This introduction to random walks on infinite graphs gives particular emphasis to graphs with polynomial volume growth. It offers an overview of analytic methods, starting with the connection between random walks and electrical resistance, and then proceeding to study the use of isoperimetric and Poincare inequalities. The book presents rough isometries and looks at the properties of a graph that are stable under these transformations. Applications include the 'type problem': determining whether a graph is transient or recurrent. The final chapters show how geometric properties of the graph can be used to establish heat kernel bounds, that is, bounds on the transition probabilities of the random walk, and it is proved that Gaussian bounds hold for graphs that are roughly isometric to Euclidean space. Aimed at graduate students in mathematics, the book is also useful for researchers as a reference for results that are hard to find elsewhere.
Martin T. Barlow is Professor in the Mathematics Department at the University of British Columbia. He was one of the founders of the mathematical theory of diffusions on fractals, and more recently has worked on random walks on random graphs. He gave a talk at the International Congress of Mathematicians (ICM) in 1990, and was elected a Fellow of the Royal Society of Canada in 1998 and a Fellow of the Royal Society in 2005. He is the winner of the Jeffrey-Williams Prize of the Canadian Mathematical Society and the CRM-Fields-PIMS Prize of the three Canadian mathematics institutes (the Centre de recherches mathematiques, the Fields Institute, and the Pacific Institute for the Mathematical Sciences).