Upper tails of self-intersection local times of random walks: survey of proof techniques

TL;DR

This paper surveys advanced proof techniques for analyzing the probability of large deviations in the self-intersection local times of random walks, highlighting their connections to statistical mechanics and large deviation theory.

Contribution

It provides a comprehensive overview of heuristics and recent methods used to establish upper bounds for these probabilities, emphasizing their complexity and sophistication.

Findings

Summarizes key heuristics and techniques in the field.

Highlights the difficulty of proving upper bounds.

Connects the problem to statistical mechanics and large deviation theory.

Abstract

The asymptotics of the probability that the self-intersection local time of a random walk on $\Z^d$ exceeds its expectation by a large amount is a fascinating subject because of its relation to some models from Statistical Mechanics, to large-deviation theory and variational analysis and because of the variety of the effects that can be observed. However, the proof of the upper bound is notoriously difficult and requires various sophisticated techniques. We survey some heuristics and some recently elaborated techniques and results. This is an extended summary of a talk held on the CIRM-conference on {\it Excess self-intersection local times, and related topics} in Luminy, 6-10 Dec., 2010.