Exact distributions of the number of distinct and common sites visited by N independent random walkers

Abstract

We study the number of distinct sites S_N(t) and common sites W_N(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme value quantities associated to N independent random walkers. Using this mapping, we compute exactly their probability distributions P_N^d(S,t) and P_N^d(W,t) for any value of N in the limit of large time t, where the random walkers can be described by Brownian motions. In the large N limit one finds that S_N(t)/\sqrt{t} \propto 2 \sqrt{\log N} + \widetilde{s}/(2 \sqrt{\log N}) and W_N(t)/\sqrt{t} \propto \widetilde{w}/N where \widetilde{s} and \widetilde{w} are random variables whose probability density functions (pdfs) are computed exactly and are found to be non trivial. We verify our results through direct numerical simulations.