Abstract
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)⋍ 1 2 1− const |x| d−2 valid for ∥ x∥ small on the scale N 1 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 ⩾ 3, on the other hand, this set is essentially distributed randomly through the lattice.
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