COVERING OF A FINITE LATTICE BY A RANDOM-WALK

We study the covering process by a simple random walk of a d-dimensional periodic hypercubic lattice of N sites. In d = 1, the probability L(N)(x) for site x to be the last site visited in this covering process does not depend on x, as long as x is not the starting point of the walk. We argue that in dimensions d > 2, the probability L(N)(x) approaches a constant value according to a Coulomb law: L(N)(x) congruent-to 1/N(1 - const/\x\d-2), valid for \x\ small on the scale N1/d, whereas it behaves logarithmically in d = 2. Also, there is a dimension-dependent characteristic time scale on which the last site is visited. The structure of the set of sites not yet visited on this characteristic time scale is fractal-like in d = 2. In d greater-than-or-equal-to 3, on the other hand, this set is essentially distributed randomly through the lattice.