Communications in Mathematical Physics

, Volume 104, Issue 1, pp 87–102 | Cite as

Multidimensional random walks in random environments with subclassical limiting behavior

  • Richard Durrett


In this paper we will describe and analyze a class of multidimensional random walks in random environments which contain the one dimensional nearest neighbor situation as a special case and have the pleasant feature that quite a lot can be said about them. Our results make rigorous a heuristic argument of Marinari et al. (1983), and show that in anyd<∞ we can have (a)X n is recurrent and (b)X n ∼(logn)2.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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. Derrida, B., Luck, J.M.: Diffusion on a random lattice: weak disorder expansion in arbitrary dimension. Phys. Rev. B28, 7183–7190 (1983)Google Scholar
  2. Durrett, R.: Oriented percolation in two dimensions. Ann. Probab.12, 999–1040 (1984a)Google Scholar
  3. Durrett, R.: Brownian motion and martingales in analysis. Monterey, CA: Wadsworth 1984bGoogle Scholar
  4. Durrett, R.: Reversible diffusion processes. In: Probability and harmonic analysis. Chao, J., Woyczynski, W. (eds.). New York: Dekker 1985aGoogle Scholar
  5. Durrett, R.: Particle systems, random media, and large deviations. Conference proceedings. AMS Contemporary Math. Series, Vol. 41 (1985b)Google Scholar
  6. Fisher, D.: Random walks in random environments. Bell Laboratories, preprint (1984)Google Scholar
  7. Gangolli, R.: Abstract harmonic analysis and Lévy's Brownian motion of several parameters, 5th Berkeley Symp., Vol. II, pp. 13–30 (1965)Google Scholar
  8. Griffeath, D., Liggett, T.: Critical phenomena for Spitzer's reversible nearest particle systems. Ann. Probab.10, 881–895 (1982)Google Scholar
  9. Kesten, H., Kozlov, M., Spitzer, F.: A limit law for random walk in a random environment. Compos. Math.30, 145–168 (1975)Google Scholar
  10. Kotani, S.: A lemma for stationary random sequences and its application to the recurrence of random walks in random media. Preprint (1985)Google Scholar
  11. Lévy, P.: Processus stochastiques et mouvement Brownien. Paris: Gauthier-Villars (1948)Google Scholar
  12. Liggett, T.: Interacting particle systems. Berlin, Heidelberg, New York: Springer 1985Google Scholar
  13. Luck, J.M.: Diffusion in a random medium: a renormalization group approach. Nucl. Phys. B225, 169–184 (1983)Google Scholar
  14. Luck, J.M.: A numerical study of diffusion and conduction in a 2D random medium. J. Phys. A17, 2069–2077 (1984)Google Scholar
  15. Marinari, E., Parisi, G., Ruelle, D., Windey, P.: Random walks in a random environment and 1/f noise. Phys. Rev. Lett.50, 1223–1225 (1983a)Google Scholar
  16. Marinari, E., Parisi, G., Ruelle, D., Windey, P.: On the interpretation of 1/f noise. Commun. Math. Phys.89, 1–12 (1983b)Google Scholar
  17. Pitt, L.D.: Local times for Gaussian vector fields. Indiana Math. J.27, 309–330 (1978)Google Scholar
  18. Ritter, G.A.: Random walk in a random environment, critical case. Ph.D. Thesis, Cornell (1976)Google Scholar
  19. Schumacher, S. (1984): Diffusions with random coefficients. Ph.D. Thesis, UCLA. A summary appears in Durrett (1985b)Google Scholar
  20. Sinai, Ya.: Limit behavior of one-dimensional random walks in random environments. Theor. Probab. Appl.27, 247–258 (1982)Google Scholar
  21. Solomon, F.: Random walks in a random environment. Ann. Probab.3, 1–31 (1975)Google Scholar
  22. Skorokhod, A.V.: Limit theorems for stochastic processes. Theor. Probab. Appl.1, 261–290 (1956)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Richard Durrett
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

Personalised recommendations