The distributions of the number of distinct sites S visited by random walks of n steps in the infinite cluster of two-dimensional lattices at the percolation threshold are studied. Different lattice sizes, different origins of the walks, and different realizations of the disorder are investigated by Monte Carlo simulations. The distribution of the mean values of (S") appears to have selfaveraging features. The probability distribution of the normalized values of (S") is investigated with

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... to its multifractal behavior. The distributions of the probabilities p($ ) for fixed S" are presented and analyzed. These distributions are wide and their moments show behavior that cannot be characterized by multifractal scaling exponents. PACS number(s): 05.40. +j, 66.30.Jt