Number of distinct sites visited by N random walkers

We study the number of distinct sites visited by N random walkers after t steps SN(t) under the condition that all the walkers are initially at the origin. We derive asymptotic expressions for the mean number of distinct sites SN(t) in one, two, and three dimensions. We find that SN(t) passes through several growth regimes; at short times SN(t)td (regime I), for txttx we find that SN(t)(t ln[N S1(t)/td/2])d/2 (regime II), and for ttx, SN(t)NS1(t) (regime III). The crossover times are txln N for all dimensions, and tx, exp N, and N2 for one, two, and three dimensions, respectively. We show that in regimes II and III SN(t) satisfies a scaling relation of the form SN(t)td/2f(x), with xNS1(t)/td/2. We also obtain asymptotic results for the complete probability distribution of SN(t) for the one-dimensional case in the limit of large N and t.