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

Abstract

We discuss the logarithmic asymptotics for the upper tails of self-intersection local times of random walks on Zd. This topic has been studied a lot in the last decade, since it is a natural question, and a rich phenemonology of critical behaviours of the random walk arises, depending on the dimension, the intersection parameter, the scale, and the type of the random process. Furthermore, the question is technically difficult to handle, due to bad continuity and boundedness properties of the self-intersection local time. A couple of different techniques for studying self-intersections have been introduced yet, wich turned out to be more or less fruitful in various situations. It is the goal of this note to survey and compare some of the most fruitful techniques used in recent years