of several chains, such as the gambler’s ruin and the couponcollector, that come up throughout probability. In Chapter 3, we discuss Glauberdynamics and the Metropolis algorithm in the context of “spin systems.” Thesechains are important in statistical mechanics and theoretical computer science.

Chapter 4 proves that, under mild conditions, Markov chains do, in fact, con-verge to their stationary distributions and defines total variation distance andmixing time, the key tools for quantifying that convergence. The techniques ofChapters 5, 6, and 7, on coupling, strong stationary times, and methods for lowerbounding distance from stationarity, respectively, are central to the area.

In Chapter 8, we pause to examine card shuffling chains. Random walks on thesymmetric group are an important mathematical area in their own right, but wehope that readers will appreciate a rich class of examples appearing at this stagein the exposition.

Chapter 9 describes the relationship between random walks on graphs andelectrical networks, while Chapters 10 and 11 discuss hitting times and cover times.

Chapter 12 introduces eigenvalue techniques and discusses the role of the re-laxation time (the reciprocal of the spectral gap) in the mixing of the chain.

In Part II, we cover more sophisticated techniques and present several detailedcase studies of particular families of chains. Much of this material appears here forthe first time in textbook form.

Chapter 13 covers advanced spectral techniques, including comparison of Dirich-let forms and Wilson’s method for lower bounding mixing.

Chapters 14 and 15 cover some of the most important families of “large” chainsstudied in computer science and statistical mechanics and some of the most impor-tant methods used in their analysis. Chapter 14 introduces the path couplingmethod, which is useful in both sampling and approximate counting. Chapter 15looks at the Ising model on several different graphs, both above and below thecritical temperature.

Chapter 16 revisits shuffling, looking at two examples—one with an applicationto genomics—whose analysis requires the spectral techniques of Chapter 13.

Chapter 17 begins with a brief introduction to martingales and then presentssome applications of the evolving sets process.

Chapter 18 considers the cutoff phenomenon. For many families of chains wherewe can prove sharp upper and lower bounds on mixing time, the distance fromstationarity drops from near 1 to near 0 over an interval asymptotically smallerthan the mixing time. Understanding why cutoff is so common for families ofinterest is a central question.

xii PREFACE

Chapter 19, on lamplighter chains, brings together methods presented through-out the book. There are many bounds relating parameters of lamplighter chainsto parameters of the original chain: for example, the mixing time of a lamplighterchain is of the same order as the cover time of the base chain.

Chapters 20 and 21 introduce two well-studied variants on finite discrete timeMarkov chains: continuous time chains and chains with countable state spaces.In both cases we draw connections with aspects of the mixing behavior of finitediscrete-time Markov chains.

Chapter 25, written by Propp and Wilson, describes the remarkable construc-tion of coupling from the past, which can provide exact samples from the stationarydistribution.

Chapter 26 closes the book with a list of open problems connected to materialcovered in the book.

For the Reader

Starred sections, results, and chapters contain material that either digressesfrom the main subject matter of the book or is more sophisticated than whatprecedes them and may be omitted.

Exercises are found at the ends of chapters. Some (especially those whoseresults are applied in the text) have solutions at the back of the book. We of courseencourage you to try them yourself first!

The Notes at the ends of chapters include references to original papers, sugges-tions for further reading, and occasionally “complements.” These generally containrelated material not required elsewhere in the book—sharper versions of lemmas orresults that require somewhat greater prerequisites.

The Notation Index at the end of the book lists many recurring symbols.Much of the book is organized by method, rather than by example. The reader

may notice that, in the course of illustrating techniques, we return again and againto certain families of chains—random walks on tori and hypercubes, simple cardshuffles, proper colorings of graphs. In our defense we offer an anecdote.

In 1991 one of us (Y. Peres) arrived as a postdoc at Yale and visited ShizuoKakutani, whose rather large office was full of books and papers, with bookcasesand boxes from floor to ceiling. A narrow path led from the door to Kakutani’s desk,which was also overflowing with papers. Kakutani admitted that he sometimes haddifficulty locating particular papers, but he proudly explained that he had found away to solve the problem. He would make four or five copies of any really interestingpaper and put them in different corners of the office. When searching, he would besure to find at least one of the copies. . . .

Cross-references in the text and the Index should help you track earlier occur-rences of an example. You may also find the chapter dependency diagrams belowuseful.

We have included brief accounts of some background material in Appendix A.These are intended primarily to set terminology and notation, and we hope youwill consult suitable textbooks for unfamiliar material.

Be aware that we occasionally write symbols representing a real number whenan integer is required (see, e.g., the

(nδk

)’s in the proof of Proposition 13.37). We

FOR THE INSTRUCTOR xiii

hope the reader will realize that this omission of floor or ceiling brackets (and thedetails of analyzing the resulting perturbations) is in her or his best interest asmuch as it is in ours.

For the Instructor

The prerequisites this book demands are a first course in probability, linearalgebra, and, inevitably, a certain degree of mathematical maturity. When intro-ducing material which is standard in other undergraduate courses—e.g., groups—weprovide definitions but often hope the reader has some prior experience with theconcepts.

In Part I, we have worked hard to keep the material accessible and engagingfor students. (Starred material is more sophisticated and is not required for whatfollows immediately; they can be omitted.)

Here are the dependencies among the chapters of Part I:

Chapters 1 through 7, shown in gray, form the core material, but there areseveral ways to proceed afterwards. Chapter 8 on shuffling gives an early richapplication but is not required for the rest of Part I. A course with a probabilisticfocus might cover Chapters 9, 10, and 11. To emphasize spectral methods andcombinatorics, cover Chapters 8 and 12 and perhaps continue on to Chapters 13and 16.

While our primary focus is on chains with finite state spaces run in discrete time,continuous-time and countable-state-space chains are both discussed—in Chapters20 and 21, respectively.

We have also included Appendix B, an introduction to simulation methods, tohelp motivate the study of Markov chains for students with more applied interests.A course leaning towards theoretical computer science and/or statistical mechan-ics might start with Appendix B, cover the core material, and then move on toChapters 14, 15, and 25.

Of course, depending on the interests of the instructor and the ambitions andabilities of the students, any of the material can be taught! Below we includea full diagram of dependencies of chapters. Its tangled nature results from theinterconnectedness of the area: a given technique can be applied in many situations,while a particular problem may require several techniques for full analysis.

xiv PREFACE

The logical dependencies of chapters. The core Chapters 1through 7 are in dark gray, the rest of Part I is in light gray,and Part II is in white.

For the Expert

Several other recent books treat Markov chain mixing. Our account is morecomprehensive than those of Haggstrom (2002), Jerrum (2003), or Montene-gro and Tetali (2006), yet not as exhaustive as Aldous and Fill (1999). Nor-ris (1998) gives an introduction to Markov chains and their applications but doesnot focus on mixing. Since this is a textbook, we have aimed for accessibility andcomprehensibility, particularly in Part I.

What is different or novel in our approach to this material?

– Our approach is probabilistic whenever possible. We also integrate “classi-cal” material on networks, hitting times, and cover times and demonstrateits usefulness for bounding mixing times.

– We provide an introduction to several major statistical mechanics models,most notably the Ising model, and collect results on them in one place.

FOR THE EXPERT xv

– We give expository accounts of several modern techniques and examples,including evolving sets, the cutoff phenomenon, lamplighter chains, andthe L-reversal chain.

– We systematically treat lower bounding techniques, including several ap-plications of Wilson’s method.

– We use the transportation metric to unify our account of path couplingand draw connections with earlier history.

– We present an exposition of coupling from the past by Propp and Wilson,the originators of the method.

Acknowledgements

The authors thank the Mathematical Sciences Research Institute, the NationalScience Foundation VIGRE grant to the Department of Statistics at the Universityof California, Berkeley, and National Science Foundation grants DMS-0244479 andDMS-0104073 for support. We also thank Hugo Rossi for suggesting we embark onthis project. Thanks to Blair Ahlquist, Tonci Antunovic, Elisa Celis, Paul Cuff,Jian Ding, Ori Gurel-Gurevich, Tom Hayes, Itamar Landau, Yun Long, KarolaMeszaros, Shobhana Murali, Weiyang Ning, Tomoyuki Shirai, Walter Sun, Sith-parran Vanniasegaram, and Ariel Yadin for corrections to an earlier version andmaking valuable suggestions. Yelena Shvets made the illustration in Section 6.5.4.The simulations of the Ising model in Chapter 15 are due to Raissa D’Souza. Wethank Laszlo Lovasz for useful discussions. We are indebted to Alistair Sinclair forhis work co-organizing the M.S.R.I. program Probability, Algorithms, and Statisti-cal Physics in 2005, where work on this book began. We thank Robert Calhounfor technical assistance.

Finally, we are greatly indebted to David Aldous and Persi Diaconis, who initi-ated the modern point of view on finite Markov chains and taught us much of whatwe know about the subject.

xvi

Bibliography

The pages on which a reference appears follow the symbol ↑.

Ahlfors, L. V. 1978. Complex analysis, 3rd ed., McGraw-Hill Book Co., New York. An introduc-tion to the theory of analytic functions of one complex variable; International Series in Pure andApplied Mathematics. ↑126Aldous, D. J. 1982. Some inequalities for reversible Markov chains, J. London Math. Soc. (2)25, no. 3, 564–576. ↑337, 348Aldous, D. J. 1983a. On the time taken by random walks on finite groups to visit every state, Z.

Wahrsch. Verw. Gebiete 62, no. 3, 361–374. ↑158Aldous, D. J. 1983b. Random walks on finite groups and rapidly mixing Markov chains, Seminaron probability, XVII, Lecture Notes in Math., vol. 986, Springer, Berlin, pp. 243–297. ↑58, 242Aldous, D. J. 1989a. An introduction to covering problems for random walks on graphs, J. The-oret. Probab. 2, no. 1, 87–89. ↑158Aldous, D. J. 1989b. Lower bounds for covering times for reversible Markov chains and randomwalks on graphs, J. Theoret. Probab. 2, no. 1, 91–100. ↑158Aldous, D. J. 1990. A random walk construction of uniform spanning trees and uniform labelledtrees, SIAM Journal on Discrete Mathematics 3, 450–465. ↑359Aldous, D. J. 1991a. Random walk covering of some special trees, J. Math. Anal. Appl. 157,no. 1, 271–283. ↑157Aldous, D. J. 1991b. Threshold limits for cover times, J. Theoret. Probab. 4, no. 1, 197–211.↑158, 279Aldous, D. J. 1995. On simulating a Markov chain stationary distribution when transition prob-abilities are unknown (D. Aldous, P. Diaconis, J. Spencer, and J. M. Steele, eds.), IMA Volumesin Mathematics and Its Applications, vol. 72, Springer-Verlag. ↑358Aldous, D. J. 1999. unpublished note. ↑199Aldous, D. J. 2004, American Institute of Mathematics (AIM) research workshop “Sharp Thresh-olds for Mixing Times” (Palo Alto, December 2004), available at http://www.aimath.org/WWN/mixingtimes. ↑271Aldous, D. and P. Diaconis. 1986. Shuffling cards and stopping times, Amer. Math. Monthly 93,no. 5, 333–348. ↑58, 86, 95, 113Aldous, D. and P. Diaconis. 1987. Strong uniform times and finite random walks, Adv. in Appl.

