Two-Dimensional Anisotropic Random Walks

  • H. Silver
  • K. E. Shuler
  • K. Lindenberg


A class of two-dimensional random walks on lattices is investigated in which the walker can always step to nearest neighbor sites in any row but can step to nearest neighbor sites in a column only for certain specified “connected” columns. Two cases are considered: 1) the vertical lattice “connections” occur periodically and 2) the vertical “connections” are placed randomly. In the first case the asymptotic behaviour of a number of walk properties is found to agree with the corresponding asymptotic behaviour for a walk on a translationally invariant regular two-dimensional lattice with appropriately chosen anisotropic stepping probabilities. For the case of random column connections we show that this problem is isomorphic to the problem of studying vibrations in randomly disordered harmonic lattices. The anisotropic two-dimensional lattice random walks studied in this paper may have important implications for the understanding of transport processes occurring in anisotropic crystals such as TCNQ and TTF-TCNQ and for fluid flow through packed columns.


Random Walk Transition Probability Matrix Random Walk Model Initial Site Dimensional Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Chandrasekhar, Rev. Mod. Phys., 15, 1 (1943).CrossRefGoogle Scholar
  2. 2.
    H. Scher and M. Lax, Phys. Rev.,B7, 4491 (1973); Phys. Rev. B7, 4502 (1973).Google Scholar
  3. 3.
    E. W. Montroll, J. Math. Phys., 10, 753 (1969).CrossRefGoogle Scholar
  4. 4.
    W. D. Metz, Science,190, 450 (1975) and references therein.Google Scholar
  5. 5.
    V. Barnett, J. App. Prob., 12, 466 (1975).CrossRefGoogle Scholar
  6. 6.
    C. Domb, A. A. Maradudin, E. W. Montroll and G. H. Weiss, Phys. Rev., 115, 18 (1959).CrossRefGoogle Scholar
  7. 7.
    A. A. Maradudin, E. W. Montroll and G. H. Weiss, “Theory of Lattice Dynamics in the Harmonic Approximation,” (Academic Press, New York 1963.)Google Scholar
  8. 8.
    E. W. Montroll in “Topics on Statistical Mechanics of Interacting Particles,” Les Houches (1959), Chaps. 4 and 5; N. V. Vdovichenko, Soviet Phys. J.E.T.P., 20, 477 (1965).Google Scholar
  9. 9.
    U. Grenander and G. Szegb, “Toeplitz Forms and Their Applications”. (University of California Press, Berkeley, California, 1958.)Google Scholar
  10. 10.
    E. W. Montroll, Proc. Sympt. App. Mathematics, 26, 193 (1964).Google Scholar
  11. 11.
    G. Darboux, J. Math., 3l, 377 (1878)Google Scholar
  12. 12.
    E. W. Montroll and G. H. Weiss, J. Math. Phys., 26, 167 (1965).CrossRefGoogle Scholar
  13. 13.
    K. E. Shuler, H. Silver and K. Lindenberg, J. Stat. Phys., (in press).Google Scholar
  14. 14.
    F. J. Dyson, Phys. Rev., 92, 1331 (1953).CrossRefGoogle Scholar
  15. 15.
    R. Bellman, Phys. Rev., 101, 19 (1958).CrossRefGoogle Scholar
  16. 16.
    A. A. Maradudin, E. W. Montroll, G. H. Weiss, R. Herman and H. W. Milnes, Acaddmie Royale de Belgique, 14, 1 (1960)Google Scholar

Copyright information

© Plenum New York 1977

Authors and Affiliations

  • H. Silver
    • 1
    • 2
  • K. E. Shuler
    • 1
  • K. Lindenberg
    • 1
  1. 1.Department of ChemistryUniversity of CaliforniaSan Diego La JollaUSA
  2. 2.Control Data Australia PTY. LimitedMelbourneAustralia

Personalised recommendations