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

G. Berkolaiko, S. Havlin, H. Larralde, and G. H. Weiss
Phys. Rev. E 53, 5774 – Published 1 June 1996
PDFExport Citation

Abstract

We calculate asymptotic forms for the expected number of distinct sites, 〈SN(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|1+α 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 〈SN(n)〉 is proportional to (Nn2)1/(1+α) for α<1 and to N1/(1+α)n1/α 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. © 1996 The American Physical Society.

  • Received 18 August 1995

DOI:https://doi.org/10.1103/PhysRevE.53.5774

©1996 American Physical Society

Authors & Affiliations

G. Berkolaiko, S. Havlin, H. Larralde, and G. H. Weiss

  • Department of Mathematics, Voronezh State University, 394693 Voronezh, Russia
  • Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel
  • Cavendish Laboratory, Cambridge University, Cambridge CBS OHE, United Kingdom
  • PSL DCRT, National Institutes of Health, Bethesda, Maryland 20205

References (Subscription Required)

Click to Expand
Issue

Vol. 53, Iss. 6 — June 1996

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review E

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×