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.
Citation
Carsten Thomassen. "Isoperimetric Inequalities and Transient Random Walks on Graphs." Ann. Probab. 20 (3) 1592 - 1600, July, 1992. https://doi.org/10.1214/aop/1176989708
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