Territory covered by N diffusing particles
Abstract
THE number of distinct sites visited by a random walker after t steps is of great interest^{121}, 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^{1316}, metallurgy^{57}, chemistry^{17,18} and physics^{1922}. Previous analyses have been limited to the number of distinct sites visited by a single random walker^{1922}, but the (nontrivial) generalization to the number of distinct sites visited by TV walkers is particularly relevant to a range of problemsfor example, the classic problem in mathematical ecology of defining the territory covered by N members of a given species^{1316}. 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 ddimensional 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.
 Publication:

Nature
 Pub Date:
 January 1992
 DOI:
 10.1038/355423a0
 Bibcode:
 1992Natur.355..423L