Territory covered by N Levy flights on d-dimensional lattices

We study the territory covered by N Levy flights by calculating the mean number of distinct sites, [S-N(n)], visited after n time steps on a d-dimensional, d greater than or equal to 2, lattice. The Levy flights are initially at the origin and each has a probability Al--(d+alpha) to perform an l-length jump in a randomly chosen direction at each time step. We obtain asymptotic results for different values of alpha. For d=2 and N-->infinity we find [S-N(n)] proportional to CalphaN2/(2+alpha)n(4/(2+alpha)), when alpha<2 and [S-N(n)]proportional to N-2/(2+alpha)n(2/alpha), when alpha>2. For d=2 and n-->infinity we find [S-N(n)]proportional to Nn for alpha<2 and [S-N(n)]proportional to Nn/1nn for alpha>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 [S-N(n)] on d greater than or equal to 3 dimensional lattices.