Exit laws from large balls of (an)isotropic random walks in random environment. (English) Zbl 1344.60098

Baur, Erich; Bolthausen, Erwin

Ann. Probab. 43, No. 6, 2859-2948 (2015).

Authors’ abstract: We study exit laws from large balls in \(\mathbb{Z}^{d},d\geq 3\), of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to a 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 environments. Our work includes the results on isotropic random walks in random environments of [the second author and O. Zeitouni, Probab. Theory Relat. Fields 138, No. 3–4, 581–645 (2007; Zbl 1126.60088)]. Since several proofs in [loc. cit.] were incomplete, a somewhat different approach was given in the first author’s thesis [Long-time behavior of random walks in random environment. Zürich: Universität Zürich (PhD Thesis) (2013)]. Here, we extend this approach to certain anisotropic walks and provide a further step towards a fully perturbative theory of random walks in random environments.

Reviewer: Marius Iosifescu (Bucureşti)

MSC:

60K37	Processes in random environments
60G50	Sums of independent random variables; random walks
82C41	Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics

Keywords:

random walks; random environments; exit measure; perturbative regime; nonballistic behaviour

Citations:

Zbl 1126.60088

References:

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