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Random walk in a random environment with correlated sites

Part of: Special processes Applications to specific types of physical systems

Published online by Cambridge University Press:  14 July 2016

T. Komorowski  and
G. Krupa
T. Komorowski*
Affiliation:
University of M. Curie-Skłodowska
G. Krupa*
Affiliation:
Catholic University of Lublin
*
∗ Postal address: Institute of Mathematics, University of M. Curie-Skłodowska, Pl. M. Curie-Skłodowskiej 1, 20–032 Lublin, Poland.
∗∗ Postal address: Department of Mathematics and Nature, Catholic University of Lublin, Al. Racławickie 14, 20-950 Lublin, Poland. Email address: grzegorz.krupa@kul.lublin.pl
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Abstract

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Zd. The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

Keywords

Random walks in random environmentslaw of large numbers

MSC classification

Primary: 60K40: Other physical applications of random processes
Secondary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses)
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 1018 - 1032
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

Supported by a grant (No. 2 PO3A 017 17) from the State Committee for Scientific Research of Poland.

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