
Book Details2009; 371 ppMSC: Primary 60; 65; 68; 82
Now available in Second Edition: MBK/107
This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. The authors develop the key tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. Markov Chains and Mixing Times is meant to bring the excitement of this active area of research to a wide audience.
ReadershipUndergraduates, graduate students, and research mathematicians interested in probability, combinatorics, simulation, computer science, and Markov chain.

Table of Contents

Part I. Basic Methods and Examples

Chapter 1. Introduction to finite Markov chains

Chapter 2. Classical (and useful) Markov chains

Chapter 3. Markov chain Monte Carlo: Metropolis and Glauber chains

Chapter 4. Introduction to Markov chain mixing

Chapter 5. Coupling

Chapter 6. Strong stationary times

Chapter 7. Lower bounds on mixing times

Chapter 8. The symmetric group and shuffling cards

Chapter 9. Random walks on networks

Chapter 10. Hitting times

Chapter 11. Cover times

Chapter 12. Eigenvalues

Part II. The Plot Thickens

Chapter 13. Eigenfunctions and comparison of chains

Chapter 14. The transportation metric and path coupling

Chapter 15. The Ising model

Chapter 16. From shuffling cards to shuffling genes

Chapter 17. Martingales and evolving sets

Chapter 18. The cutoff phenomenon

Chapter 19. Lamplighter walks

Chapter 20. Continuoustime chains

Chapter 21. Countable state space chains

Chapter 22. Coupling from the past

Chapter 23. Open problems

Appendix A. Background material

Appendix B. Introduction to simulation

Appendix C. Solutions to selected exercises


Additional Material

Reviews

Markov Chains and Mixing Times is a magical book, managing to be both friendly and deep. It gently introduces probabilistic techniques so that an outsider can follow. At the same time, it is the first book 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 
A superb introduction to Markov chains which treats riffle shuffling and stationary times...
Sami Assaf, University of Southern California, Persi Diaconis, Stanford University, and Kannan Soundararajan, Stanford University, in their paper "Riffle Shuffles with Biased Cuts" 
In this book, [the authors] rapidly take a wellprepared undergraduate to the frontiers of research. Short, focused chapters with clear logical dependencies allow readers to use the book in multiple ways.
CHOICE Magazine 
This book is a beautiful introduction to Markov chains and the analysis of their convergence towards a stationary distribution. Personally, I enjoyed very much the lucid and clear writing style of the exposition in combination with full mathematical rigor and the fascinating relations of the theory of Markov chains to several other mathematical areas.
Zentralblatt MATH 
Throughout the book, the authors generously provide concrete examples that motivate theory and illustrate ideas. I expect this superb book to be widely used in graduate courses around the world, and to become a standard reference.
Mathematical Reviews


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 Book Details
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Now available in Second Edition: MBK/107
This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. The authors develop the key tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. Markov Chains and Mixing Times is meant to bring the excitement of this active area of research to a wide audience.
Undergraduates, graduate students, and research mathematicians interested in probability, combinatorics, simulation, computer science, and Markov chain.

Part I. Basic Methods and Examples

Chapter 1. Introduction to finite Markov chains

Chapter 2. Classical (and useful) Markov chains

Chapter 3. Markov chain Monte Carlo: Metropolis and Glauber chains

Chapter 4. Introduction to Markov chain mixing

Chapter 5. Coupling

Chapter 6. Strong stationary times

Chapter 7. Lower bounds on mixing times

Chapter 8. The symmetric group and shuffling cards

Chapter 9. Random walks on networks

Chapter 10. Hitting times

Chapter 11. Cover times

Chapter 12. Eigenvalues

Part II. The Plot Thickens

Chapter 13. Eigenfunctions and comparison of chains

Chapter 14. The transportation metric and path coupling

Chapter 15. The Ising model

Chapter 16. From shuffling cards to shuffling genes

Chapter 17. Martingales and evolving sets

Chapter 18. The cutoff phenomenon

Chapter 19. Lamplighter walks

Chapter 20. Continuoustime chains

Chapter 21. Countable state space chains

Chapter 22. Coupling from the past

Chapter 23. Open problems

Appendix A. Background material

Appendix B. Introduction to simulation

Appendix C. Solutions to selected exercises

Markov Chains and Mixing Times is a magical book, managing to be both friendly and deep. It gently introduces probabilistic techniques so that an outsider can follow. At the same time, it is the first book 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 
A superb introduction to Markov chains which treats riffle shuffling and stationary times...
Sami Assaf, University of Southern California, Persi Diaconis, Stanford University, and Kannan Soundararajan, Stanford University, in their paper "Riffle Shuffles with Biased Cuts" 
In this book, [the authors] rapidly take a wellprepared undergraduate to the frontiers of research. Short, focused chapters with clear logical dependencies allow readers to use the book in multiple ways.
CHOICE Magazine 
This book is a beautiful introduction to Markov chains and the analysis of their convergence towards a stationary distribution. Personally, I enjoyed very much the lucid and clear writing style of the exposition in combination with full mathematical rigor and the fascinating relations of the theory of Markov chains to several other mathematical areas.
Zentralblatt MATH 
Throughout the book, the authors generously provide concrete examples that motivate theory and illustrate ideas. I expect this superb book to be widely used in graduate courses around the world, and to become a standard reference.
Mathematical Reviews