How uniformly a random walker covers a finite lattice

We study the distribution of the number of visits a random walker makes at a given site on a finite lattice with N sites, during a very long walk which visits each site a large number of times. For regular hypercubic lattices in all dimensions we find normal central limit behavior, but with anomalously large variance in ⩽2 dimensions. In 2 dimensions, the ratio of the variance over the average number of visits increases logarithmically with N, while it increases ≈N in one dimension. We confront this with the case of self-repelling (also called “true self-avoiding”) walks. There, the variance remains bounded for all times, and increases logarithmically with N in 2 dimensions.