The Annals of Probability

A Law of Large Numbers for Random Walks in Random Environment

Alain-Sol Sznitman and Martin Zerner

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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.

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

First available in Project Euclid: 31 May 2002

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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.

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