Territory covered by N Lévy flights on d-dimensional lattices

We study the territory covered by N L\'evy flights by calculating the mean number of distinct sites, 〈SN(n)〉, visited after n time steps on a d-dimensional, d{}2, lattice. The L\'evy flights are initially at the origin and each has a probability A^-(d+α) to perform an {}-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 〈SN(n)〉{∝}C_α{N}^{2/(2+{{α}})}n^4/(2+α), when {α}2 and 〈SN(n)〉{∝}N^2/(2+α){n}^{2/{{α}}}, when α>2. For d=2 and n→∞ we find 〈{S}N(n)〉∝Nn for α2 and 〈{S}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 〈SN(n)〉 on d{}3 dimensional lattices.