Territory covered by<i>N</i>random walkers

Abstract

The problem of evaluating the number of distinct sites S(N)(t) covered up to time t by N random walkers is revisited. For the nontrivial time regime and for N>>1 we show how to get the asymptotic behavior of S(N)(t) and we calculate the main and first two corrective terms. The mth corrective term decays mildly as 1/ln(m) N. For d-dimensional (d=1,2,3) simple cubic lattices, the main term is the volume of the hypersphere of radius [(ln N(2))2Dt/d](1/2), D being the diffusion constant, and the corrective terms account for the roughening of the surface of the set of visited sites.