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We study the number of distinct sites S_N(t) and common sites W_N(t) visitedby 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 ontoextreme value quantities associated to N independent random walkers. Using thismapping, we compute exactly their probability distributions P_N^d(S,t) andP_N^d(W,t) for any value of N in the limit of large time t, where the randomwalkers can be described by Brownian motions. In the large N limit one findsthat 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 resultsthrough direct numerical simulations.

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