A quenched limit theorem for the local time of random walks on Z2

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, (X, Y)l 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. (C) 2008 Elsevier B.V. All rights reserved.