Large Deviations of Cover Time of Tori in Dimensions $d\geq 3$

TL;DR

This paper investigates the probabilities of large deviations in the cover time of high-dimensional tori by simple random walks, establishing bounds and asymptotics that deepen understanding of random walk cover times.

Contribution

It provides matching bounds for the probability of unusually small cover times and derives precise asymptotics in certain regimes, utilizing a novel coupling with random interlacements.

Findings

Established lower bounds matching known upper bounds for cover time deviations.

Derived sharp asymptotics for cover times when the deviation parameter exceeds a certain threshold.

Utilized a new coupling technique between random walks and random interlacements.

Abstract

We consider large deviations of the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, $d \geq 3$ by simple random walk. We prove a lower bound on the probability that the cover time is smaller than $\gamma\in (0,1)$ times its expected value, with exponents matching the upper bound from [Goodman-den Hollander, Probab. Theory Related Fields (2014)] and [Comets-Gallesco-Popov-Vachkovskaia, Electron. J. Probab. (2013)]. Moreover, we derive sharp asymptotics for $\gamma \in (\frac{d+2}{2d},1)$. The strong coupling of the random walk on the torus and random interlacements developed in a recent work [Pr\'evost-Rodriguez-Sousi, arXiv:2309.03192] serves as an important ingredient in the proofs.