Large deviations for self-intersection local times in subcritical dimensions

TL;DR

This paper extends large deviations results for self-intersection local times of random walks on integer lattices to a broader subcritical domain, unifying proof techniques across different regimes.

Contribution

It broadens the scope of large deviations principles for SILT to the entire subcritical domain and unifies proof methods across critical and supercritical cases.

Findings

Extended large deviations results to the full subcritical domain p(d-2)<d.

Unified proof techniques for different regimes of random walks.

Provided a broader scale of deviations for SILT.

Abstract

Let $(X_t,t\geq 0)$ be a random walk on $\mathbb{Z}^d$. Let $ l_t(x)= \int_0^t \delta_x(X_s)ds$ be the local time at site $x$ and $ I_t= \sum\limits_{x\in\mathbb{Z}^d} l_t(x)^p $ the p-fold self-intersection local time (SILT). Becker and K\"onig have recently proved a large deviations principle for $I_t$ for all $(p,d)\in\mathbb{R}^d\times\mathbb{Z}^d$ such that $p(d-2)<2$. We extend these results to a broader scale of deviations and to the whole subcritical domain $p(d-2)<d$. Moreover we unify the proofs of the large deviations principle using a method introduced by Castell for the critical case $p(d-2)=d$ and developed by Laurent for the critical and supercritical case $p(d-\alpha)\geq d$ of $\alpha$-stable random walk.