Isoperimetric Inequalities and Transient Random Walks on Graphs
Thomassen, Carsten
Ann. Probab., Tome 20 (1992) no. 4, p. 1592-1600 / Harvested from Project Euclid
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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.
Publié le : 1992-07-14
Classification:  Isoperimetric inequalities,  transient trees,  60J15,  94C15 
@article{1176989708,
     author = {Thomassen, Carsten},
     title = {Isoperimetric Inequalities and Transient Random Walks on Graphs},
     journal = {Ann. Probab.},
     volume = {20},
     number = {4},
     year = {1992},
     pages = { 1592-1600},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176989708}
}

Thomassen, Carsten. Isoperimetric Inequalities and Transient Random Walks on Graphs. Ann. Probab., Tome 20 (1992) no. 4, pp.  1592-1600. http://gdmltest.u-ga.fr/item/1176989708/