On the Domination of Random Walk on a Discrete Cylinder by Random Interlacements

TL;DR

This paper establishes a stochastic domination of the local behavior of random walk on a discrete cylinder by random interlacements, leading to bounds on disconnection time and extending previous results to lower dimensions.

Contribution

It introduces a new domination control of random walk local pictures by random interlacements in lower dimensions, improving understanding of disconnection times.

Findings

Domination control of random walk by interlacements in dimensions d≥2.

Lower bounds on disconnection time of the discrete cylinder.

Proof of tightness of the ratio of base size squared to disconnection time.

Abstract

We consider simple random walk on a discrete cylinder with base a large d-dimensional torus of side-length N, when d is two or more. We develop a stochastic domination control on the local picture left by the random walk in boxes of side-length almost of order N, at certain random times comparable to the square of the number of sites in the base. We show a domination control in terms of the trace left in similar boxes by random interlacements in the infinite (d+1)-dimensional cubic lattice at a suitably adjusted level. As an application we derive a lower bound on the disconnection time of the discrete cylinder, which as a by-product shows the tightness of the laws of the ratio of the square of the number of sites in the base to the disconnection time. This fact had previously only been established when d is at least 17, in arXiv: math/0701414.