Journal of Statistical Physics

, Volume 23, Issue 1, pp 11–25 | Cite as

Lattice random walks for sets of random walkers. First passage times

  • Katja Lindenberg
  • V. Seshadri
  • K. E. Shuler
  • George H. Weiss


We have studied the mean first passage time for the first of aset of random walkers to reach a given lattice point on infinite lattices ofD dimensions. In contrast to the well-known result ofinfinite mean first passage times for one random walker in all dimensionsD, we findfinite mean first passage times for certain well-specified sets of random walkers in all dimensions, exceptD = 2. The number of walkers required to achieve a finite mean time for the first walker to reach the given lattice point is a function of the lattice dimensionD. ForD > 4, we find that only one random walker is required to yield a finite first passage time, provided that this random walker reaches the given lattice point with unit probability. We have thus found a simple random walk property which “sticks” atD > 4.

Key words

Random walks infinite lattices first passage times 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Polya,Math. Ann. 84:149 (1921).Google Scholar
  2. 2.
    E. W. Montroll,SIAM J. Math. 4:241 (1956).Google Scholar
  3. 3.
    E. W. Montroll,Proc. Symp. Appl. Math. XVI:193 (1964).Google Scholar
  4. 4.
    E. W. Montroll and G. H. Weiss,J. Math. Phys. 6:167 (1965).Google Scholar
  5. 5.
    E. W. Montroll,J. Soc. Indust. Appl. Math. 4:241 (1956).Google Scholar
  6. 6.
    G. H. Hardy and J. E. Littlewood,Proc. Land. Math. Soc. 30:23 (1929); G. H. Hardy,Divergent Series (Oxford, 1949).Google Scholar
  7. 7.
    A. A. Maradudin, E. W. Montroll, G. H. Weiss, R. Hermann, and H. Milnes,Acad. Roy. Belg. Cl. Sci. Mem., Vol.14 (1960).Google Scholar
  8. 8.
    M. E. Fisher,Rev. Mod. Phys. 46:597 (1974).Google Scholar
  9. 9.
    D. Bedeaux, K. Lakatos-Lindenberg, and K. E. Shuler,J. Math. Phys. 12:2116 (1971).Google Scholar
  10. 10.
    S. Foldes and G. Gabor,Discr. Math. 24:103 (1978).Google Scholar
  11. 11.
    G. Doetsch,Theorie und Anwendungen der Laplace-Transformation (Dover, 1945).Google Scholar
  12. 12.
    M. Abramowitz and I. Stegun,Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964).Google Scholar

Copyright information

© Plenum Publishing Corporation 1980

Authors and Affiliations

  • Katja Lindenberg
    • 1
  • V. Seshadri
    • 1
  • K. E. Shuler
    • 1
  • George H. Weiss
    • 2
  1. 1.Department of ChemistryUniversity of California-San DiegoLa Jolla
  2. 2.Division of Computer TechnologyNational Institutes of HealthBethesda

Personalised recommendations