Interlacement percolation on transient weighted graphs

TL;DR

This paper extends the concept of random interlacements to transient weighted graphs, analyzes percolation properties, and provides explicit critical values for certain graph classes, advancing understanding of percolation phenomena in complex networks.

Contribution

It generalizes the construction of random interlacements to weighted graphs and establishes key percolation thresholds and properties for various graph classes.

Findings

Critical value u_* is finite for non-amenable graphs.

u_* is positive for graphs satisfying IS_6 when combined with Z.

Explicit formula for u_* on regular trees.

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 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_*.