Number of distinct sites visited by<b><i>N</i></b>random walkers on a Euclidean lattice

Abstract

The evaluation of the average number ${S}_{N}(t)$ of distinct sites visited up to time t by N-independent random walkers all starting from the same origin on an Euclidean lattice is addressed. We find that, for the nontrivial time regime and for large $N, {S}_{N}(t)\ensuremath{\approx}{S}_{N}(t)(1\ensuremath{-}\ensuremath{\Delta}),$ where ${S}_{N}(t)$ is the volume of a hypersphere of radius $(4\mathrm{Dt}\\mathrm{ln}{N)}^{1/2},$ $\ensuremath{\Delta}=\frac{1}{2}{\ensuremath{\sum}}_{n=1}^{\ensuremath{\infty}}{\mathrm{ln}}^{\ensuremath{-}n}N{\ensuremath{\sum}}_{m=0}^{n}{s}_{m}^{(n)}{\mathrm{ln}}^{m}\mathrm{ln}N,$ d is the dimension of the lattice, and the coefficients ${s}_{m}^{(n)}$ depend on the dimension and time. The first three terms of these series are calculated explicitly and the resulting expressions are compared with other approximations and with simulation results for dimensions 1, 2, and 3. Some implications of these results on the geometry of the set of visited sites are discussed.