Isoperimetric Inequalities and Transient Random Walks on Graphs

Abstract

recurrent. We show that any graph satisfying an isoperimetric inequality only slightly stronger than that of Z2 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 E f(k)-2 converges. This can be applied to parabolicity versus hyperbolicity of surfaces. 1. Introduction. Let G be a connected graph which is locally finite, that is, all vertices have finite degree. We consider a random walk starting at a vertex v, say, such that at any vertex u, the walk proceeds to a neighbour with probability 1/d(u), where d(u) is the degree (i.e., the number of neighbours) of u. The graph G is recurrent if we revisit v with probability 1. Otherwise G is transient. It is well known that the three-dimensional grid Z3 is transient while Z2 is recurrent. More generally, Nash-Williams [13] (see also [4, 10]) proved that any graph with smaller growth rate than Z2 is recurrent. Lyons [10] showed that certain subgraphs of grids are transient provided they grow just a little faster than Z2. Other results, in terms of isoperimetric inequalities, supporting the statement that Z2 is, in a sense, an extreme recurrent graph, can be derived from work of Fernandez [5], Grigor'yan [7] and Varopoulos [15]. (Varopoulos [16] used results of Gromov [8] to characterize completely the recurrent Cayley graphs.) We shall carry these results further. If V is a vertex set in G, then aV will denote the boundary of V, that is, the set of vertices of V having neighbours outside of V. Let f be a nondecreasing positive real function defined on the natural numbers. We say that G satisfies an f-isoperimetric inequality if there exists a constant c > 0 such that, for each finite vertex set V of G,