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Markov Chains and Mixing Times
David A. Levin, University of Oregon, Eugene, OR, Yuval Peres, Microsoft Research, Redmond, WA, and University of California, Berkeley, CA, and Elizabeth L. Wilmer, Oberlin College, OH
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2009; 371 pp; hardcover
ISBN-10: 0-8218-4739-2
ISBN-13: 978-0-8218-4739-8
List Price: US$67 Member Price: US$53.60
Order Code: MBK/58

Zeros of Gaussian Analytic Functions and Determinantal Point Processes - J Ben Hough, Manjunath Krishnapur, Yuval Peres and Balint Virag

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.

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Undergraduates, graduate students, and research mathematicians interested in probability, combinatorics, simulation, computer science, and Markov chain.

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

"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

"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