Expected number of distinct sites visited by N Lvy 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 Lvy flights in one dimension. By a Lvy 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 0 are considered. In our analysis each Lvy 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 . When n is fixed and N we find that [Formula Presented](n) is proportional to ([Formula Presented][Formula Presented] for <1 and to [Formula Presented][Formula Presented] for 1. When 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. 1996 The American Physical Society.

Expected number of distinct sites visited by N Lvy flights on a one-dimensional lattice Academic Article