A quenched limit theorem for the local time of random walks on Z2Z2

Let XX and YY be two independent random walks on Z2Z2 with zero mean and finite variances, and let Lt(X,Y)Lt(X,Y) be the local time of X−YX−Y at the origin at time tt. We show that almost surely with respect to YY, Lt(X,Y)/logtLt(X,Y)/logt conditioned on YY converges in distribution to an exponential random variable with the same mean as the distributional limit of Lt(X,Y)/logtLt(X,Y)/logt 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.