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. This
topic has important connections to combinatorics, statistical physics,
and theoretical computer science. Many of the techniques presented
originate in these disciplines.
The central tools for estimating convergence times, including
coupling, strong stationary times, and spectral methods, are
developed. The authors discuss many examples, including card shuffling 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.
The first edition has been used in courses in mathematics and
computer science departments 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.
Undergraduate and graduate students interested in the modern theory of Markov chains.
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
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 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.
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 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