Infinite Graphs with a Nontrivial Bond Percolation Threshold: Some Sufficient Conditions |
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Authors: | Procacci Aldo Scoppola Benedetto |
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Institution: | 1. Departamento de Matemática, UFMG, Caixa Postal 1621, 30161-970, Belo Horizonte, MG, Brazil 2. Dipartimento di Matematica, Universitá di Roma “Tor Vergata,”, Via della Ricerca Scientifica, 00133, Roma, Italy
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Abstract: | We consider Bernoulli bond percolation on infinite graphs and we identify a class of graphs for which the critical percolation probability is strictly less than 1. The graphs in this class have to fulfill conditions stated in terms of a minimal cut set property and a logarithmic isoperimetric inequality. For the particular case of planar graphs the condition on minimal cut sets can be be replaced by the assumption that the dual of the graph is bounded degree. |
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