Math. 8, no. 1, 69–97. ↑86Aldous, D. and P. Diaconis. 2002. The asymmetric one-dimensional constrained Ising model:Rigorous results, J. Statist. Phys. 107, no. 5-6, 945–975. ↑98Aldous, D. and J. Fill. 1999. Reversible Markov chains and random walks on graphs, available athttp://www.stat.berkeley.edu/~aldous/RWG/book.html. ↑xiv, 19, 58, 141, 147, 149, 158, 360Aleliunas, R., R. M. Karp, R. J. Lipton, L. Lovasz, and C. Rackoff. 1979. Random walks, uni-versal traversal sequences, and the complexity of maze problems, 20th Annual Symposium onFoundations of Computer Science (San Juan, Puerto Rico, 1979), IEEE, New York, pp. 218–223.↑158Alon, N. 1986. Eigenvalues and expanders, Combinatorica 6, no. 2, 83–96. ↑199Alon, N. and V. D. Milman. 1985. λ1, isoperimetric inequalities for graphs, and superconcentra-tors, J. Combin. Theory Ser. B 38, no. 1, 73–88. ↑199Alon, N. and J. H. Spencer. 2008. The probabilistic method, 3rd ed., Wiley-Interscience Seriesin Discrete Mathematics and Optimization, John Wiley & Sons Inc., Hoboken, NJ. With anappendix on the life and work of Paul Erdos. ↑319

425

426 BIBLIOGRAPHY

Anantharam, V. and P. Tsoucas. 1989. A proof of the Markov chain tree theorem, Statistics andProbability Letters 8, 189–192. ↑359Angel, O., Y. Peres, and D. B. Wilson. 2008. Card shuffling and Diophantine approximation,Ann. Appl. Probab. 18, no. 3, 1215–1231. ↑Archer, A. F. 1999. A modern treatment of the 15 puzzle, Amer. Math. Monthly 106, no. 9,793–799. ↑404Artin, M. 1991. Algebra, Prentice Hall Inc., Englewood Cliffs, NJ. ↑35, 112Asmussen, S., P. Glynn, and H. Thorisson. 1992. Stationary detection in the initial transientproblem, ACM Transactions on Modeling and Computer Simulation 2, 130–157. ↑358Balazs, M. and A. Folly. 2016. An electric network for nonreversible Markov chains, Amer. Math.Monthly 123, no. 7, 657–682. ↑148Barlow, M. T. 2017. Random walks and heat kernels on graphs, Cambridge University Press.↑305Barlow, M. T., T. Coulhon, and T. Kumagai. 2005. Characterization of sub-Gaussian heat kernelestimates on strongly recurrent graphs, Comm. Pure Appl. Math. 58, no. 12, 1642–1677. ↑305Barnes, G. and U. Feige. 1996. Short random walks on graphs, SIAM J. Discrete Math. 9, no. 1,19–28. ↑158Barrera, J., B. Lachaud, and B. Ycart. 2006. Cut-off for n-tuples of exponentially convergingprocesses, Stochastic Process. Appl. 116, no. 10, 1433–1446. ↑291Basharin, G. P., A. N. Langville, and V. A. Naumov. 2004. The life and work of A. A. Markov,Linear Algebra Appl. 386, 3–26. ↑19Basu, R., J. Hermon, and Y. Peres. 2017. Characterization of cutoff for reversible Markov chains,Ann. Probab. 45, no. 3, 1448–1487. ↑272Baxter, J. R. and R. V. Chacon. 1976. Stopping times for recurrent Markov processes, Illinois J.Math. 20, no. 3, 467–475. ↑Baxter, R. J. 1982. Exactly solved models in statistical mechanics, Academic Press. ↑350Bayer, D. and P. Diaconis. 1992. Trailing the dovetail shuffle to its lair, Ann. Appl. Probab. 2,no. 2, 294–313. ↑111, 113, 114, 178Ben-Hamou, A. and J. Salez. 2017. Cutoff for non-backtracking random walks on sparse randomgraphs, Ann. Probab. 45, no. 3, 1752–1770. ↑272Benjamin, A. T. and J. J. Quinn. 2003. Proofs that really count: The art of combinatorial proof,Dolciani Mathematical Expositions, vol. 27, Math. Assoc. Amer., Washington, D. C. ↑210Benjamini, I., N. Berger, C. Hoffman, and E. Mossel. 2005. Mixing times of the biased cardshuffling and the asymmetric exclusion process, Trans. Amer. Math. Soc. 357, no. 8, 3013–3029(electronic). ↑335Berestycki, N., E. Lubetzky, Y. Peres, and A. Sly. 2018. Random walks on the random graph,Annals of Probability, to appear. ↑272Berger, N., C. Kenyon, E. Mossel, and Y. Peres. 2005. Glauber dynamics on trees and hyperbolicgraphs, Probab. Theory Related Fields 131, no. 3, 311–340. ↑230Bernstein, M. and E. Nestoridi. 2017. Cutoff for random to random card shuffle, available athttps://arxiv.org/abs/1703.06210. ↑363Billingsley, P. 1995. Probability and measure, 3rd ed., Wiley Series in Probability and Mathe-matical Statistics, John Wiley & Sons Inc., New York. ↑365, 371Boczkowski, L., Y. Peres, and P. Sousi. 2018. Sensitivity of mixing times in Eulerian digraphs,SIAM Journal Discrete Math, to appear. ↑158Borgs, C., J. T. Chayes, A. Frieze, J. H. Kim, P. Tetali, E. Vigoda, and V. H. Vu. 1999. Tor-pid mixing of some Monte Carlo Markov chain algorithms in statistical physics, 40th Annual

Symposium on Foundations of Computer Science (New York, 1999), IEEE Computer Soc., LosAlamitos, CA, pp. 218–229, DOI 10.1109/SFFCS.1999.814594. MR1917562 ↑232Borgs, C., J. T. Chayes, and P. Tetali. 2012. Tight bounds for mixing of the Swendsen-Wangalgorithm at the Potts transition point, Probab. Theory Related Fields 152, no. 3-4, 509–557,DOI 10.1007/s00440-010-0329-0. MR2892955 ↑232Borovkov, A. A. and S. G. Foss. 1992. Stochastically recursive sequences and their generalizations,Siberian Advances in Mathematics 2, 16–81. ↑350Bodineau, T. 2005. Slab percolation for the Ising model, Probab. Theory Related Fields 132,no. 1, 83–118. ↑231Bollobas, B. 1988. The isoperimetric number of random regular graphs, European J. Combin. 9,no. 3, 241–244, DOI 10.1016/S0195-6698(88)80014-3. MR947025 ↑200

BIBLIOGRAPHY 427

Bollobas, B. 1998. Modern graph theory, Graduate Texts in Mathematics, vol. 184, Springer-Verlag, New York. ↑126Boucheron, S., G. Lugosi, and P. Massart. 2013. Concentration inequalities, Oxford UniversityPress, Oxford. A nonasymptotic theory of independence. With a foreword by Michel Ledoux.↑200Bremaud, P. 1999. Markov chains, Texts in Applied Mathematics, vol. 31, Springer-Verlag, NewYork. Gibbs fields, Monte Carlo simulation, and queues. ↑45Broder, A. 1989. Generating random spanning trees, 30th Annual Symposium on Foundationsof Computer Science, 442–447. ↑359Broder, A. Z. and A. R. Karlin. 1989. Bounds on the cover time, J. Theoret. Probab. 2, no. 1,101–120. ↑158Bubley, R. and M. Dyer. 1997. Path coupling: A technique for proving rapid mixing in Markovchains, Proceedings of the 38th Annual Symposium on Foundations of Computer Science,pp. 223–231. ↑203, 204Cancrini, N., P. Caputo, and F. Martinelli. 2006. Relaxation time of L-reversal chains and otherchromosome shuffles, Ann. Appl. Probab. 16, no. 3, 1506–1527. ↑242Cancrini, N., F. Martinelli, C. Roberto, and C. Toninelli. 2008. Kinetically constrained spinmodels, Probab. Theory Related Fields 140, no. 3-4, 459–504. ↑98Caputo, P., T. M. Liggett, and T. Richthammer. 2010. Proof of Aldous’ spectral gap conjecture,J. Amer. Math. Soc. 23, no. 3, 831–851. ↑335, 363Caputo, P., E. Lubetzky, F. Martinelli, A. Sly, and F. L. Toninelli. 2014. Dynamics of (2 + 1)-dimensional SOS surfaces above a wall: Slow mixing induced by entropic repulsion, Ann. Probab.42, no. 4, 1516–1589. ↑323Carne, T. K. 1985. A transmutation formula for Markov chains (English, with French summary),Bull. Sci. Math. (2) 109, no. 4, 399–405. ↑98Cerf, R. and A. Pisztora. 2000. On the Wulff crystal in the Ising model, Ann. Probab. 28, no. 3,947–1017. ↑Cesi, F., G. Guadagni, F. Martinelli, and R. H. Schonmann. 1996. On the two-dimensionalstochastic Ising model in the phase coexistence region near the critical point, J. Statist. Phys.85, no. 1-2, 55–102. ↑231Chandra, A. K., P. Raghavan, W. L. Ruzzo, R. Smolensky, and P. Tiwari. 1996. The electricalresistance of a graph captures its commute and cover times, Comput. Complexity 6, no. 4, 312–340. Extended abstract originally published in Proc. 21st ACM Symp. Theory of Computing(1989) 574–586. ↑147Chayes, J. T., L. Chayes, and R. H. Schonmann. 1987. Exponential decay of connectivities in thetwo-dimensional Ising model, J. Statist. Phys. 49, no. 3-4, 433–445. ↑230Cheeger, J. 1970.A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis(Papers dedicated to Salomon Bochner, 1969), Princeton Univ. Press, Princeton, pp. 195–199.↑199Chen, F., L. Lovasz, and I. Pak. 1999. Lifting Markov chains to speed up mixing, Annual ACMSymposium on Theory of Computing (Atlanta, GA, 1999), ACM, New York, pp. 275–281 (elec-tronic). ↑98Chen, G.-Y. and L. Saloff-Coste. 2008. The cutoff phenomenon for ergodic Markov processes,Electron. J. Probab. 13, no. 3, 26–78. ↑270Chen, G.-Y. and L. Saloff-Coste. 2013. Comparison of cutoffs between lazy walks and Markoviansemigroups, J. Appl. Probab. 50, no. 4, 943–959. ↑291Chen, M.-F. 1998. Trilogy of couplings and general formulas for lower bound of spectral gap,Probability towards 2000 (New York, 1995), Lecture Notes in Statist., vol. 128, Springer, NewYork, pp. 123–136. ↑180, 274Chung, F., P. Diaconis, and R. Graham. 2001. Combinatorics for the East model, Adv. in Appl.Math. 27, no. 1, 192–206. ↑98Chyakanavichyus, V. and P. Vaıtkus. 2001. Centered Poisson approximation by the Stein method

(Russian, with Russian and Lithuanian summaries), Liet. Mat. Rink. 41, no. 4, 409–423; Englishtransl.,. 2001, Lithuanian Math. J. 41, no. 4, 319–329. ↑291Coppersmith, D., U. Feige, and J. Shearer. 1996. Random walks on regular and irregular graphs,SIAM J. Discrete Math. 9, no. 2, 301–308. ↑158Coppersmith, D., P. Tetali, and P. Winkler. 1993. Collisions among random walks on a graph,SIAM J. Discrete Math. 6, no. 3, 363–374. ↑148

428 BIBLIOGRAPHY

Cox, J. T., Y. Peres, and J. E. Steif. 2016. Cutoff for the noisy voter model, Ann. Appl. Probab.26, no. 2, 917–932. ↑230Cuff, P., J. Ding, O. Louidor, E. Lubetzky, Y. Peres, and A. Sly. 2012. Glauber dynamics for themean-field Potts model, J. Stat. Phys. 149, no. 3, 432–477. ↑232Dembo, A., J. Ding, J. Miller, and Y. Peres. 2013. Cut-off for lamplighter chains on tori: Di-mension interpolation and phase transition, arXiv preprint arXiv:1312.4522. ↑279Dembo, A., Y. Peres, J. Rosen, and O. Zeitouni. 2004. Cover times for Brownian motion and

random walk in two dimensions, Ann. Math. 160, 433–464. ↑158Denardo, E. V. 1977. Periods of connected networks and powers of nonnegative matrices, Math-ematics of Operations Research 2, no. 1, 20–24. ↑20Devroye, L. 1986. Nonuniform random variate generation, Springer-Verlag, New York. ↑391Diaconis, P. 1988a. Group representations in probability and statistics, Lecture Notes—Monograph Series, vol. 11, Inst. Math. Stat., Hayward, CA. ↑35, 105, 112, 113, 177Diaconis, P. 1988b. Applications of noncommutative Fourier analysis to probability problems,

