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

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/162461Let $X$ 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.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 problemProbability60J15 (Primary) 60K37, 60J55, 60F05 (Secondary)A quenched limit theorem for the local time of random walks on \Z^2text