Disconnection, random walks, and random interlacements

TL;DR

This paper establishes asymptotic upper bounds on the probability that random interlacements or simple random walks disconnect large boxes in high-dimensional lattices, capturing the main exponential decay rate.

Contribution

It provides the first asymptotic upper bounds for disconnection probabilities in the context of random interlacements and simple random walks in dimensions d ≥ 3.

Findings

Derived asymptotic upper bounds for disconnection probabilities

Bounds match the principal exponential decay rate

Applicable to both random interlacements and simple random walks

Abstract

We consider random interlacements on Z^d, with d bigger or equal to 3, when their vacant set is in a strongly percolative regime. We derive an asymptotic upper bound on the probability that the random interlacements disconnect a box of large side-length from the boundary of a larger homothetic box. As a corollary, we obtain an asymptotic upper bound on a similar quantity, where the random interlacements are replaced by the simple random walk. It is plausible, but open at the moment, that these asymptotic upper bounds match the asymptotic lower bounds obtained by Xinyi Li and the author in arXiv:1310.2177, for random interlacements, and by Xinyi Li in a recent article, for the simple random walk. In any case, our bounds capture the principal exponential rate of decay of these probabilities, in any dimension d bigger or equal to 3.