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Multiscale analysis of exit distributions for random walks in random environments
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  • Published: 01 December 2006

Multiscale analysis of exit distributions for random walks in random environments

  • Erwin Bolthausen1 &
  • Ofer Zeitouni2,3 

Probability Theory and Related Fields volume 138, pages 581–645 (2007)Cite this article

  • 154 Accesses

  • 18 Citations

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Abstract

We present a multiscale analysis for the exit measures from large balls in \(\mathbb{Z}^d, d\geq 3\), 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.

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Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, Zürich, CH-8057, Switzerland

    Erwin Bolthausen

  2. Department of Mathematics, University of Minnesota, 206 Church St SE, Minneapolis, MN, 55455, USA

    Ofer Zeitouni

  3. Department of Mathematics and Department of Electrical Engineering, Technion, Haifa, Israel

    Ofer Zeitouni

Authors
  1. Erwin Bolthausen
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  2. Ofer Zeitouni
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Corresponding author

Correspondence to Ofer Zeitouni.

Additional information

Partially supported by NSF grant DMS-0503775.

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Cite this article

Bolthausen, E., Zeitouni, O. Multiscale analysis of exit distributions for random walks in random environments. Probab. Theory Relat. Fields 138, 581–645 (2007). https://doi.org/10.1007/s00440-006-0032-3

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  • Received: 26 March 2006

  • Revised: 30 August 2006

  • Published: 01 December 2006

  • Issue Date: July 2007

  • DOI: https://doi.org/10.1007/s00440-006-0032-3

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Keywords

  • Random walk
  • Random environment
  • Multiscale analysis
  • Exit measure

Mathematics Subject Classification (2000)

  • Primary 60K37
  • Secondary 82C41
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