Markov Chains and Mixing Times

David A. Levin, University of Oregon, Eugene, OR, and Yuval Peres, Microsoft Research, Redmond, WA

• Basic methods and examplesIntroduction to finite Markov chainsClassical (and useful) Markov chainsMarkov chain Monte Carlo: Metropolis and Glauber chainsIntroduction to Markov chain mixingCouplingStrong stationary timesLower bounds on mixing timesThe symmetric group and shuffling cardsRandom walks on networksHitting timesCover timesEigenvaluesThe plot thickensEigenfunctions and comparison of chainsThe transportation metric and path couplingThe Ising modelFrom shuffling cards to shuffling genesMartingales and evolving setsThe cutoff phenomenonLamplighter walksContinuous-time chainsCountable state space chainsMonotone chainsThe exclusion processCesaro mixing time, stationary times, and hitting large setsCoupling from the pastOpen problemsBackground materialIntroduction to simulationErgodic theoremSolutions to selected exercises

Taking a recently emerged topic with a massive research literature and writing a textbook that can take a student from basic undergraduate mathematics to the ability to read current research papers is a hugely impressive achievement. This book will long remain the definitive required reading for anyone wishing to engage the topic more than superficially."" — David Aldous, Mathematical Intelligencer ""Mixing times are an active research topic within many fields from statistical physics to the theory of algorithms, as well as having intrinsic interest within mathematical probability and exploiting discrete analogs of important geometry concepts. The first edition became an instant classic, being accessible 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 ""The study of finite Markov chains has recently attracted increasing interest from a variety of researchers. 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 bring the reader up to date on the latest research. I have used the first edition in a graduate course and I look forward to using this edition for the same purpose in the near future."" —Alan Frieze, Carnegie Mellon University Praise for the first edition: ""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 ""In this book, [the authors] rapidly take a well-prepared undergraduate to the frontiers of research. Short, focused chapters with clear logical dependencies allow readers to use the book in multiple ways."" —CHOICE Magazine