Large deviations for intersection local times in critical dimension

TL;DR

This paper establishes a large deviations principle for the self-intersection local times of a simple random walk in critical dimension, revealing the probability of rare intersection events in high-dimensional settings.

Contribution

It provides the first large deviations results for intersection local times in the critical case, extending to multiple walks for integer q.

Findings

Large deviations principle for self-intersection local times at critical dimension

Results for intersection local times of multiple independent walks when q is integer

Extension of large deviations results to critical dimension in high-dimensional random walks

Abstract

Let $(X_t,t\geq0)$ be a continuous time simple random walk on $\mathbb{Z}^d$ ($d\geq3$), and let $l_T(x)$ be the time spent by $(X_t,t\geq0)$ on the site $x$ up to time $T$. We prove a large deviations principle for the $q$-fold self-intersection local time $I_T=\sum_{x\in\mathbb{Z}^d}l_T(x)^q$ in the critical case $q=\frac{d}{d-2}$. When $q$ is integer, we obtain similar results for the intersection local times of $q$ independent simple random walks.