Ecole d’Ete de Probabilites de Saint-Flour XV–XVII, 1985–87, Lecture Notes in Math., vol. 1362,Springer, Berlin, pp. 51–100. ↑272Diaconis, P. 1996. The cutoff phenomenon in finite Markov chains, Proc. Nat. Acad. Sci. U.S.A.93, no. 4, 1659–1664. ↑Diaconis, P. 2003. Mathematical developments from the analysis of riffle shuffling, Groups, com-binatorics & geometry (Durham, 2001), World Sci. Publ., River Edge, NJ, pp. 73–97. ↑114Diaconis, P. 2013. Some things we’ve learned (about Markov chain Monte Carlo), Bernoulli 19,no. 4, 1294–1305. ↑323Diaconis, P. and J. A. Fill. 1990. Strong stationary times via a new form of duality, Ann. Probab.18, no. 4, 1483–1522. ↑86, 177, 258, 270Diaconis, P. and D. Freedman. 1999. Iterated random functions, SIAM Review 41, 45–76. ↑350Diaconis, P., M. McGrath, and J. Pitman. 1995. Riffle shuffles, cycles, and descents, Combina-torica 15, no. 1, 11–29. ↑114Diaconis, P. and L. Saloff-Coste. 1993a. Comparison theorems for reversible Markov chains, Ann.Appl. Probab. 3, no. 3, 696–730. ↑200Diaconis, P. and L. Saloff-Coste. 1993b. Comparison techniques for random walk on finite groups,Ann. Probab. 21, no. 4, 2131–2156. ↑200, 242Diaconis, P. and L. Saloff-Coste. 1996a. Nash inequalities for finite Markov chains, J. Theoret.Probab. 9, no. 2, 459–510. ↑148Diaconis, P. and L. Saloff-Coste. 1996b. Logarithmic Sobolev inequalities for finite Markov chains,Ann. Appl. Probab. 6, no. 3, 695–750. ↑291Diaconis, P. and L. Saloff-Coste. 1996c. Walks on generating sets of abelian groups, Probab.Theory Related Fields 105, no. 3, 393–421. ↑362Diaconis, P. and L. Saloff-Coste. 1998. What do we know about the Metropolis algorithm?, J.Comput. System Sci. 57, no. 1, 20–36. 27th Annual ACM Symposium on the Theory of Com-puting (STOC’95) (Las Vegas, NV). ↑45Diaconis, P. and L. Saloff-Coste. 2006. Separation cut-offs for birth and death chains, Ann. Appl.Probab. 16, no. 4, 2098–2122. ↑270, 272, 363Diaconis, P. and M. Shahshahani. 1981. Generating a random permutation with random trans-positions, Z. Wahrsch. Verw. Gebiete 57, no. 2, 159–179. ↑101, 164, 177, 242Diaconis, P. and M. Shahshahani. 1987. Time to reach stationarity in the Bernoulli-Laplacediffusion model, SIAM J. Math. Anal. 18, no. 1, 208–218. ↑272Diaconis, P. and D. Stroock. 1991. Geometric bounds for eigenvalues of Markov chains, Ann.

Appl. Probab. 1, no. 1, 36–61. ↑199Ding, J., J. R. Lee, and Y. Peres. 2012. Cover times, blanket times, and majorizing measures,Ann. of Math. (2) 175, no. 3, 1409–1471. ↑158Ding, J., E. Lubetzky, and Y. Peres. 2009. The mixing time evolution of Glauber dynamics forthe mean-field Ising model, Comm. Math. Phys. 289, no. 2, 725–764. ↑230Ding, J., E. Lubetzky, and Y. Peres. 2010a. Total variation cutoff in birth-and-death chains,Probab. Theory Related Fields 146, no. 1-2, 61–85. ↑271, 363Ding, J., E. Lubetzky, and Y. Peres. 2010b. Mixing time of critical Ising model on trees ispolynomial in the height, Comm. Math. Phys. 295, no. 1, 161–207. ↑323

BIBLIOGRAPHY 429

Ding, J. and Y. Peres. 2011. Mixing time for the Ising model: A uniform lower bound for allgraphs, Ann. Inst. Henri Poincare Probab. Stat. 47, no. 4, 1020–1028, available at arXiv:0909.5162v2. ↑323, 362Dobrushin, R., R. Kotecky, and S. Shlosman. 1992. Wulff construction. A global shape fromlocal interaction, Translations of Mathematical Monographs, vol. 104, American MathematicalSociety, Providence, RI. Translated from the Russian by the authors. ↑231Dobrushin, R. L. and S. B. Shlosman. 1987. Completely analytical interactions: Constructive

description, J. Statist. Phys. 46, no. 5-6, 983–1014. ↑231Doeblin, W. 1938. Espose de la theorie des chaınes simple constantes de Markov a un nombrefini d’etats, Rev. Math. Union Interbalkan. 2, 77–105. ↑73Doob, J. L. 1953. Stochastic processes, John Wiley & Sons Inc., New York. ↑260Doyle, P. G. and E. J. Snell. 1984. Random walks and electrical networks, Carus Math. Mono-graphs, vol. 22, Math. Assoc. Amer., Washington, D. C. ↑126, 303, 305Doyle, P. G. and J. Steiner. 2011. Commuting time geometry of ergodic Markov chains, availableat arxiv:1107.2612. ↑148Dubins, L. E. 1968. On a theorem of Skorohod, Ann. Math. Statist. 39, 2094–2097. ↑86Dudley, R. M. 2002. Real analysis and probability, Cambridge Studies in Advanced Mathematics,vol. 74, Cambridge University Press, Cambridge. Revised reprint of the 1989 original. ↑215Durrett, R. 2003. Shuffling chromosomes, J. Theoret. Probab. 16, 725–750. ↑237, 242Durrett, R. 2005. Probability: Theory and examples, 3rd ed., Brooks/Cole, Belmont, CA. ↑284,365Dyer, M., L. A. Goldberg, and M. Jerrum. 2006a. Dobrushin conditions and systematic scan,Approximation, randomization and combinatorial optimization, Lecture Notes in Comput. Sci.,vol. 4110, Springer, Berlin, pp. 327–338. ↑361Dyer, M., L. A. Goldberg, and M. Jerrum. 2006b. Systematic scan for sampling colorings, Ann.Appl. Probab. 16, no. 1, 185–230. ↑361Dyer, M., L. A. Goldberg, M. Jerrum, and R. Martin. 2006. Markov chain comparison, Probab.Surv. 3, 89–111 (electronic). ↑200Dyer, M. and C. Greenhill. 2000. On Markov chains for independent sets, J. Algorithms 35,no. 1, 17–49. ↑358Dyer, M., C. Greenhill, and M. Molloy. 2002. Very rapid mixing of the Glauber dynamics forproper colorings on bounded-degree graphs, Random Structures Algorithms 20, no. 1, 98–114.↑214Dyer, M., C. Greenhill, and M. Ullrich. 2014. Structure and eigenvalues of heat-bath Markovchains, Linear Algebra Appl. 454, 57–71, DOI 10.1016/j.laa.2014.04.018. ↑178Eggenberger, F. and G Polya. 1923. Uber die Statistik vorketter vorgange, Zeit. Angew. Math.Mech. 3, 279–289. ↑35Einstein, A. 1934. On the method of theoretical physics, Philosophy of Science 1, no. 2, 163–169.↑1Elias, P. 1972. The efficient construction of an unbiased random sequence, Ann. Math. Statist.43, 865–870. ↑391Erdos, P. and A. Renyi. 1961. On a classical problem of probability theory (English, with Russiansummary), Magyar Tud. Akad. Mat. Kutato Int. Kozl. 6, 215–220. ↑35Feige, U. 1995a. A tight lower bound on the cover time for random walks on graphs, RandomStructures Algorithms 6, no. 4, 433–438. ↑158Feige, U. 1995b. A tight upper bound on the cover time for random walks on graphs, RandomStructures Algorithms 6, no. 1, 51–54. ↑Feige, U. 1997. Collecting coupons on trees, and the cover time of random walks, Comput.Complexity 6, no. 4, 341–356. ↑158Feller, W. 1968. An introduction to probability theory and its applications, 3rd ed., Vol. 1, Wiley,New York. ↑19, 35, 177, 305, 365Fill, J. A. 1991. Eigenvalue bounds on convergence to stationarity for nonreversible Markovchains, with an application to the exclusion process, Ann. Appl. Probab. 1, no. 1, 62–87. ↑98Fill, J. A. 1998. An interruptible algorithm for perfect sampling via Markov chains, Annals ofApplied Probability 8, 131–162. ↑350, 354Fill, J. A. 2009. On hitting times and fastest strong stationary times for skip-free and moregeneral chains, J. Theoret. Probab. 22, no. 3, 587–600. ↑177

430 BIBLIOGRAPHY

Fill, J. A. and M. Huber. 2000. The randomness recycler: A new technique for perfect sampling,41st Annual Symposium on Foundations of Computer Science, 503–511. ↑350Fill, J. A., M. Machida, D. J. Murdoch, and J. S. Rosenthal. 2000. Extension of Fill’s perfectrejection sampling algorithm to general chains, Random Structure and Algorithms 17, 290–316.↑350Folland, G. B. 1999. Real analysis, 2nd ed., Pure and Applied Mathematics (New York), JohnWiley & Sons, Inc., New York. Modern techniques and their applications; A Wiley-Interscience

Publication. ↑59Frieze, A. and E. Vigoda. 2007. A survey on the use of Markov chains to randomly samplecolourings, Combinatorics, complexity, and chance, Oxford Lecture Ser. Math. Appl., vol. 34,Oxford Univ. Press, Oxford, pp. 53–71. ↑215Ganapathy, M. K. 2007. Robust mixing, Electronic Journal of Probability 12, 262–299. ↑113Ganguly, S., E. Lubetzky, and F. Martinelli. 2015. Cutoff for the east process, Comm. Math.Phys. 335, no. 3, 1287–1322. ↑272Ganguly, S. and F. Martinelli. 2016. Upper triangular matrix walk: Cutoff for finitely manycolumns, available at arXiv:1612.08741. ↑364Gaudilliere, A. and C. Landim. 2014. A Dirichlet principle for non reversible Markov chainsand some recurrence theorems, Probab. Theory Related Fields 158, no. 1-2, 55–89, DOI10.1007/s00440-012-0477-5. MR3152780 ↑148Gheissari, R. and E. Lubetzky. 2016. Mixing times of critical 2D Potts models, available atarxiv:1607.02182. ↑232Godbole, A. P. and S. G. Papastavridis (eds.) 1994. Runs and patterns in probability: Selected pa-pers, Mathematics and Its Applications, vol. 283, Kluwer Academic Publishers Group, Dordrecht.↑158Graham, R. L., D. E. Knuth, and O. Patashnik. 1994. Concrete mathematics: A foundation forcomputer science, 2nd ed., Addison Wesley, Reading, Massachusetts. ↑111, 210Greenberg, S., A. Pascoe, and D. Randall. 2009. Sampling biased lattice configurations usingexponential metrics, ACM-SIAM Symposium on Discrete Algorithms (New York, New York,2009), pp. 76-85. ↑335Griffeath, D. 1974/75. A maximal coupling for Markov chains, Z. Wahrscheinlichkeitstheorie und

