The Annals of Probability

A Law of Large Numbers for Random Walks in Random Environment

Alain-Sol Sznitman and Martin Zerner

Full-text: Open access


We derive a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of Kalikow. The approach is based on the existence of a renewal structure under an assumption of “transience in the direction $l$ .” This extends, to a multidimensional context, previous work of Kesten. Our results also enable proving the convergence of the law of the environment viewed from the particle toward a limiting distribution.

Article information

Ann. Probab., Volume 27, Number 4 (1999), 1851-1869.

First available in Project Euclid: 31 May 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K40: Other physical applications of random processes
Secondary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Random walk in random environment law of large numbers Kalikow’s condition renewal structure


Sznitman, Alain-Sol; Zerner, Martin. A Law of Large Numbers for Random Walks in Random Environment. Ann. Probab. 27 (1999), no. 4, 1851--1869. doi:10.1214/aop/1022874818.

Export citation


  • [1] Alon, N., Spencer, J. and Erd os, P. (1992). The Probabilistic Method. Wiley, New York.
  • [2] Bricmont, J. and Kupiainen, A. (1991). Random walks in asymmetric random environments. Comm. Math. Phys. 142 345-420.
  • [3] Durrett, R. (1991). Probability: Theory and Examples. Wadsworth and Brooks/Cole, Pacific Grove, CA.
  • [4] Feller, W. (1957). An Introduction to Probability Theory and Its Applications 1, 3rd ed. Wiley, New York.
  • [5] Kalikow, S. A. (1981). Generalized random walk in a random environment. Ann. Probab. 9 753-768.
  • [6] Kesten, H. (1977). A renewal theorem for random walk in a random environment. Proc. Sympos. Pure Math. 31 67-77.
  • [7] Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limitlaw for random walk in random environment. Compositio Math. 30 145-168.
  • [8] Kozlov, S. M. (1985). The method of averaging and walk in inhomogeneous environments. Russian Math. Surveys 40 73-145.
  • [9] Molchanov, S. A. (1994). Lectures on random media. Ecole d'´et´e de Probabilit´es de St. Flour XXII. Lecture Notes in Math. 1581 242-411. Springer, Berlin.
  • [10] Solomon, F. (1975). Random walks in a random environment. Ann. Probab. 3 1-31.
  • [11] Sznitman, A.-S. (1999). Slowdown and neutral pockets for a random walk in random environment. Probab. Theory Related Fields. To appear.
  • [12] Zerner, M. P. W. (1998). Lyapunov exponents and quenched large deviation for multidimensional random walk in random environment. Ann. Probab. 26 1446-1476.