Gumbel fluctuations for cover times in the discrete torus

TL;DR

This paper proves that the fluctuations of cover times for simple random walks on high-dimensional discrete tori follow the Gumbel distribution, confirming a long-standing conjecture and introducing a new coupling method with random interlacements.

Contribution

It establishes the Gumbel distribution as governing the cover time fluctuations and develops a novel coupling between random interlacements and random walk in the torus.

Findings

Cover time fluctuations follow Gumbel distribution in high dimensions.

New coupling method between random interlacements and random walk.

Results confirm longstanding conjecture in the field.

Abstract

This work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side-length are governed by the Gumbel extreme value distribution. This result was conjectured for example in the book by Aldous & Fill. We also derive some corollaries which qualitatively describe "how" covering happens. In addition, we develop a new and stronger coupling of the model of random interlacements, introduced by Sznitman, and random walk in the torus. This coupling is used to prove the cover time result and is also of independent interest.