A quenched limit theorem for the local time of random walks on \Z^2

TL;DR

This paper proves a quenched limit theorem for the local time of the difference of two independent random walks on , showing that conditioned on one walk, the normalized local time converges to an exponential distribution, relevant to models like the parabolic Anderson model.

Contribution

It establishes a quenched limit theorem for the local time of two independent  random walks, revealing the distributional convergence conditioned on one walk.

Findings

Normalized local time converges to an exponential distribution.

Almost sure convergence conditioned on one walk.

Relevance to the parabolic Anderson model with a moving catalyst.

Abstract

Let $X$ and $Y$ be two independent random walks on $\Z^2$ with zero mean and finite variances, and let $L_t(X,Y)$ be the local time of $X-Y$ at the origin at time $t$. We show that almost surely with respect to $Y$, $L_t(X,Y)/\log t$ conditioned on $Y$ converges in distribution to an exponential random variable with the same mean as the distributional limit of $L_t(X,Y)/\log t$ without conditioning. This question arises naturally from the study of the parabolic Anderson model with a single moving catalyst, which is closely related to a pinning model.