Verw. Gebiete 31, 95–106. ↑73Griffiths, S., R. J. Kang, R. Imbuzeiro Oliveira, and V. Patel. 2012. Tight inequalities among sethitting times in Markov chains, Proc. Amer. Math. Soc. ↑348Grinstead, C. and L. Snell. 1997. Introduction to probability, 2nd revised ed., American Mathe-matical Society, Providence. ↑35, 58Guo, H. and M. Jerrum. 2017. Random cluster dynamics for the Ising model is rapidly mix-ing, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms,pp. 1818–1827. ↑231Haggstrom, O. 2002. Finite Markov chains and algorithmic applications, London MathematicalSociety Student Texts, vol. 52, Cambridge University Press, Cambridge. ↑xiv, 58Haggstrom, O. 2007. Problem solving is often a matter of cooking up an appropriate Markovchain, Scand. J. Statist. 34, no. 4, 768–780. ↑46Haggstrom, O. and J. Jonasson. 1997. Rates of convergence for lamplighter processes, StochasticProcess. Appl. 67, no. 2, 227–249. ↑279Haggstrom, O. and K. Nelander. 1998. Exact sampling from anti-monotone systems, Statist.Neerlandica 52, no. 3, 360–380. ↑359Haggstrom, O. and K. Nelander. 1999. On exact simulation of Markov random fields usingcoupling from the past, Scand. J. Statist. 26, no. 3, 395–411. ↑356, 357Hajek, B. 1988. Cooling schedules for optimal annealing, Math. Oper. Res. 13, no. 2, 311–329.↑45Handjani, S. and D. Jungreis. 1996. Rate of convergence for shuffling cards by transpositions, J.Theoret. Probab. 9, no. 4, 983–993. ↑363Hastings, W. K. 1970. Monte Carlo sampling methods using Markov chains and their applica-tions, Biometrika 57, no. 1, 97 –109. ↑45Hayes, T. P. and A. Sinclair. 2007. A general lower bound for mixing of single-site dynamics ongraphs, Ann. Appl. Probab. 17, no. 3, 931–952. ↑98, 323, 362

BIBLIOGRAPHY 431

Hayes, T. P and E. Vigoda. 2003. A non-Markovian coupling for randomly sampling colorings,Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 618–627.↑74Hermon, J., H. Lacoin, and Y. Peres. 2016. Total variation and separation cutoffs are not equiv-alent and neither one implies the other, Electron. J. Probab. 21, Paper No. 44. ↑272, 363Herstein, I. N. 1975. Topics in algebra, 2nd ed., John Wiley and Sons, New York. ↑35, 112Hoeffding, W. 1963. Probability inequalities for sums of bounded random variables, J. Amer.

Statist. Assoc. 58, 13–30. MR0144363 (26 #1908) ↑369Holley, R. 1974. Remarks on the FKG inequalities, Comm. Math. Phys. 36, 227–231. ↑323Holley, R. and D. Stroock. 1988. Simulated annealing via Sobolev inequalities, Comm. Math.Phys. 115, no. 4, 553–569. ↑45Holroyd, A. E. 2011. Some circumstances where extra updates can delay mixing, J. Stat. Phys.145, no. 6, 1649–1652. ↑323Hoory, S., N. Linial, and A. Wigderson. 2006. Expander graphs and their applications, Bull.Amer. Math. Soc. (N.S.) 43, no. 4, 439–561 (electronic). ↑200Horn, R. A. and C. R. Johnson. 1990. Matrix analysis, Cambridge University Press, Cambridge.↑177, 376Huber, M. 1998. Exact sampling and approximate counting techniques, Proceedings of the 30thAnnual ACM Symposium on the Theory of Computing, 31–40. ↑356Ioffe, D. 1995. Exact large deviation bounds up to Tc for the Ising model in two dimensions,Probab. Theory Related Fields 102, no. 3, 313–330. ↑231Ising, E. 1925. Beitrag zur theorie der ferromagnetismus, Zeitschrift Fur Physik 31, 253–258.↑232Jerison, D. 2013. General mixing time bounds for finite Markov chains via the absolute spectralgap, available at arxiv:1310.8021. ↑177Jerrum, M. R. 1995. A very simple algorithm for estimating the number of k-colorings of alow-degree graph, Random Structures Algorithms 7, no. 2, 157–165. ↑214Jerrum, M. R. 2003. Counting, sampling and integrating: Algorithms and complexity, Lecturesin Mathematics ETH Zurich, Birkhauser Verlag, Basel. ↑xiv, 58Jerrum, M. R. and A. J. Sinclair. 1989. Approximating the permanent, SIAM Journal on Com-

puting 18, 1149–1178. ↑199, 231Jerrum, M. and A. Sinclair. 1996. The Markov chain Monte Carlo method: An approach toapproximate counting and integration, Approximation Algorithms for NP-hard Problems. ↑210,211Jerrum, M., A. Sinclair, and E. Vigoda. 2004. A polynomial-time approximation algorithm forthe permanent of a matrix with nonnegative entries, J. ACM 51, no. 4, 671–697 (electronic).↑215Jerrum, M. R., L. G. Valiant, and V. V. Vazirani. 1986. Random generation of combinatorialstructures from a uniform distribution, Theoret. Comput. Sci. 43, no. 2-3, 169–188. ↑215Johnson, N. L. and S. Kotz. 1977. Urn models and their application, John Wiley & Sons, NewYork-London-Sydney. An approach to modern discrete probability theory; Wiley Series in Prob-ability and Mathematical Statistics. ↑35Jonasson, J. 2012. Mixing times for the interchange process, ALEA Lat. Am. J. Probab. Math.Stat. 9, no. 2, 667–683. ↑335, 363Kahn, J., J. H. Kim, L. Lovasz, and V. H. Vu. 2000. The cover time, the blanket time, and theMatthews bound, 41st Annual Symposium on Foundations of Computer Science (Redondo Beach,CA, 2000), IEEE Comput. Soc. Press, Los Alamitos, CA, pp. 467–475. ↑158Kahn, J. D., N. Linial, N. Nisan, and M. E. Saks. 1989. On the cover time of random walks ongraphs, J. Theoret. Probab. 2, no. 1, 121–128. ↑158Kaımanovich, V. A. and A. M. Vershik. 1983. Random walks on discrete groups: Boundary andentropy, Ann. Probab. 11, no. 3, 457–490. ↑279Kakutani, S. 1948. On equivalence of infinite product measures, Ann. of Math. (2) 49, 214–224.

↑291Kandel, D., Y. Matias, R. Unger, and P. Winkler. 1996. Shuffling biological sequences, DiscreteApplied Mathematics 71, 171–185. ↑243Kantorovich, L. V. 1942. On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N.S.)37, 199–201. ↑203, 214

432 BIBLIOGRAPHY

Kantorovich, L. V. and G. S. Rubinstein. 1958. On a space of completely additive functions(Russian, with English summary), Vestnik Leningrad. Univ. 13, no. 7, 52–59. ↑215Karlin, S. and J. McGregor. 1959. Coincidence properties of birth and death processes, Pacific J.Math. 9, 1109–1140. ↑177Karlin, S. and H. M. Taylor. 1975. A first course in stochastic processes, 2nd ed., AcademicPress, New York. ↑19Karlin, S. and H. M. Taylor. 1981. A second course in stochastic processes, Academic Press Inc.

[Harcourt Brace Jovanovich Publishers], New York. ↑177Kassabov, M. 2005. Kazhdan constants for SLn(Z), Internat. J. Algebra Comput. 15, no. 5-6,971–995. ↑362Kasteleyn, P. W. 1961. The statistics of dimers on a lattice I. The number of dimer arrangementson a quadratic lattice, Physica 27, no. 12, 1209–1225. ↑386Keilson, J. 1979. Markov chain models—rarity and exponentiality, Applied Mathematical Sci-ences, vol. 28, Springer-Verlag, New York. ↑177Kemeny, J. G., J. L. Snell, and A. W. Knapp. 1976. Denumerable Markov chains, 2nd ed.,Springer-Verlag, New York. With a chapter on Markov random fields, by David Griffeath; Grad-uate Texts in Mathematics, No. 40. ↑305Kendall, W. S. and J. Møller. 2000. Perfect simulation using dominating processes on orderedspaces, with application to locally stable point processes, Adv. in Appl. Probab. 32, no. 3, 844–865. ↑350Kenyon, C., E. Mossel, and Y. Peres. 2001.Glauber dynamics on trees and hyperbolic graphs, 42ndIEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), IEEE ComputerSoc., Los Alamitos, CA, pp. 568–578. ↑230, 231Knuth, D. 1997. The art of computer programming, 3rd ed., Vol. 2: Seminumerical Algorithms,Addison-Wesley, Reading, Massachusetts. ↑385Kobe, S. 1997. Ernst Ising—physicist and teacher, J. Statist. Phys. 88, no. 3-4, 991–995. ↑232Kolata, G. January 9, 1990. In shuffling cards, 7 is winning number, New York Times, C1. ↑110,113Komjathy, J., J. Miller, and Y. Peres. 2014. Uniform mixing time for random walk on lamplightergraphs (English, with English and French summaries), Ann. Inst. Henri Poincare Probab. Stat.

50, no. 4, 1140–1160. ↑279Komjathy, J. and Y. Peres. 2013. Mixing and relaxation time for random walk on wreath productgraphs, Electron. J. Probab. 18, no. 71, 23, DOI 10.1214/EJP.v18-2321. MR3091717 ↑279Labbe, C. and H. Lacoin. 2016. Cutoff phenomenon for the asymmetric simple exclusion processand the biased card shuffling, available at arxiv:1610.07383v1. ↑335Lacoin, H. 2015. A product chain without cutoff, Electron. Commun. Probab. 20, no. 19, 9. ↑272,291Lacoin, H. 2016a. Mixing time and cutoff for the adjacent transposition shuffle and the simpleexclusion, Ann. Probab. 44, no. 2, 1426–1487. ↑242, 334, 363Lacoin, H. 2016b. The cutoff profile for the simple exclusion process on the circle, Ann. Probab.44, no. 5, 3399–3430. ↑334Laslier, B. and F. L. Toninelli. 2015. How quickly can we sample a uniform domino tiling of the2L × 2L square via Glauber dynamics?, Probab. Theory Related Fields 161, no. 3-4, 509–559.↑323Lawler, G. F. 1991. Intersections of random walks, Probability and Its Applications, BirkhauserBoston, Inc., Boston, MA. ↑126Lawler, G. and A. Sokal. 1988. Bounds on the L2 spectrum for Markov chains and Markovprocesses: A generalization of Cheeger’s inequality, Trans. Amer. Math. Soc. 309, 557–580. ↑183,199Lee, J. R. and T. Qin. 2012. A note on mixing times of planar random walks, available at arxiv:1205.3980. ↑Leon, C. A. and F. Perron. 2004. Optimal Hoeffding bounds for discrete reversible Markov chains,

Ann. Appl. Probab. 14, no. 2, 958–970. ↑178Letac, G. 1986. A contraction principle for certain Markov chains and its applications, Contem-porary Mathematics 50, 263–273. ↑350Levin, D. A., M. J. Luczak, and Y. Peres. 2010. Glauber dynamics for the mean-field Ising model:Cut-off, critical power law, and metastability, Probab. Theory Related Fields 146, no. 1-2, 223–265. ↑230, 271, 362

