NUMBER OF DISTINCT SITES VISITED BY N-RANDOM WALKERS

We study the number of distinct sites visited by N random walkers after t steps S(N)(t) under the condition that all the walkers are initially at the origin. We derive asymptotic expressions for the mean number of distinct sites [S(N)(t)] in one, two, and three dimensions. We find that [S(N)(t)] passes through several growth regimes, at short times [S(N)(t)] approximately t(d) (regime I), for t(x) << t << t'x we find that [S(N)(t)] approximately (t ln[N S1(t)/t(d/2)])d/2 (regime II), and for t >> t'x, [S(N)(t)) approximately NS1(t) (regime III). The crossover times are t(x) approximately ln N for all dimensions, and t'x approximately infinity, exp N, and N2 for one, two, and three dimensions, respectively. We show that in regimes II and III [S(N)(t)] satisfies a scaling relation of the form [S(N)(t)] approximately t(d/2) f(x), with x = N[S1(t)]/t(d/2). We also obtain asymptotic results for the complete probability distribution of S(N)(t) for the one-dimensional case in the limit of large N and t.