No abstract

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.

Reviewed by David Aldous Unlike most books reviewed in the Intelligencer this is definitely a textbook. It assumes knowledge one might acquire in the first two years of an undergraduate mathematics program-basic mathematical probability, plus linear algebra, a little graph theory and the infamous concept of "mathematical maturity". It has the theorem-proof style of pure mathematics, but with friendly explanations of intuition and motivation.