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November 2015 Exit laws from large balls of (an)isotropic random walks in random environment
Erich Baur, Erwin Bolthausen
Ann. Probab. 43(6): 2859-2948 (November 2015). DOI: 10.1214/14-AOP948

Abstract

We study exit laws from large balls in $\mathbb{Z}^{d}$, $d\geq3$, of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to simple random walk. Under a centering condition on the measure governing the environment, we prove that the exit laws are close to those of a symmetric random walk, which we identify as a perturbed simple random walk. We obtain bounds on total variation distances as well as local results comparing exit probabilities on boundary segments. As an application, we prove transience of the random walks in random environment.

Our work includes the results on isotropic random walks in random environment of Bolthausen and Zeitouni [Probab. Theory Related Fields 138 (2007) 581–645]. Since several proofs in Bolthausen and Zeitouni (2007) were incomplete, a somewhat different approach was given in the first author’s thesis [Long-time behavior of random walks in random environment (2013) Zürich Univ.]. Here, we extend this approach to certain anisotropic walks and provide a further step towards a fully perturbative theory of random walks in random environment.

Citation

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Erich Baur. Erwin Bolthausen. "Exit laws from large balls of (an)isotropic random walks in random environment." Ann. Probab. 43 (6) 2859 - 2948, November 2015. https://doi.org/10.1214/14-AOP948

Information

Received: 1 September 2013; Revised: 1 May 2014; Published: November 2015
First available in Project Euclid: 11 December 2015

zbMATH: 1344.60098
MathSciNet: MR3433574
Digital Object Identifier: 10.1214/14-AOP948

Subjects:
Primary: 60K37
Secondary: 82C41

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 6 • November 2015
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