Loop pruning and downward deviations for maximum local time of discrete-time simple random walks

TL;DR

This paper establishes the precise asymptotic probability of downward deviations of the maximum local time for simple random walks on high-dimensional integer lattices, introducing a novel loop-pruned random walk structure.

Contribution

It proves the lower bound for downward deviation probabilities and introduces the loop-pruned random walk as a new analytical tool.

Findings

Sharp asymptotic formula for downward deviation probability.

Introduction of the loop-pruned random walk structure.

Completion of the asymptotic analysis for maximum local time deviations.

Abstract

We study downward deviations of the maximum local time of the discrete-time simple random walk on $\mathbb{Z}^d$, $d\ge 3$. In our previous paper \cite{li2026ldmaxlocal}, the corresponding upper bound was established, while the matching lower bound was left open. In the present paper, we prove this lower bound and hence obtain the sharp asymptotic formula for the downward-deviation probability. To provide a discrete-time analogue of the jump-chain/holding-time structure used in the continuous-time argument, we introduce a new random structure which we name as {\it loop-pruned random walk} and the associated loop-pruning decomposition, which is also of independent interest.