Multiscale analysis of exit distributions for random walks in random environments |
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Authors: | Erwin Bolthausen Ofer Zeitouni |
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Institution: | 1. Institut für Mathematik, Universit?t Zürich, Winterthurerstrasse 190, Zürich, CH-8057, Switzerland 2. Department of Mathematics, University of Minnesota, 206 Church St SE, Minneapolis, MN, 55455, USA 3. Department of Mathematics and Department of Electrical Engineering, Technion, Haifa, Israel
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Abstract: | We present a multiscale analysis for the exit measures from large balls in , of random walks in certain i.i.d. random environments which are small perturbations of the fixed environment corresponding
to simple random walk. Our main assumption is an isotropy assumption on the law of the environment, introduced by Bricmont
and Kupiainen. Under this assumption, we prove that the exit measure of the random walk in a random environment from a large
ball, approaches the exit measure of a simple random walk from the same ball, in the sense that the variational distance between
smoothed versions of these measures converges to zero. We also prove the transience of the random walk in random environment.
The analysis is based on propagating estimates on the variational distance between the exit measure of the random walk in
random environment and that of simple random walk, in addition to estimates on the variational distance between smoothed versions
of these quantities.
Partially supported by NSF grant DMS-0503775. |
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Keywords: | Random walk Random environment Multiscale analysis Exit measure |
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