Isoperimetric Inequalities and Transient Random Walks on Graphs

Abstract

The two-dimensional grid $Z^2$ and any graph of smaller growth rate is recurrent. We show that any graph satisfying an isoperimetric inequality only slightly stronger than that of $Z^2$ is transient. More precisely, if $f(k)$ denotes the smallest number of vertices in the boundary of a connected subgraph with $k$ vertices, then the graph is transient if the infinite sum $\sum f(k)^{-2}$ converges. This can be applied to parabolicity versus hyperbolicity of surfaces.