Territory covered by N diffusing particles

Abstract

THE number of distinct sites visited by a random walker after t steps is of great interest^1–21, 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 ecology^13–16, metallurgy^5–7, chemistry^17,18 and physics^19–22. Previous analyses have been limited to the number of distinct sites visited by a single random walker^19–22, 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 species^13–16. Here we present an analytical solution to the problem of calculating S_N(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 S_N(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.