This book is concerned with certain aspects of discrete probability on infinite graphs that are currently in vigorous development. Of course, finite graphs are analyzed as well, but usually with the aim of understanding infinite graphs and networks. These areas of discrete probability are full of interesting, beautiful, and surprising results, many of which connect to other areas of mathematics and theoretical computer science. Numerous fascinating questions are still open.

Our major topics include random walks and their intimate connection to electrical networks; uniform spanning trees, their limiting forests, and their marvelous relationships with random walks and electrical networks; branching processes; percolation and the powerful, elegant mass-transport technique; isoperimetric inequalities and how they relate to both random walks and percolation; minimal spanning trees and forests and their connections to percolation; Hausdorff dimension, capacity, and how to understand them via trees; and random walks on Galton-Watson trees. Connections among our topics are pervasive and rich, making for surprising and enjoyable proofs.

There are three main classes of graphs on which discrete probability is most interesting, namely, trees, Cayley graphs of groups (or, more generally, transitive, or even quasi-transitive, graphs), and planar graphs. More classical discrete probability has tended to focus on the special and important case of the Euclidean lattices, Zd, which are prototypical Cayley graphs. This book develops the general theory of various probabilistic processes on graphs and then specializes to the three broad classes listed, always seeing what we can say in the case of Zd.

Besides their intrinsic interest, there are several reasons for a special study of trees. Since in most cases, analysis is easier on trees, analysis can be carried further. Then one can often either apply the results from trees to other situations or transfer to other situations the techniques developed by working on trees. Trees also occur naturally in many situations, either combinatorially or as descriptions of compact sets in Euclidean space, Rd.

In choosing our topics, we have been swayed by those results we find most striking as well as by those that do not require extensive background. Thus, the only prerequisite is basic knowledge of Markov chains and conditional expectation with respect to a σ-algebra.