On large intersection and self-intersection local times in dimension five or more

TL;DR

This paper explores the similarities in large deviation behaviors of intersection and self-intersection local times in high dimensions, providing new estimates and unifying their rate functionals.

Contribution

It demonstrates the equivalence of large deviation principles for intersection and self-intersection local times in dimensions five or more, and introduces a new estimate for high level set distributions.

Findings

Same rate functional for large deviation principles of both objects

New estimate for the distribution of high level sets

Application to the geometry of intersection sets of random walks

Abstract

We show a remarkable similarity between strategies to realize a large intersection or self-intersection local times in dimension five or more. This leads to the same rate functional for large deviation principles for the two objects obtained respectively by Chen and Morters, and by the present author. We also present a new estimate for the distribution of high level sets for a random walk, with application to the geometry of the intersection set of two high level sets of the local times of two independent random walks.