Expected number of distinct sites visited by N Levy flights on a one-dimensional lattice

We calculate asymptotic forms for the expected number of distinct sites, [S-N(n)], visited by N noninteracting n-step symmetric Levy flights in one dimension. By a Levy flight we mean one in which the probability of making a step of j sites is proportional to 1/\j\(1+alpha) in the limit j-->infinity. All values of alpha>O are considered. In our analysis each Levy flight is initially at the origin and both N and n are assumed to be large. Different asymptotic results are obtained for different ranges in cu. When n is fixed and N-->infinity we find that [S-N(n)] is proportional to (Na-2)(1/(1+alpha)) for alpha<1 and to N-1/(1+alpha)n(1/alpha) for alpha>1. When alpha exceeds 2 the second moment is finite and one expects the results of Larralde et al. [Phys. Rev. A 45, 7128 (1992)] to be valid. We give results for both fixed n and N-->infinity and N fixed and n-->infinity. In the second case the analysis leads to the behavior predicted by Larralde et al.