Interlacement percolation on transient weighted graphs

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 IS6 (see (1.5)), u∗ is positive for the product G×Z (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∗.