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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 54 open journal systems 


Interlacement percolation on transient weighted graphs

Augusto Teixeira, Eidgenössische Technische Hochschule Zürich


Abstract
In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [14], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and analyze some of its properties on specific classes of graphs. For the case of non-amenable graphs, we prove that the critical value u* for the percolation of the vacant set is finite. We also prove that, once G satisfies the isoperimetric inequality IS_6 (see (1.5)), u* is positive for the product GxZ (where we endow Z with unit weights). When the graph under consideration is a tree, we are able to characterize the vacant cluster containing some fixed point in terms of a Bernoulli independent percolation process. For the specific case of regular trees, we obtain an explicit formula for the critical value u*.


Full text: PDF

Pages: 1604-1627

Published on: July 9, 2009
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Electronic Journal of Probability. ISSN: 1083-6489