BIBLIOGRAPHY 433

Levin, D. A. and Y. Peres. 2010. Polya’s theorem on random walks via Polya’s urn, Amer. Math.Monthly 117, no. 3, 220–231. ↑305Levin, D. A. and Y. Peres. 2016. Mixing of the exclusion process with small bias, J. Stat. Phys.165, no. 6, 1036–1050. ↑335Li, S.-Y. R. 1980. A martingale approach to the study of occurrence of sequence patterns inrepeated experiments, Ann. Probab. 8, no. 6, 1171–1176. ↑260Liggett, T. M. 1985. Interacting particle systems, Springer-Verlag, New York. ↑126, 335Liggett, T. M. 1999. Stochastic interacting systems: Contact, voter and exclusion processes,Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sci-ences], vol. 324, Springer-Verlag, Berlin. ↑335Lindvall, T. 2002. Lectures on the coupling method, Dover, Mineola, New York. ↑74Littlewood, J. E. 1948. Large numbers, Mathematical Gazette 32, no. 300. ↑99Logan, B. F. and L. A. Shepp. 1977. A variational problem for random Young tableaux, Advancesin Math. 26, no. 2, 206–222. ↑Long, Y., A. Nachmias, W. Ning, and Y. Peres. 2014. A power law of order 1/4 for critical meanfield Swendsen-Wang dynamics, Mem. Amer. Math. Soc. 232, no. 1092, vi+84. ↑231Lovasz, L. 1993. Random walks on graphs: A survey, Combinatorics, Paul Erdos is Eighty, pp. 1–46. ↑58, 149Lovasz, L. and R. Kannan. 1999. Faster mixing via average conductance, Annual ACM Sympo-sium on Theory of Computing (Atlanta, GA, 1999), ACM, New York, pp. 282–287 (electronic).↑261Lovasz, L. and P. Winkler. 1993. On the last new vertex visited by a random walk, J. GraphTheory 17, 593–596. ↑86Lovasz, L. and P. Winkler. 1995a. Exact mixing in an unknown Markov chain, Electronic Journalof Combinatorics 2. Paper #R15. ↑358Lovasz, L. and P. Winkler. 1995b. Efficient stopping rules for Markov chains, Proc. 27th ACMSymp. on the Theory of Computing, 76–82. ↑86, 337, 348Lovasz, L. and P. Winkler. 1998. Mixing times, Microsurveys in discrete probability (Princeton,NJ, 1997), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 41, Amer. Math. Soc.,Providence, RI, pp. 85–133. ↑58, 84, 86, 336, 337, 348Loynes, R. M. 1962. The stability of a queue with non-independent inter-arrival and servicetimes, Proceedings of the Cambridge Philosophical Society 58, 497–520. ↑350Lubetzky, E., F. Martinelli, A. Sly, and F. L. Toninelli. 2013. Quasi-polynomial mixing of the2D stochastic Ising model with “plus” boundary up to criticality, J. Eur. Math. Soc. (JEMS) 15,no. 2, 339–386. ↑360Lubetzky, E. and Y. Peres. 2016. Cutoff on all Ramanujan graphs, Geom. Funct. Anal. 26, no. 4,1190–1216, DOI 10.1007/s00039-016-0382-7. MR3558308 ↑272Lubetzky, E. and A. Sly. 2010. Cutoff phenomena for random walks on random regular graphs,Duke Math. J. 153, no. 3, 475–510. ↑272Lubetzky, E. and A. Sly. 2012. Critical Ising on the square lattice mixes in polynomial time,Comm. Math. Phys. 313, no. 3, 815–836. ↑231Lubetzky, E. and A. Sly. 2013. Cutoff for the Ising model on the lattice, Invent. Math. 191,no. 3, 719–755. ↑230, 231, 362Lubetzky, E. and A. Sly. 2014a. Cutoff for general spin systems with arbitrary boundary condi-tions, Comm. Pure Appl. Math. 67, no. 6, 982–1027. ↑231, 291, 362Lubetzky, E. and A. Sly. 2014b. Universality of cutoff for the Ising model. Preprint. ↑362Lubetzky, E. and A. Sly. 2016. Information percolation and cutoff for the stochastic Ising model,J. Amer. Math. Soc. 29, no. 3, 729–774. ↑231, 362Lubotzky, A. 1994. Discrete groups, expanding graphs and invariant measures, Progress in Math-ematics, vol. 125, Birkhauser Verlag, Basel. With an appendix by Jonathan D. Rogawski. ↑200Luby, M., D. Randall, and A. Sinclair. 1995. Markov chain algorithms for planar lattice struc-tures, Proceedings of the 36th IEEE Symposium on Foundations of Computing, 150–159. ↑73,386Luby, M., D. Randall, and A. Sinclair. 2001. Markov chain algorithms for planar lattice struc-tures, SIAM J. Comput. 31, 167–192. ↑73Luby, M. and E. Vigoda. 1995. Approximately counting up to four, Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, 150–159. Extended abstract. ↑74

434 BIBLIOGRAPHY

Luby, M. and E. Vigoda. 1999. Fast convergence of the Glauber dynamics for sampling indepen-dent sets, Random Structures and Algorithms 15, no. 3-4, 229–241. ↑74Lyons, R. 2005. Asymptotic enumeration of spanning trees, Combin. Probab. Comput. 14, no. 4,491–522. ↑255Lyons, R and S. Oveis Gharan. 2017. Sharp bounds on random walk eigenvalues via spectralembedding, International Mathematics Research Notices. ↑148Lyons, R. and Y. Peres. 2016. Probability on trees and networks, Cambridge University Press,

Cambridge. ↑98, 126, 148, 305Lyons, T. 1983. A simple criterion for transience of a reversible Markov chain, Ann. Probab.11, no. 2, 393–402. ↑296, 305Madras, N. and D. Randall. 1996. Factoring graphs to bound mixing rates, Proceedings of the37th IEEE Symposium on Foundations of Computing, 194–203. ↑200Madras, N. and G. Slade. 1993. The self-avoiding walk, Birkhauser, Boston. ↑386, 387Mann, B. 1994. How many times should you shuffle a deck of cards?, UMAP J. 15, no. 4,303–332. ↑113Markov, A. A. 1906. Rasprostranenie zakona bol’shih chisel na velichiny, zavisyaschie drug otdruga, Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 2-ya seriya 15,135–156. ↑19Martinelli, F. 1994. On the two-dimensional dynamical Ising model in the phase coexistenceregion, J. Statist. Phys. 76, no. 5-6, 1179–1246. ↑360Martinelli, F. 1999. Lectures on Glauber dynamics for discrete spin models, Lectures on prob-ability theory and statistics (Saint-Flour, 1997), Lecture Notes in Math., vol. 1717, Springer,Berlin, pp. 93–191. ↑227, 232, 323, 361Martinelli, F. and E. Olivieri. 1994. Approach to equilibrium of Glauber dynamics in the onephase region. I. The attractive case, Comm. Math. Phys. 161, no. 3, 447–486. ↑231Martinelli, F., E. Olivieri, and R. H. Schonmann. 1994. For 2-D lattice spin systems weak mixingimplies strong mixing, Comm. Math. Phys. 165, no. 1, 33–47. ↑231Martinelli, F., A. Sinclair, and D. Weitz. 2004. Glauber dynamics on trees: Boundary conditionsand mixing time, Comm. Math. Phys. 250, no. 2, 301–334. ↑230Martinelli, F. and M. Wouts. 2012.Glauber dynamics for the quantum Ising model in a transverse

field on a regular tree, J. Stat. Phys. 146, no. 5, 1059–1088. ↑323Matthews, P. 1988a. Covering problems for Markov chains, Ann. Probab. 16, 1215–1228. ↑149,150, 157Matthews, P. 1988b. A strong uniform time for random transpositions, J. Theoret. Probab. 1,no. 4, 411–423. ↑112, 164Matthews, P. 1989. Some sample path properties of a random walk on the cube, J. Theoret.Probab. 2, no. 1, 129–146. ↑157McKay, B. D. and C. E. Praeger. 1996. Vertex-transitive graphs that are not Cayley graphs. II,J. Graph Theory 22, no. 4, 321–334. ↑29Menshikov, M., S. Popov, and A. Wade. 2017. Non-homogeneous random walks: Lyapunov func-tion methods for near-critical stochastic systems, Vol. 209, Cambridge University Press. ↑305Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. 1953. Equationof state calculations by fast computing machines, J. Chem. Phys. 21, 1087–1092. ↑45Mihail, M. 1989. Conductance and convergence of Markov chains—A combinatorial treatmentof expanders, Proceedings of the 30th Annual Conference on Foundations of Computer Science,1989, pp. 526–531. ↑98Miller, J. and Y. Peres. 2012. Uniformity of the uncovered set of random walk and cutoff forlamplighter chains, Ann. Probab. 40, no. 2, 535–577. ↑279, 363Mironov, I. 2002. (Not so) random shuffles of RC4, Advances in cryptology—CRYPTO 2002,Lecture Notes in Comput. Sci., vol. 2442, Springer, Berlin, pp. 304–319. ↑113Montenegro, R. and P. Tetali. 2006. Mathematical aspects of mixing times in Markov chains,Vol. 1. ↑xiv, 58Mori, T. F. 1987. On the expectation of the maximum waiting time, Ann. Univ. Sci. Budapest.Sect. Comput. 7, 111–115 (1988). ↑158Morris, B. 2006. The mixing time for simple exclusion, Ann. Appl. Probab. 16, no. 2, 615–635.↑334Morris, B. 2009. Improved mixing time bounds for the Thorp shuffle and L-reversal chain, Ann.Probab. 37, no. 2, 453–477. ↑243

BIBLIOGRAPHY 435

Morris, B., W. Ning, and Y. Peres. 2014. Mixing time of the card-cyclic-to-random shuffle, Ann.Appl. Probab. 24, no. 5, 1835–1849. ↑361Morris, B. and Y. Peres. 2005. Evolving sets, mixing and heat kernel bounds, Probab. TheoryRelated Fields 133, no. 2, 245–266. ↑250, 261Morris, B. and C. Qin. 2017. Improved bounds for the mixing time of the random-to-randominsertion shuffle, Electron. Commun. Probab. 22. ↑363Mossel, E., Y. Peres, and A. Sinclair. 2004. Shuffling by semi-random transpositions, Proceedings

of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS’04) October17–19, 2004, Rome, Italy, pp. 572–581. ↑113, 200, 361Nacu, S. 2003. Glauber dynamics on the cycle is monotone, Probab. Theory Related Fields 127,no. 2, 177–185. ↑323, 360Nash-Williams, C. St. J. A. 1959. Random walks and electric currents in networks, Proc. Cam-bridge Philos. Soc. 55, 181–194. ↑126von Neumann, J. 1951. Various techniques used in connection with random digits, NationalBureau of Standards Applied Mathematics Series 12, 36–38. ↑378, 391Norris, J. R. 1998.Markov chains, Cambridge Series in Statistical and Probabilistic Mathematics,vol. 2, Cambridge University Press, Cambridge. Reprint of 1997 original. ↑xiv, 305Oliveira, R. I. 2012. Mixing and hitting times for finite Markov chains, Electron. J. Probab. 17,no. 70, 12. ↑348Oliveira, R. I. 2013. Mixing of the symmetric exclusion processes in terms of the correspondingsingle-particle random walk, Ann. Probab. 41, no. 2, 871–913. ↑335Pak, I. 1997. Random walks on groups : Strong uniform time approach, Ph.D. thesis, HarvardUniversity. ↑86Pemantle, R. 2007. A survey of random processes with reinforcement, Probab. Surv. 4, 1–79(electronic). ↑35Peres, Y. 1992. Iterating von Neumann’s procedure for extracting random bits, Ann. Stat. 20,no. 1, 590–597. ↑379, 391Peres, Y. 1999. Probability on trees: An introductory climb, Lectures on Probability Theory and

Statistics, Ecole d’Ete de Probabilites de Saint-Flour XXVII — 1997, pp. 193–280. ↑126Peres, Y. 2002. Brownian intersections, cover times and thick points via trees, (Beijing, 2002),Higher Ed. Press, Beijing, pp. 73–78. ↑157Peres, Y., S. Popov, and P. Sousi. 2013. On recurrence and transience of self-interacting randomwalks, Bull. Braz. Math. Soc. (N.S.) 44, no. 4, 841–867. ↑305Peres, Y. and D. Revelle. 2004. Mixing times for random walks on finite lamplighter groups,Electron. J. Probab. 9, no. 26, 825–845 (electronic). ↑279Peres, Y. and A. Sly. 2013. Mixing of the upper triangular matrix walk, Probab. Theory RelatedFields 156, no. 3-4, 581–591. ↑364Peres, Y. and P. Sousi. 2015a. Mixing times are hitting times of large sets, J. Theoret. Probab.28, no. 2, 488–519. ↑336, 348Peres, Y. and P. Sousi. 2015b. Total variation cutoff in a tree (English, with English and Frenchsummaries), Ann. Fac. Sci. Toulouse Math. (6) 24, no. 4, 763–779, DOI 10.5802/afst.1463.MR3434255 ↑272Peres, Y. and P. Winkler. 2013. Can extra updates delay mixing?, Comm. Math. Phys. 323,no. 3, 1007–1016. ↑323, 362Pinsker, M. S. 1973. On the complexity of a concentrator, Proc. 7th Int. Teletraffic Conf., Stock-holm, Sweden, pp. 318/1–318/4. ↑196, 200Pisztora, A. 1996. Surface order large deviations for Ising, Potts and percolation models, Probab.

