On large deviations for the cover time of two-dimensional torus

TL;DR

This paper establishes large deviation probabilities for the cover time of a two-dimensional discrete torus, revealing the exponential decay rate depending on a parameter, using decoupling techniques via soft local times.

Contribution

It introduces a novel approach to analyze large deviations in cover times by decoupling the walker's trace into independent excursions using soft local times.

Findings

Derived precise large deviation probabilities for cover times

Established exponential decay rates depending on the parameter gamma

Applied decoupling method to analyze walker's trace

Abstract

Let $\mathcal{T}_n$ be the cover time of two-dimensional discrete torus $\mathbb{Z}^2_n=\mathbb{Z}^2/n\mathbb{Z}^2$. We prove that $\mathbb{P}[\mathcal{T}_n\leq \frac{4}{\pi}\gamma n^2\ln^2 n]=\exp(-n^{2(1-\sqrt{\gamma})+o(1)})$ for $\gamma\in (0,1)$. One of the main methods used in the proofs is the decoupling of the walker's trace into independent excursions by means of soft local times.