Interlacement percolation on transient weighted graphs. (English) Zbl 1192.60108

Teixeira, Augusto

Electron. J. Probab. 14, 1604-1627 (2009).

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