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

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