Percolation on Infinite Graphs and Isoperimetric Inequalities |
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Authors: | Rogério G Alves Aldo Procacci Remy Sanchis |
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Institution: | 1. Departamento de Matem??tica, UFMG, 30161-970, Belo Horizonte, MG, Brazil 2. Departamento de Matem??tica, UFOP, 35400-000, Ouro Preto, MG, Brazil
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Abstract: | 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 when 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. |
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