We present a multiscale analysis for the exit measures from large balls in Zd, d ≥ 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.

Bolthausen, E; Zeitouni, O (2007). *Multiscale analysis of exit distributions for random walks in random environments.* Probability Theory and Related Fields, 138(3-4):581-645.

## Abstract

We present a multiscale analysis for the exit measures from large balls in Zd, d ≥ 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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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | Random walk - Random environment - Multiscale analysis - Exit measure |

Language: | English |

Date: | 2007 |

Deposited On: | 07 Dec 2009 10:38 |

Last Modified: | 05 Apr 2016 13:23 |

Publisher: | Springer |

ISSN: | 0178-8051 |

Additional Information: | The original publication is available at www.springerlink.com |

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

Related URLs: | http://arxiv.org/abs/math/0607192 |

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