Territory covered by N diffusing particles

Abstract

THE number of distinct sites visited by a random walker after t steps is of great interest1–21, as it provides a direct measure of the territory covered by a diffusing particle. Thus, this quantity appears in the description of many phenomena of interest in ecology13–16, metallurgy5–7, chemistry17,18 and physics19–22. Previous analyses have been limited to the number of distinct sites visited by a single random walker19–22, but the (nontrivial) generalization to the number of distinct sites visited by TV walkers is particularly relevant to a range of problems-for example, the classic problem in mathematical ecology of defining the territory covered by N members of a given species13–16. Here we present an analytical solution to the problem of calculating SN(t), the mean number of distinct sites visited by N random walkers on a d-dimensional lattice, for d = 1, 2, 3 in the limit of large N. We confirm the analytical arguments by Monte Carlo and exact enumeration methods. We find that there are three distinct time regimes, and we determine SN(t) in each regime. Moreover, we also find a remarkable transition, for dimensions ≳2, in the geometry of the set of visited sites. This set initially grows as a disk with a relatively smooth surface until it reaches a certain size, after which the surface becomes increasingly rough.