Self-intersection local times of random walks: Exponential moments in subcritical dimensions

TL;DR

This paper analyzes the exponential moments of the p-fold self-intersection local times of a simple random walk on  lattice, deriving precise asymptotics and large deviation principles linked to inequalities like Gagliardo-Nirenberg.

Contribution

It provides the first detailed asymptotic analysis of exponential moments of self-intersection local times in subcritical dimensions, connecting variational formulas to large deviations.

Findings

Derived precise logarithmic asymptotics for exponential moments.

Established a large deviation principle for normalized local times.

Connected asymptotics to the Gagliardo-Nirenberg inequality.

Abstract

Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $\Z^d$ up to time $t$. This is the $p$-norm of the vector of the walker's local times, $\ell_t$. We derive precise logarithmic asymptotics of the expectation of $\exp\{\theta_t \|\ell_t\|_p\}$ for scales $\theta_t>0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $\theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the {\it Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation principle for $\|\ell_t\|_p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_t\gg\E[\|\ell_t\|_p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $\ll t^{1/d}$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.