Number of distinct sites visited byNrandom walkers

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Abstract
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 tx≪t≪tx′ we find that 〈SN(t)〉∼(t ln[N S1(t)/td/2])d/2 (regime II), and for t≫tx′, 〈SN(t)〉∼NS1(t) (regime III). The crossover times are tx∼ln 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 x≡N〈S1(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.