Theory and Related Fields 104, no. 4, 427–466. ↑231Pitman, J. W. 1974. Uniform rates of convergence for Markov chain transition probabilities, Z.Wahrscheinlichkeitstheorie und Verw. Gebiete 29, 193–227. ↑73Pitman, J. W. 1976. On coupling of Markov chains, Z. Wahrscheinlichkeitstheorie und Verw.Gebiete 35, no. 4, 315–322. ↑73Polya, G. 1931. Sur quelques points de la theorie des probabilites, Ann. Inst. H. Poincare 1,117–161. ↑35Pourmiri, A. and T. Sauerwald. 2014. Cutoff phenomenon for random walks on Kneser graphs,Discrete Appl. Math. 176, 100–106. ↑272Preston, C. J. 1974. Gibbs states on countable sets, Cambridge University Press, London-NewYork. Cambridge Tracts in Mathematics, No. 68. ↑

436 BIBLIOGRAPHY

Propp, J. and D. Wilson. 1996. Exact sampling with coupled Markov chains and applications tostatistical mechanics, Random Structure and Algorithms 9, 223–252. ↑320, 349, 350, 355Propp, J. and D. Wilson. 1998. How to get a perfectly random sample from a generic Markovchain and generate a random spanning tree of a directed graph, Journal of Algorithms (SODA’96 special issue) 27, no. 2, 170–217. ↑358, 359Quastel, J. 1992. Diffusion of color in the simple exclusion process, Comm. Pure Appl. Math.45, no. 6, 623–679. ↑200Randall, D. 2006. Slow mixing of Glauber dynamics via topological obstructions, SODA (2006).Available at http://www.math.gatech.edu/~randall/reprints.html. ↑227, 229Randall, D. and A. Sinclair. 2000. Self-testing algorithms for self-avoiding walks, Journal ofMathematical Physics 41, no. 3, 1570–1584. ↑386, 388Randall, D. and P. Tetali. 2000. Analyzing Glauber dynamics by comparison of Markov chains,J. Math. Phys. 41, no. 3, 1598–1615. Probabilistic techniques in equilibrium and nonequilibriumstatistical physics. ↑200Restrepo, R., D. Stefankovic, J. C. Vera, E. Vigoda, and L. Yang. 2014. Phase transition forGlauber dynamics for independent sets on regular trees, SIAM J. Discrete Math. 28, no. 2,835–861. ↑323Riordan, J. 1944. Three-line Latin rectangles, Amer. Math. Monthly 51, 450–452. ↑198Rollin, A. 2007. Translated Poisson approximation using exchangeable pair couplings, Ann. Appl.Probab. 17, no. 5-6, 1596–1614. ↑291Salas, J. and A. D. Sokal. 1997. Absence of phase transition for antiferromagnetic Potts modelsvia the Dobrushin uniqueness theorem, J. Statist. Phys. 86, no. 3-4, 551–579. ↑214Saloff-Coste, L. 1997. Lectures on finite Markov chains, Lectures on Probability Theory and

Statistics, Ecole d’Ete de Probabilites de Saint-Flour XXVI - 1996, pp. 301–413. ↑58, 148Saloff-Coste, L. and J. Zuniga. 2007. Convergence of some time inhomogeneous Markov chains

via spectral techniques, Stochastic Process. Appl. 117, no. 8, 961–979. ↑113Saloff-Coste, L. and J. Zuniga. 2008. Refined estimates for some basic random walks on thesymmetric and alternating groups, ALEA Lat. Am. J. Probab. Math. Stat. 4, 359–392. ↑363Sarnak, P. 2004. What is. . . an expander?, Notices Amer. Math. Soc. 51, no. 7, 762–763. ↑200Scarabotti, F. and F. Tolli. 2008. Harmonic analysis of finite lamplighter random walks, J. Dyn.Control Syst. 14, no. 2, 251–282, available at arXiv:math.PR/0701603. ↑279Schonmann, R. H. 1987. Second order large deviation estimates for ferromagnetic systems in thephase coexistence region, Comm. Math. Phys. 112, no. 3, 409–422. ↑227, 230Seneta, E. 2006. Non-negative matrices and Markov chains, Springer Series in Statistics,Springer, New York. Revised reprint of the second (1981) edition, Springer-Verlag, New York.↑19, 58Simon, B. 1993. The statistical mechanics of lattice gases. Vol. I, Princeton Series in Physics,Princeton University Press, Princeton, NJ. ↑231, 232Sinclair, A. 1992. Improved bounds for mixing rates of Markov chains and multicommodity flow,Combin. Probab. Comput. 1, no. 4, 351–370. ↑Sinclair, A. 1993. Algorithms for random generation and counting, Progress in Theoretical Com-puter Science, Birkhauser Boston Inc., Boston, MA. A Markov chain approach. ↑58, 215Sinclair, A. and M. Jerrum. 1989. Approximate counting, uniform generation and rapidly mixingMarkov chains, Inform. and Comput. 82, no. 1, 93–133. ↑183, 199, 227Snell, J. L. 1997. A conversation with Joe Doob, Statist. Sci. 12, no. 4, 301–311. ↑365Soardi, P. M. 1994. Potential theory on infinite networks, Lecture Notes in Mathematics,vol. 1590, Springer-Verlag, Berlin. ↑126, 305Spielman, D. A. and S.-H. Teng. 1996. Spectral partitioning works: Planar graphs and finiteelement meshes, 37th Annual Symposium on Foundations of Computer Science (Burlington,VT, 1996), IEEE Comput. Soc. Press, Los Alamitos, CA. ↑177Spitzer, F. 1976. Principles of random walks, 2nd ed., Springer-Verlag, New York. GraduateTexts in Mathematics, Vol. 34. ↑291Stanley, R. P. 1986. Enumerative combinatorics, Vol. 1, Wadsworth & Brooks/Cole, Belmont,California. ↑210Stanley, R. P. 1999. Enumerative combinatorics, Vol. 2, Cambridge University Press. ↑35Stanley, R. P. 2008. Catalan addendum. Available at http://www-math.mit.edu/~rstan/ec/

catadd.pdf. ↑35

BIBLIOGRAPHY 437

Steele, J. M. 1997. Probability theory and combinatorial optimization, CBMS-NSF Regional Con-ference Series in Applied Mathematics, vol. 69, Society for Industrial and Applied Mathematics(SIAM), Philadelphia, PA. ↑369Strassen, V. 1965.The existence of probability measures with given marginals, Ann. Math. Statist.36, 423–439. MR0177430 (31 #1693) ↑323Stroock, D. W. and B. Zegarlinski. 1992. The equivalence of the logarithmic Sobolev inequalityand the Dobrushin-Shlosman mixing condition, Comm. Math. Phys. 144, no. 2, 303–323. ↑231Subag, E. 2013. A lower bound for the mixing time of the random-to-random insertions shuffle,Electron. J. Probab. 18, no. 20, 20. ↑363Sugimine, N. 2002. A lower bound on the spectral gap of the 3-dimensional stochastic Isingmodels, J. Math. Kyoto Univ. 42, no. 4, 751–788. ↑360Tetali, P. 1999. Design of on-line algorithms using hitting times, SIAM J. Comput. 28, no. 4,1232–1246 (electronic). ↑148Thomas, L. E. 1989. Bound on the mass gap for finite volume stochastic Ising models at lowtemperature, Comm. Math. Phys. 126, no. 1, 1–11. ↑227, 231Thorisson, H. 1988. Backward limits, Annals of Probability 16, 914–924. ↑350Thorisson, H. 2000. Coupling, stationarity, and regeneration, Probability and Its Applications(New York), Springer-Verlag, New York. ↑74, 305Thorp, E. O. 1965. Elementary problem E1763, Amer. Math. Monthly 72, no. 2, 183. ↑112, 113Thurston, W. P. 1990. Conway’s tiling groups, Amer. Math. Monthly 97, no. 8, 757–773. ↑8Uyemura-Reyes, J. C. 2002. Random walk, semidirect products, and card shuffling, Ph.D. thesis,Stanford University. ↑363Varopoulos, N. Th. 1985. Long range estimates for Markov chains (English, with French sum-mary), Bull. Sci. Math. (2) 109, no. 3, 225–252. ↑98Vasershtein, L. N. 1969. Markov processes over denumerable products of spaces describing largesystem of automata (Russian), Problemy Peredaci Informacii 5, no. 3, 64–72; English transl.,.1969, Problems of Information Transmission 5, no. 3, 47–52. ↑214Vershik, A. M. 2004. The Kantorovich metric: The initial history and little-known applications(Russian, with English and Russian summaries), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat.Inst. Steklov. (POMI) 312, no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 11, 69–85,

311; English transl.,. 2004, J. Math. Sci. (N. Y.) 133, no. 4, 1410–1417, available at arxiv:math.FA/0503035. ↑214Versik, A. M. and S. V. Kerov. 1977. Asymptotic behavior of the Plancherel measure of thesymmetric group and the limit form of Young tableaux (Russian), Dokl. Akad. Nauk SSSR 233,no. 6, 1024–1027. ↑Vigoda, E. 2000. Improved bounds for sampling colorings, J. Math. Phys. 41, no. 3, 1555–1569.↑214Vigoda, E. 2001. A note on the Glauber dynamics for sampling independent sets, Electron. J.Combin. 8, no. 1, Research Paper 8, 8 pp. (electronic). ↑74, 358Villani, C. 2003. Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58,American Mathematical Society, Providence, RI. ↑214Wilf, H. S. 1989. The editor’s corner: The white screen problem, Amer. Math. Monthly 96, 704–707. ↑157Williams, D. 1991. Probability with martingales, Cambridge Mathematical Textbooks, CambridgeUniversity Press, Cambridge. ↑260Wilson, D. B. 2000a. Layered multishift coupling for use in perfect sampling algorithms (with aprimer on CFTP) (N. Madras, ed.), Fields Institute Communications, vol. 26, American Math-ematical Society. ↑359Wilson, D. B. 2000b. How to couple from the past using a read-once source of randomness,Random Structures and Algorithms 16, 85–113. ↑350, 354Wilson, D. B. 2003. Mixing time of the Rudvalis shuffle, Electron. Comm. Probab. 8, 77–85(electronic). ↑200Wilson, D. B. 2004a. Mixing times of lozenge tiling and card shuffling Markov chains, Ann.Appl. Probab. 14, no. 1, 274–325. ↑193, 200, 242, 323, 386Wilson, D. B. 2004b. Perfectly random sampling with Markov chains. Available at http://

dbwilson/exact. ↑359Woess, W. 2000. Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics,vol. 138, Cambridge University Press, Cambridge. ↑305

438 BIBLIOGRAPHY

Zeitouni, O. 2004. Random walks in random environment, Lectures on probability theory andstatistics, Lecture Notes in Math., vol. 1837, Springer, Berlin, pp. 189–312. ↑305Zuckerman, D. 1989. Covering times of random walks on bounded degree trees and other graphs,J. Theoret. Probab. 2, no. 1, 147–157. ↑158Zuckerman, D. 1992. A technique for lower bounding the cover time, SIAM J. Discrete Math. 5,81–87. ↑157, 158van Zuylen, A. and F. Schalekamp. 2004. The Achilles heel of the GSR shuffle. A note on new

age solitaire, Probab. Engrg. Inform. Sci. 18, no. 3, 315–328. ↑114

Notation Index

The symbol := means defined as.

