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Interlacement percolation on transient weighted graphs. (English) Zbl 1192.60108
Summary: In this article, we first extend the construction of random interlacements, introduced by A. S. Sznitman in [Upper bound on the disconnection time of discrete cylinders and random interlacements, Preprint http://www.math.ethz.ch/u/sznitman/ (2008)], 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 $${\mathcal G}$$ satisfies the isoperimetric inequality $$IS_6$$ (see (1.5)), $$u_{*}$$ is positive for the product $${\mathcal G\times\mathbb Z}$$ (where we endow $$\mathbb Z$$ 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 $$u_*$$.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
##### Keywords:
random walks; random interlacements; percolation
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