Territory covered by N Lévy flights on d-dimensional lattices

We study the territory covered by N Lévy flights by calculating the mean number of distinct sites, 〈${\mathrm{S}}_{\mathrm{N}}$(n)〉, visited after n time steps on a d-dimensional, d⩾2, lattice. The Lévy flights are initially at the origin and each has a probability A${\mathrm{\ell }}^{\mathrm{-}(\mathrm{d}+\mathrm{\alpha })}$ to perform an ℓ-length jump in a randomly chosen direction at each time step. We obtain asymptotic results for different values of α. For d=2 and N→∞ we find 〈${\mathrm{S}}_{\mathrm{N}}$(n)〉∝${\mathrm{C}}_{\mathrm{\alpha }}$ ${\mathrm{N}}^{2\mathrm{/}(2+\mathrm{\alpha })}$ ${\mathrm{n}}^{4\mathrm{/}(2+\mathrm{\alpha })}$, when α<2 and 〈${\mathrm{S}}_{\mathrm{N}}$(n)〉∝${\mathrm{N}}^{2\mathrm{/}(2+\mathrm{\alpha })}$ ${\mathrm{n}}^{2\mathrm{/}\mathrm{\alpha }}$, when α>2. For d=2 and n→∞ we find 〈${\mathrm{S}}_{\mathrm{N}}$(n)〉∝Nn for α<2 and 〈${\mathrm{S}}_{\mathrm{N}}$(n)〉∝Nn/ln n for α>2. The last limit corresponds to the result obtained by Larralde et al. [Phys. Rev. A 45, 7128 (1992)] for bounded jumps. We also present asymptotic results for 〈${\mathrm{S}}_{\mathrm{N}}$(n)〉 on d⩾3 dimensional lattices.