Markov Chains and Mixing Times

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 ...

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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.

Simon Asher Levin (born April 22, 1941) is an American ecologist. He is a James S. McDonnell Distinguished University Professor in the Department of Ecology and Evolution and the Director of the Center for BioComplexity at Princeton University.[1] He specializes in using mathematical modeling and empirical studies in the understanding of macroscopic patterns of ecosystems and b...

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Simon Asher Levin (born April 22, 1941) is an American ecologist. He is a James S. McDonnell Distinguished University Professor in the Department of Ecology and Evolution and the Director of the Center for BioComplexity at Princeton University.[1] He specializes in using mathematical modeling and empirical studies in the understanding of macroscopic patterns of ecosystems and biological diversities.

Introduction to finite Markov chains --
Classical (and useful) Markov chains --
Markov chain Monte Carlo : metropolis and Glauber chains --
Introduction to Markov chain mixing --
Coupling --
Strong stationary times --
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Introduction to finite Markov chains --
Classical (and useful) Markov chains --
Markov chain Monte Carlo : metropolis and Glauber chains --
Introduction to Markov chain mixing --
Coupling --
Strong stationary times --
Lower bounds on mixing times --
The symmetric group and shuffling cards --
Random walks on networks --
Hitting times --
Cover times --
Eigenvalues --
Eigenfunctions and comparison of chains --
The transportation metric and path coupling --
The Ising model --
From shuffling cards to shuffling genes --
Martingales and evolving sets --
The cutoff phenomenon --
Lamplighter walks --
Continuous-time chains --
Countable state space chains --
Coupling from the past --
Open problems --
Appendix A : Background material --
Appendix B : Introduction to simulation
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