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

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.