Cover times in the discrete cylinder

TL;DR

This paper demonstrates that the rescaled and recentered cover times of finite sets in the discrete cylinder by simple random walk converge to the Gumbel distribution, using a novel approach involving random interlacements.

Contribution

It introduces a new method using random interlacements and develops a stronger coupling between simple random walk and interlacements for analyzing cover times.

Findings

Cover times converge to Gumbel distribution as set size increases.

New coupling technique between random walk and interlacements.

Applications to various covering problems in the discrete cylinder.

Abstract

This article proves that, in terms of local times, the rescaled and recentered cover times of finite subsets of the discrete cylinder by simple random walk converge in law to the Gumbel distribution, as the cardinality of the set goes to infinity. As applications we obtain several other results related to covering in the discrete cylinder. Our method is new and involves random interlacements, which were introduced by Sznitman in arXiv:0704.2560. To enable the proof we develop a new stronger coupling of simple random walk in the cylinder and random interlacements, which is also of independent interest.