Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

‘This long-awaited work focuses on one of the most interesting and important parts of probability theory. Half a century ago, most work on models such as random walks, Ising, percolation and interacting particle systems concentrated on processes defined on the d-dimensional Euclidean lattice. In the intervening years, interest has broadened dramatically to include processes on more general graphs, with trees being a particularly important case. This led to new problems and richer behavior, and as a result, to the development of new techniques. The authors are two of the major developers of this area; their expertise is evident throughout.’

Thomas M. Liggett - University of California, Los Angeles

‘Masterly, beautiful, encyclopaedic, and yet browsable - this great achievement is obligatory reading for anyone working near the conjunction of probability and network theory.’

Geoffrey Grimmett - University of Cambridge

‘For the last ten years, I have not let a doctoral student graduate without reading this [work]. Sadly, the earliest of those students are missing a considerable amount of material that the bound and published edition contains. Not only are the classical topics of random walks, electrical theory, and uniform spanning trees covered in more coherent fashion than in any other source, but this book is also the best place to learn about a number of topics for which the other choices for textual material are limited. These include mass transport, random walk boundaries, and dimension and capacity in the context of Markov processes.’

Robin Pemantle - University of Pennsylvania

‘Lyons and Peres have done an amazing job of motivating their material and of explaining it in a conversational and accessible fashion. Even though the book emphasizes probability on infinite graphs, it is one of my favorite references for probability on finite graphs. If you want to understand random walks, isoperimetry, random trees, or percolation, this is where you should start.’

Daniel Spielman - Yale University, Connecticut

‘This long-awaited book offers a splendid account of several major areas of discrete probability. Both authors have made outstanding contributions to the subject, and the exceptional quality of the book is largely due to their high level of mastery of the field. Although the only prerequisites are basic probability theory and elementary Markov chains, the book succeeds in providing an elegant presentation of the most beautiful and deepest results in the various areas of probability on graphs. The powerful techniques that made these results available, such as the use of isoperimetric inequalities or the mass-transport principle, are also presented in a detailed and self-contained manner. This book will be indispensable to any researcher working in probability on graphs and related topics, and it will also be a must for anybody interested in the recent developments of probability theory.’

Jean-François Le Gall - Université Paris-Sud

'This is a very timely book about a circle of actively developing subjects in discrete probability. No wonder that it became very popular two decades before publication, while still in development. Not only a comprehensive reference source, but also a good textbook to learn the subject, it will be useful for specialists and newcomers alike.'

Stanislav Smirnov - Université of Genève

'A glorious labor of love, compiled over more than two decades of work, that brilliantly surveys the deep and expansive relationships between random trees and other areas of mathematics. Rarely does one encounter a text so exquisitely well written or enjoyable to read. One cannot take more than a few steps in modern probability without encountering one of the topics surveyed here. A truly essential resource.'

Scott Sheffield - Massachusetts Institute of Technology

'There is much to be learned from studying this book. Many of the ideas and tools are useful in a wide variety of different contexts … Geoff Grimmett’s quote on the cover calls the book ‘Masterly, beautiful, encyclopedic and yet browsable.’ I totally agree. Even though it is freely available on the web, you should buy a copy of the book.'

Richard Durrett Source: Mathematical Association of America Reviews (www.maa.org)

'This is a monumental book covering a lot of interesting problems in discrete probability, written by two experts in the field … The authors have done a great job of providing full proofs of all main results, hence creating a self-contained reference in this area.'

Abbas Mehrabian Source: Zentralblatt MATH

'This long-awaited book, a project that started in 1993, is bound to be the main reference in the fascinating field of probability on trees and weighted graphs. The authors are the leading experts behind the tremendous developments experienced in the subject in recent decades, where the underlying networks evolved from classical lattices to general graphs … This pedagogically written book is a marvelous support for several courses on topics from combinatorics, Markov chains, geometric group theory, etc., as well as on their inspiring relationships. The wealth of exercises (with comments provided at the end of the book) will enable students and researchers to check their understanding of this fascinating mathematics.'

Laurent Miclo Source: MathSciNet