Large deviations for maximum local time of simple random walk in dimensions $d\ge 3$

TL;DR

This paper derives precise asymptotic probabilities and fluctuation results for the maximum local time of simple random walks in dimensions three and higher.

Contribution

It provides sharp asymptotic estimates and Gumbel-type fluctuation results for the maximum local time in high-dimensional simple random walks.

Findings

Sharp asymptotic probabilities for large deviations of maximum local time

Gumbel-type fluctuations around the logarithmic scale

Results applicable for dimensions d ≥ 3

Abstract

We obtain sharp asymptotic probabilities for upward and downward large deviations of the maximum local time of simple random walks on $\mathbb{Z}^d$, $d \ge 3$. We also obtain Gumbel-type fluctuations around the logarithmic scale of the maximum local time.