How a random walk covers a finite lattice

A random walker is confined to a finite periodic d-dimensional lattice of N initially white sites. When visited by the walk a site is colored black. After t steps of the walk, for t scaled appropriately with N, we determined the structure of the set of white sites. The variance of their number has a line of critical points in the td plane, which separates a mean-field region from a region with enhanced fluctuations. At d = 2 the critical point becomes a critical interval. Moreover, for d = 2 the set of white sites is fractal with a fractal dimensionality whose t-dependence we determine.