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

We calculate asymptotic forms for the expected number of distinct sites, 〈[Formula Presented](n)〉, visited by N noninteracting n-step symmetric Lévy flights in one dimension. By a Lévy flight we mean one in which the probability of making a step of j sites is proportional to 1/j[Formula Presented] in the limit j→∞. All values of α≳0 are considered. In our analysis each Lévy 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 α. When n is fixed and N→∞ we find that 〈[Formula Presented](n)〉 is proportional to ([Formula Presented][Formula Presented] for α<1 and to [Formula Presented][Formula Presented] for α≳1. When α 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→∞ and N fixed and n→∞. In the second case the analysis leads to the behavior predicted by Larralde et al.