Decoupling inequalities and interlacement percolation on G x Z

TL;DR

This paper investigates percolation properties of random interlacements on product graphs G x Z, establishing decoupling inequalities and critical thresholds, with new results even for standard lattice cases.

Contribution

It develops decoupling inequalities and determines the finiteness and positivity of the critical level u_* for percolation on G x Z, extending known results to broader graph classes.

Findings

Critical level u_* is finite for the model.

u_* is positive when alpha ≥ 1 + beta/2.

Several new exponential decay bounds in the percolation phases.

Abstract

We study the percolative properties of random interlacements on the product of G with the integer line Z, when G is a weighted graph satisfying certain sub-Gaussian estimates attached to the parameters alpha > 1, measuring the volume growth on G, and beta between 2 and alpha + 1, measuring the sub-diffusive nature of the random walk on G. We develop decoupling inequalities, which are a key tool in showing that the critical level u_* for the percolation of the vacant set of random interlacements is always finite in our set-up, and that it is positive when alpha \geq 1 + beta/2. We also obtain several stretched exponential controls both in the percolative and non-percolative phases of the model. Even in the case where G = Z^d, d \geq 2, several of these results are new.