Journal of Statistical Physics

, Volume 92, Issue 3–4, pp 713–725 | Cite as

Area Distribution for Directed Random Walks

  • Thordur Jonsson
  • John F. Wheater


We study the probability distribution for the area under a directed random walk in the plane. The walk can serve as a simple model for avalanches based on the idea that the front of an avalanche can be described by a random walk and the size is given by the area enclosed. This model captures some of the qualitative features of earthquakes, avalanches, and other self-organized critical phenomena in one dimension. By finding nonlinear functional relations for the generating functions we calculate directly the exponent in the size distribution law and find it to be 4/3.

Directed random walks 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W. Feller, An introduction to probability theory and its applications, Vol. 1 (John Wiley and Sons, New York, 1968).Google Scholar
  2. 2.
    T. Prellberg and R. Brak, J. Stat. Phys. 78:701 (1995).Google Scholar
  3. 3.
    D. B. Abraham and E. R. Smith, J. Stat. Phys. 43:621 (1986).Google Scholar
  4. 4.
    G. Louchard, J. Appl. Prob. 21:479 (1984).Google Scholar
  5. 5.
    R. Burridge and L. Knopoff, Bull. Seismol. Soc. Am. 57:341 (1967).Google Scholar
  6. 6.
    J. M. Carlson and J. S. Langer, Phys. Rev. A 40:6470 (1989).Google Scholar
  7. 7.
    J. M. Carlson, J. S. Langer, and B. E. Shaw, Rev. Mod. Phys. 66:657 (1994).Google Scholar
  8. 8.
    H. Nakanishi, Phys. Rev. A 41:7086 (1990).Google Scholar
  9. 9.
    T. Jonsson and S. F. Marinosson, Physics Letters A 207:165 (1995).Google Scholar
  10. 10.
    C. Tang and P. Bak, J. Stat. Phys. 51:797–802 (1988).Google Scholar
  11. 11.
    D. Dhar and S. N. Majumdar, J. Phys. A: Math. Gen. 23:4333 (1990).Google Scholar
  12. 12.
    S. A. Janowsky and C. A. Laberge, J. Phys. A: Math. Gen. 26:L973 (1993).Google Scholar
  13. 13.
    H. Flyvbjerg, K. Sneppen, and P. Bak, Phys. Rev. Lett. 71:4087 (1993).Google Scholar
  14. 14.
    S. Zapperi, K. B. Lauritsen, and H. E. Stanley, Phys. Rev. Lett. 75:4071 (1995).Google Scholar
  15. 15.
    D. Dhar and R. Ramaswamy, Phys. Rev. Lett. 63:1659 (1989).Google Scholar
  16. 16.
    B. Gutenberg and C. F. Richter, Seismicity of the Earth (Princeton University Press, Princeton, 1954).Google Scholar
  17. 17.
    T. Jonsson and J. F. Wheater, Gutenberg-Richter law in a random walk model for earthquakes (Oxford University preprint, OUTP 9639P).Google Scholar
  18. 18.
    V. Privman and Svrakic, J. Stat. Phys. 51:1091 (1988).Google Scholar
  19. 19.
    R. Brak and A. J. Guttmann, J. Phys. A 23:4581 (1990).Google Scholar
  20. 20.
    H. N. V. Temperley, Phys. Rev. 103:1 (1956).Google Scholar
  21. 21.
    T. Prellberg, J. Phys. A 28:1289 (1995).Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • Thordur Jonsson
    • 1
  • John F. Wheater
    • 1
  1. 1.Department of Theoretical PhysicsUniversity of OxfordOxfordUnited Kingdom

Personalised recommendations