The set {. . . ,−1, 0, 1, . . . } of integers is denoted Z and the set ofreal numbers is denoted R.

For sequences (an) and (bn), the notation an = O(bn) means thatfor some c > 0 we have an/bn ≤ c for all n, while an = o(bn) meansthat limn→∞ an/bn = 0, and an � bn means both an = O(bn) andbn = O(an) are true.

An (alternating group), 100

B (congestion ratio), 188

C(a ↔ z) (effective conductance), 118E(f, h) (Dirichlet form), 181

E(f) (Dirichlet form), 181

E (edge set), 8

E (expectation), 367

Eμ (expectation from initial distributionμ), 4

Ex (expectation from initial state x), 4

Eμ (expectation w.r.t. μ), 92, 392

G (graph), 8

G� (lamplighter graph), 273

I (current flow), 117

P (transition matrix), 2PA (transition matrix of induced chain),

186P (time reversal), 14

P{X ∈ B} (probability of event), 366

Pμ (probability from initial distribution μ),4

Px (probability from initial state x), 4

Px,y (probability w.r.t. coupling startedfrom x and y), 61

Q(x, y) (edge measure), 88

R(a ↔ z) (effective resistance), 118

Sn (symmetric group), 75

SV (configuration set), 41

V (vertex set), 8

Var (variance), 367Varμ (variance w.r.t. μ), 92

W (voltage), 117

Zn (n-cycle), 63

Zdn (torus), 64

c(e) (conductance), 115

d(t) (total variation distance), 53

d(t) (total variation distance), 53

dH (Hellinger distance), 58, 287

id (identity element), 27

i.i.d. (independent and identicallydistributed), 19, 60

r(e) (resistance), 115

sx(t) (separation distance started from x),79

s(t) (separation distance), 79

tcov (worst case mean cover time), 149

thit (maximal hitting time), 128

tmix(ε) (mixing time), 54

tCes (Cesaro mixing time), 83

tcontmix (continuous mixing time), 283

trel (relaxation time), 162

t� (target time), 128

β (inverse temperature), 44

δx (Dirac delta), 4

Δ (maximum degree), 70

Γxy (path), 188

γ (spectral gap), 162

γ� (absolute spectral gap), 162

λj (eigenvalue of transition matrix), 162

λ� (maximal non-trivial eigenvalue), 162

X (state space), 2

ω (root of unity), 164

Φ(S) (bottleneck ratio of set), 88

Φ� (bottleneck ratio), 88

π (stationary distribution), 9

ρ (metric), 201, 375

ρK(μ, ν) (transportation metric), 201

439

440 NOTATION INDEX

ρi,j (reversal), 237σ (Ising spin), 44τA (hitting time for set), 77, 116, 127τa,b (commute time), 130τcouple (coupling time), 62τcov (cover time variable), 149τAcov (cover time for set), 150

τx (hitting time), 10, 127

τ+x (first return time), 10, 127θ (flow), 117

∧ (min), 39(ijk) (cycle (permutation)), 100∂S (boundary of S), 89�2(π) (inner product space), 160[x] (equivalence class), 25〈·, ·〉 (standard inner product), 160〈·, ·〉π (inner product w.r.t. π), 160μ (reversed distribution), 551A (indicator function), 14∼ (adjacent to), 8‖μ− ν‖TV (total variation distance), 47

Index

Italics indicate that the reference is to an exercise.

absolute spectral gap, 162

absorbing state, 15

acceptance-rejection sampling, 380

alternating group, 100, 109

aperiodic chain, 7

approximate counting, 210

averaging over paths, 190

ballot theorem, 33

binary tree, 65

Ising model on, 221

random walk on

bottleneck ratio lower bound, 91

commute time, 132

coupling upper bound, 66

cover time, 151

hitting time, 145

no cutoff, 268

birth-and-death chain, 26, 260, 300

stationary distribution, 26

block dynamics

for Ising model, 224, 362

bottleneck ratio, 88, 89

bounds on relaxation time, 183

lower bound on mixing time, 88

boundary, 89

Bounded Convergence Theorem, 371

Catalan number, 32

Cayley graph, 29

censoring inequality, 315

Central Limit Theorem, 369

Cesaro mixing time, 83, 336

CFTP, see also coupling from the past

Chebyshev’s inequality, 367

Cheeger constant, 98

children (in tree), 65

coin tossing patterns, see also patterns incoin tossing

colorings, 38

approximate counting of, 210

Glauber dynamics for, 42, 361

exponential lower bound on star, 90

lower bound on empty graph, 97

path coupling upper bound, 207Metropolis dynamics for

grand coupling upper bound, 70relaxation time, 180

communicating classes, 15commute time, 130

Identity, 131comparison of Markov chains, 185

canonical paths, 188on groups, 190randomized paths, 190theorem, 188, 224, 233, 240

complete graph, 81Ising model on, 219lamplighter chain on, 278

conductance, 115bottleneck ratio, 98

configuration, 41congestion ratio, 188, 190connected graph, 17connective constant, 227continuous-time chain, 281

Convergence Theorem, 283product chains, 286relation to lazy chain, 283relaxation time, 285

Convergence Theorem, 52continuous time, 283coupling proof, 73

null recurrent chain, 301positive recurrent chain, 299

convolution, 141, 146counting lower bound, 87coupling

bound on d(t), 62characterization of total variation

distance, 50from the past, 349grand, 69, 70, 352, 355Markovian, 61, 73of distributions, 49, 50, 201of Markov chains, 61of random variables, 49, 201optimal, 50, 202

441

442 INDEX

coupon collector, 22, 62, 81, 82, 94cover time variable, 149current flow, 117cutoff, 262

open problems, 361window, 263

cutset

edge, 122cycle

biased random walk on, 14Ising model on

mixing time pre-cutoff, 220random walk on, 5, 8, 17, 28, 34, 78

bottleneck ratio, 183coupling upper bound, 63cover time, 149, 157eigenvalues and eigenfunctions, 164hitting time upper bound, 142last vertex visited, 86lower bound, 63no cutoff, 268spectral gap, 165strong stationary time upper bound,

82, 86cycle law, 118cycle notation, 100cyclic-to-random shuffle, 113

degree of vertex, 8density function, 366depth (of tree), 65descendant (in tree), 91detailed balance equations, 13diameter, 88, 201diameter lower bound, 88dimer system, 385Dirichlet form, 181distinguishing statistic, 91distribution function, 366divergence

of flow, 117Dominated Convergence Theorem, 371domino tiling, 385Doob h-transform, 257Doob decomposition, 260Durrett chain

comparison upper bound, 240distinguishing statistic lower bound, 238

East model, 363lower bound, 97

edge cutset, 122edge measure, 88effective conductance, 118effective resistance, 118

gluing nodes, 119, 122of grid graph, 123of tree, 120Parallel Law, 119

Series Law, 119triangle inequality, 125, 131

Ehrenfest urn, 24, 34, 266eigenvalues of transition matrix, 160, 176empty graph, 97energy

of flow, 121

of Ising configuration, 44ergodic theorem, 392escape probability, 118essential state, 15Eulerian graphs, 135even permutation, 100event, 365evolving-set process, 250exclusion process, 324

biased, 330monotonicity of, 326on path

mixing time, 329expander graph, 196

Ising model on, 229ExpanderMixingLemma, 177expectation, 367

Fibonacci numbers, 213FIFO queue, 304“fifteen” puzzle, 110first return time, 10, 127flow, 117fugacity, 44fully polynomial randomized approximation

scheme, 210

gambler’s ruin, 21, 34, 125, 247Gaussian elimination chain, 363generating function, 141generating set, 28geometric mixing time, 336Gibbs distribution, 44Gibbs sampler, 41Glauber dynamics

