Sharp asymptotics of disconnection time of large cylinders by simple and biased random walks

TL;DR

This paper establishes precise asymptotic bounds for the time it takes for simple and biased random walks to disconnect large cylinders, using a novel coupling technique with random interlacements.

Contribution

It provides sharp asymptotic bounds for disconnection times of large cylinders by simple and biased random walks, improving previous bounds and introducing a strong coupling method.

Findings

Sharp lower bound matching previous upper bound for simple random walk

Asymptotic bounds for biased walks when bias is moderate

Development of a strong coupling between walk trace and random interlacements

Abstract

We investigate the asymptotic disconnection time of a large discrete cylinder $(\mathbb{Z}/N\mathbb{Z})^{d}\times \mathbb{Z}$, $d\geq 2$, by simple and biased random walks. For simple random walk, we derive a sharp asymptotic lower bound that matches the upper bound from [Sznitman, Ann. Probab., 2009]. For biased walks, we obtain bounds that asymptotically match in the principal order when the bias is not too strong, which greatly improves non-matching bounds from [Windisch, Ann. Appl. Probab., 2008]. As a crucial tool in the proof, we also obtain a "very strong" coupling between the trace of random walk on the cylinder and random interlacements, which is of independent interest.