Maximum and minimum of local times for two-dimensional random walk

TL;DR

This paper analyzes the maximum and minimum local times of a two-dimensional random walk on a torus, providing leading order estimates and comparing them to Gaussian free field results, revealing different exponents.

Contribution

It offers new asymptotic estimates for local times of 2D random walks and highlights differences from Gaussian free field behaviors.

Findings

Derived leading order of maximum local times.

Estimated the number of points with extreme local times.

Identified different exponents from Gaussian free field case.

Abstract

We obtain the leading orders of the maximum and the minimum of local times for the simple random walk on the two-dimensional torus at time proportional to the cover time. We also estimate the number of points with large (or small) values of the local times. These are analogues of estimates on the two-dimensional Gaussian free fields by Bolthausen, Deuschel, and Giacomin [Ann. Probab., 29 (2001)] and Daviaud [Ann. Probab., 34 (2006)], but we have different exponents from the case of the Gaussian free field.