definition, 42for colorings, 42, 361

path coupling upper bound, 207for hardcore model, 44, 72

coupling from the past, 356relaxation time, 181

for Ising model, 44, 185, 216coupling from the past, 351

for product measure, 169glued graphs, 143

complete, 81lower bound, 86strong stationary time upper bound,

81hypercube

hitting time upper bound, 145strong stationary time, 146

INDEX 443

torus

bottleneck ratio lower bound, 90

hitting time upper bound, 127, 144

gluing (in networks), 119, 122

grand coupling, 69, 70, 352, 355

graph, 8

Cayley, 29

colorings, see also colorings

complete, 81

connected, 17

degree of vertex, 8

diameter, 88

empty, 97

expander, 196, 229

glued, see also glued graphs

grid, 123

ladder, 225

loop, 9

multiple edges, 9

oriented, 117

proper coloring of, 38, see also colorings

regular, 10

counting lower bound, 87

simple random walk on, 8

Green’s function, 119, 293

grid graph, 123

Ising model on, 227

group, 27

generating set of, 28

random walk on, 28, 75, 99, 190

symmetric, 75

halting state, 79

Hamming weight, 24

hardcore model, 42

Glauber dynamics for, 44

coupling from the past, 356

grand coupling upper bound, 72

relaxation time, 181

monotone, 310

with fugacity, 44, 72

harmonic function, 12, 18, 116, 257

Harris inequality, 311

heat bath algorithm, see also Glauberdynamics

heat kernel, 282

Hellinger distance, 58, 287, 291

hill climb algorithm, 40

hitting time, 10, 77, 116, 127

cycle identity, 132

upper bound on mixing time, 139

worst case, 128

hypercube, 23

lamplighter chain on, 278

random walk on, 28

�2 upper bound, 172

bottleneck ratio, 183

coupling upper bound, 62

cover time, 157

cutoff, 172, 266distinguishing statistic lower bound, 94

eigenvalues and eigenfunctions of, 170hitting time, 145relaxation time, 181

separation cutoff, 269strong stationary time upper bound,

76, 78, 81

Wilson’s method lower bound, 193

i.i.d., 19, 60increment distribution, 28

independent, 367indicator function, 14

induced chain, 186, 302inessential state, 15

interchange process, 324, 363inverse distribution, 107

method of simulation, 380irreducible chain, 7Ising model, 44, 216

block dynamics, 224, 362comparison of Glauber and Metropolis,

185

energy, 44fast mixing at high temperature, 216

Gibbs distribution for, 44Glauber dynamics for, 44

coupling from the past, 350infinite temperature, 44

inverse temperature, 44monotone, 306on complete graph

mixing time bounds, 219on cycle

mixing time pre-cutoff, 220relaxation time, 315

on expander, 229on grid

relaxation time lower bound, 227on tree, 229

mixing time upper bound, 221

open problems, 360partial order on configurations, 351

partition function, 44positive correlations, 312

isoperimetric constant, 98

k-fuzz, 303Kac lemma, 297

Kirchhoff’s node law, 117

�p(π) distance, 171L-reversal chain, see also Durrett chain

ladder graph, 225lamplighter chain, 273, 363

mixing time, 276on cycle, 278

444 INDEX

on hypercube, 278

on torus, 278

relaxation time, 274

separation cutoff, 279

Laws of Large Numbers, 368

lazy version of a Markov chain, 8, 176, 283

leaf, 18, 65

level (of tree), 65

linear congruential sequence, 384

Lipschitz constant, 180, 213

loop, 9

lower bound methods

bottleneck ratio, 88, 89

counting bound, 87

diameter bound, 88

distinguishing statistic, 91

Wilson’s method, 193

lozenge tiling, 352

lumped chain, see also projection

Markov chain

aperiodic, 7

birth-and-death, 26

communicating classes of, 15

comparison of, see also comparison ofMarkov chains

continuous time, 281

Convergence Theorem, 52, 73

coupling, 61

definition of, 2

ergodic theorem, 392

exclusion process, see also exclusionprocess

irreducible, 7

lamplighter, see also lamplighter chain

lazy version of, 8

mixing time of, 54

Monte Carlo method, 38, 349

null recurrent, 297

periodic, 7, 176

positive recurrent, 297

product, see also product chain

projection of, 25, 34

random mapping representation of, 6, 69

recurrent, 293

reversible, 14, 116

stationary distribution of, 9

time averages, 172

time reversal of, 14, 34

time-inhomogeneous, 19, 113, 203

transient, 293

transitive, 29, 34

unknown, 358

Markov property, 2

Markov’s inequality, 367

Markovian coupling, 61, 73

martingale, 244

Matthews method

lower bound on cover time, 150

upper bound on cover time, 150maximum principle, 18, 116

MCMC, see also Markov chain, MonteCarlo method

metric space, 201, 375

Metropolis algorithm, 38arbitrary base chain, 40for colorings, 70, 180

for Ising model, 185symmetric base chain, 38

minimum expectation of a stationary time,337

mixing time, 54

�2 upper bound, 171Cesaro, 83

continuous time, 283coupling upper bound, 62

hitting time upper bound, 139path coupling upper bound, 204

relaxation time lower bound, 163relaxation time upper bound, 162

monotone chains, 306

positive correlations, 313Monotone Convergence Theorem, 371

monotone spin system, 310Monte Carlo method, 38, 349

move-to-front chain, 82

Nash-Williams inequality, 123, 295

network, 115infinite, 294

node, 115node law, 117null recurrent, 297

odd permutation, 100Ohm’s law, 118

optimal coupling, 50, 202Optional Stopping Theorem, 246

order statistic, 389oriented edge, 117

Parallel Law, 119parity (of permutation), 100partition function, 44

pathmetric, 203

random walk on, see alsobirth-and-death chain, see alsogambler’s ruin, 60, 120, 263

eigenvalues and eigenfunctions, 166,167

path coupling, 201upper bound on mixing time, 204, 216

patterns in coin tossingcover time, 155

hitting time, 145, 248perfect sampling, see also sampling, exact

INDEX 445

periodic chain, 7

eigenvalues of, 176

pivot chain for self-avoiding walk, 386

Polya’s urn, 25, 124, 125, 138

positive correlations

definition of, 310

of product measures, 311

positive recurrent, 296

pre-cutoff, 263, 271

mixing time of Ising model on cycle, 220

previsible sequence, 245

probability

distribution, 366

measure, 366

space, 365

product chain

eigenvalues and eigenfunctions of, 168,176

in continuous time, 286

spectral gap, 169

Wilson’s method lower bound, 195

projection, 25, 34, 165

onto coordinate, 201

proper colorings, see also colorings

pseudorandom number generator, 384

random adjacent transpositions, 233

comparison upper bound, 233

coupling upper bound, 234

single card lower bound, 235

Wilson’s method lower bound, 236

random colorings, 90

random mapping representation, 6, 69

random number generator, see alsopseudorandom number generator

random sample, 38

Random Target Lemma, 128

random transposition shuffle, 101, 111

coupling upper bound, 103

lower bound, 102

relaxation time, 164

strong stationary time upper bound, 104,112

random variable, 366

random walk

on R, 244

on Z, 30, 293, 304

biased, 245

null recurrent, 296

on Zd, 292

recurrent for d = 2, 295

transient for d = 3, 295

on binary tree

bottleneck ratio lower bound, 91

commute time, 132

coupling upper bound, 66

cover time, 151

hitting time, 145

no cutoff, 268

on cycle, 5, 8, 17, 28, 34, 78

bottleneck ratio, 183

coupling upper bound, 63

cover time, 149, 157

eigenvalues and eigenfunctions, 164

hitting time upper bound, 142

last vertex visited, 86

lower bound, 63

no cutoff, 268

spectral gap, 165

strong stationary time upper bound,

82, 86

on group, 27, 75, 99, 190

on hypercube, 23, 28

�2 upper bound, 172

bottleneck ratio, 183

coupling upper bound, 62

cover time, 157

cutoff, 172, 266

distinguishing statistic lower bound, 94

eigenvalues and eigenfunctions of, 170

hitting time, 145

relaxation time, 181

separation cutoff, 269

strong stationary time upper bound,76, 78, 81

Wilson’s method lower bound, 193

on path, see also birth-and-death chain,see also gambler’s ruin, 60, 120, 263

eigenvalues and eigenfunctions, 166,167

on torus, 64

coupling upper bound, 64, 73

cover time, 152, 157

hitting time, 136

perturbed, 190, 198

self-avoiding, 385

simple, 8, 14, 115, 189

weighted, 115

randomized paths, 190

Rayleigh’s Monotonicity Law, 122, 295

Rayleigh-Ritz theorem, 376

recurrent, 293, 303

reflection principle, 30, 34, 35

regular graph, 10

counting lower bound, 87

relaxation time, 162

bottleneck ratio bounds, 183

continuous time, 285

coupling upper bound, 180

mixing time lower bound, 162

mixing time upper bound, 163

variational characterization of, 182

resistance, 115

return probability, 141, 255, 302

reversal, see also Durrett chain, 237

reversed chain, see also time reversal

446 INDEX

reversed distribution, 55

reversibility, 14, 116

detailed balance equations, 13

riffle shuffle, 106, 113

counting lower bound, 109

generalized, 111

strong stationary time upper bound, 108

rising sequence, 106

rooted tree, 65

roots of unity, 164

sampling, 379

and counting, 209

exact, 209, 355

self-avoiding walk, 385, 386, 391

semi-random transpositions, 113

separation distance, 79, 80, 85, 363

total variation upper bound, 80

upper bound on total variation, 80

Series Law, 119

shift chain, see also patterns in coin tossing

shuffle

cyclic-to-random, 113

move-to-front, 82

open problems, 361

random adjacent transposition, 233

comparison upper bound, 233

coupling upper bound, 234

single card lower bound, 235

Wilson’s method lower bound, 236

random transposition, 101, 111

coupling upper bound, 103

lower bound, 102

relaxation time, 164

strong stationary time upper bound,104, 112

riffle, 106, 113

counting lower bound, 109

generalized, 111

strong stationary time upper bound,108

semi-random transpositions, 113

top-to-random, 75

cutoff, 262

lower bound, 95

strong stationary time upper bound,78, 82, 85

simple random walk, 8, 115, 189

stationary distribution of, 9

simplex, 384

simulation

of random variables, 377, 379

sink, 117

source, 117

spectral gap, 162, see also relaxation time

absolute, 162

bottleneck ratio bounds, 183

variational characterization of, 182

spectral theorem for symmetric matrices,376

spin system, 44

montone, 310

star, 90

stationary distribution, 9

uniqueness of, 13, 17

stationary time, 77, 83

strong, 78, 258

Stirling’s formula, 376

stochastic domination, 307

stochastic flow, see also grand coupling

stopping time, 85, 246

Strassen’s theorem, 308

strength

of flow, 117

Strong Law of Large Numbers, 368

strong stationary time, 78, 258

submartingale, 244

submultiplicativity

of d(t), 54

of s(t), 85

supermartingale, 244, 260

support, 366

symmetric group, 75, 99

symmetric matrix, 376

systematic updates, 361

target time, 128, 129

tensor product, 168

Thomson’s Principle, 121, 294

tiling

domino, 385

lozenge, 352

time averages, 172

time reversal, 14, 34, 55, 58, 68, 82, 107

time-inhomogeneous Markov chain, 19, 113,203

top-to-random shuffle, 75

cutoff, 262

lower bound, 95

strong stationary time upper bound, 78,82, 85

torus

definition of, 64

glued

bottleneck ratio lower bound, 90

hitting time upper bound, 144

lamplighter chain on, 278

random walk on

coupling upper bound, 64, 73

cover time, 152, 157

hitting time, 136

perturbed, 190, 198

total variation distance, 47

coupling characterization of, 50

Hellinger distance upper bound, 288

monotonicity of, 57

INDEX 447

separation distance upper bound, 80standardized (d(t), d(t)), 53upper bound on separation distance, 80

transient, 293transition matrix

definition of, 2eigenvalues of, 160, 176

multiply on left, 5multiply on right, 5spectral representation of, 160

transition probabilities, t-step, 5transition times, 281transitive

chain, 29, 34, 58, 361network, 131

transportation metric, 201, 213transpose (of a matrix), 376transposition, 100tree, 17, 65

binary, see also binary tree, 65effective resistance, 120Ising model on, 221, 229rooted, 65

triangle inequality, 375

unbiasingvon Neumann, 378

unit flow, 117unity

roots of, 164unknown chain

sampling from, 358up-right path, 33urn model

Ehrenfest, 24, 34, 266Polya, 25, 124, 125, 138

variance, 367voltage, 117von Neumann unbiasing, 378

Wald’s identity, 86Weak Law of Large Numbers, 368weighted random walk, 115Wilson’s method, 193, 220, 236window (of cutoff), 263winning streak, 55, 66

time reversal, 68wreath product, 273

MBK/107

For additional informationand updates on this book, visit

www.ams.org/bookpages/mbk-107

AMS on the Web www.ams.org

This book is an introduction to the modern theory of Markov chains, whose goal is to determine the rate of convergence to the stationary distribution, as a function of state space size and geometry. Thistopic has important connections to combinatorics,statistical physics, and theoretical computer science. Many of the techniques presented originate in thesedisciplines.

The central tools for estimating convergence times, including coupling, strong stationary times, and

and the Ising model, from statistical mechanics, and present the connection of random walks to electrical networks and apply it to estimate hitting and cover times.

of numerous universities. The second edition features three new chapters (on monotone chains, the exclusion process, and stationary times) and also includes smaller additions and corrections throughout. Updated notes at the end of each chapter inform the reader of recent research developments.

Markov Chains and Mixing Times is a magical book, managing to be both friendly and deep. It gently

covering the geometric theory of Markov chains and has much that will be new to experts. It is certainly THE book that I will use to teach from. I recommend it to all comers, an amazing achievement.

—Persi Diaconis, Mary V. Sunseri Professor of Statistics and Mathematics, Stanford University

algorithms, as well as having intrinsic interest within mathematical probability and exploiting discrete

to advanced undergraduates and yet bringing readers close to current research frontiers. This second edition adds chapters on monotone chains, the exclusion process and hitting time parameters. Having both exercises and citations to important research papers it makes an outstanding basis for either a lecture course or self-study. —David Aldous, University of California, Berkeley

Mixing time is the key to Markov chain Monte Carlo, the queen of approximation techniques. With new chapters on monotone chains, exclusion processes, and set-hitting, Markov Chains and Mixing Times is more comprehensive and thus more indispensable than ever. Prepare for an eye-opening mathematical tour! —Peter Winkler, Dartmouth College

This is the second edition of a very valuable book on the subject. The main focus is on the mixing time of Markov chains, but there is a lot of additional material.

In this edition, the authors have taken the opportunity to add new material and

graduate course and I look forward to using this edition for the same purpose in the near future. —Alan Frieze, Carnegie Mellon University

Phot

o co

urte

sy o

f Dav

id A

. Lev

in

Phot

o co

urte

sy o

f Yu

val P

eres