Tuesday, November 6, 2012

1211.0948 (Rogério G. Alves et al.)

Percolation on infinite graphs and isoperimetric inequalities    [PDF]

Rogério G. Alves, Aldo Procacci, Remy Sanchis
We consider the Bernoulli bond percolation process (with parameter $p$) on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non-trivial percolation threshold based either on a single isoperimetric inequality if the graph has a bi-infinite geodesic, or two isoperimetric inequalities if the graph has not a bi-infinite geodesic. This new criterion extends previous criteria and brings together a large class of amenable graphs (such as regular lattices) and non-amenable graphs (such trees). We also study the finite connectivity in graphs satisfying the new general criterion and show that graphs in this class with a bi-infinite geodesic always have finite connectivity functions with exponential decay as $p$ is sufficiently close to one. On the other hand, we show that there are graphs in the same class with no bi-infinite geodesic for which the finite connectivity decays sub-exponentially (down to polynomially) in the highly supercritical phase even for $p$ arbitrarily close to one.
View original: http://arxiv.org/abs/1211.0948

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