Interlacement percolation on transient weighted graphs - Mathematics > ProbabilityReport as inadecuate

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Abstract: In this article, we first extend the construction of random interlacements,introduced by A.S. Sznitman in arXiv:0704.2560, to the more general settingof transient weighted graphs. We prove the Harris-FKG inequality for this modeland analyze some of its properties on specific classes of graphs. For the caseof non-amenable graphs, we prove that the critical value u * for thepercolation of the vacant set is finite. We also prove that, once G satisfiesthe isoperimetric inequality IS 6 see 1.5, u * is positive for the productGxZ where we endow Z with unit weights. When the graph under consideration isa tree, we are able to characterize the vacant cluster containing some fixedpoint in terms of a Bernoulli independent percolation process. For the specificcase of regular trees, we obtain an explicit formula for the critical valueu *.

Author: Augusto Teixeira


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