## The Annals of Probability

### Isoperimetric Inequalities and Transient Random Walks on Graphs

Carsten Thomassen

#### 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.

#### Article information

Source
Ann. Probab. Volume 20, Number 3 (1992), 1592-1600.

Dates
First available: 19 April 2007

http://projecteuclid.org/euclid.aop/1176989708

JSTOR

Digital Object Identifier
doi:10.1214/aop/1176989708

Mathematical Reviews number (MathSciNet)
MR1175279

Zentralblatt MATH identifier
0756.60065

Subjects
Primary: 60J15