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Percolation on Infinite Graphs and Isoperimetric Inequalities

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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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Acknowledgements

We would like to thanks two anonymous referees who helped us, with their remarks and criticisms, to improve the presentation of the paper. We are also grateful to Yuval Peres for pointing out formula (A.3) in Ref. [7]. A.P. and R.S. have been partially supported by CNPq and FAPEMIG (Programa de Pesquisador Mineiro).

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Authors and Affiliations

  1. Departamento de Matemática, UFMG, 30161-970, Belo Horizonte, MG, Brazil

    Rogério G. Alves, Aldo Procacci & Remy Sanchis

  2. Departamento de Matemática, UFOP, 35400-000, Ouro Preto, MG, Brazil

    Rogério G. Alves

Authors
  1. Rogério G. Alves
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  2. Aldo Procacci
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  3. Remy Sanchis
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Correspondence to Aldo Procacci.

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Alves, R.G., Procacci, A. & Sanchis, R. Percolation on Infinite Graphs and Isoperimetric Inequalities. J Stat Phys 149, 831–845 (2012). https://doi.org/10.1007/s10955-012-0644-1

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  • DOI: https://doi.org/10.1007/s10955-012-0644-1

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