The Annals of Probability

Exit laws from large balls of (an)isotropic random walks in random environment

Erich Baur and Erwin Bolthausen

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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.

Article information

Source
Ann. Probab., Volume 43, Number 6 (2015), 2859-2948.

Dates
Received: September 2013
Revised: May 2014
First available in Project Euclid: 11 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1449843623

Digital Object Identifier
doi:10.1214/14-AOP948

Mathematical Reviews number (MathSciNet)
MR3433574

Zentralblatt MATH identifier
1344.60098

Subjects
Primary: 60K37: Processes in random environments
Secondary: 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Random walk random environment exit measure perturbative regime nonballistic behavior

Citation

Baur, Erich; Bolthausen, Erwin. Exit laws from large balls of (an)isotropic random walks in random environment. Ann. Probab. 43 (2015), no. 6, 2859--2948. doi:10.1214/14-AOP948. https://projecteuclid.org/euclid.aop/1449843623


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References

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