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

Abstract

Let $X$ and $Y$ be two independent random walks on ${\mathbb{Z}}^2$ 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.

To appear in Stochastic Processes and Their Applications. Updated version. 16 pages. Added discussion on d=1 and d\geq 3 as well as an open problem