Long-time Behavior of Random Walks in Random Environment

TL;DR

This paper investigates the long-term behavior of random walks in isotropic, small-perturbation random environments on Z^d for d>=3, extending previous exit law results and analyzing mean sojourn times.

Contribution

It develops an extended, more accessible version of exit law analysis for random walks in random environments, including generalizations to anisotropic cases.

Findings

Extended analysis of exit laws from large balls.

Results on mean sojourn times in specified regions.

Generalization to anisotropic random walks.

Abstract

We study behavior in space and time of random walks in an i.i.d. random environment on Z^d, d>=3. It is assumed that the measure governing the environment is isotropic and concentrated on environments that are small perturbations of the fixed environment corresponding to simple random walk. We develop a revised and extended version of the paper of Bolthausen and Zeitouni (2007) on exit laws from large balls, which, as we hope, is easier to follow. Further, we study mean sojourn times in balls. This work is part of the author's PhD thesis under the supervision of Erwin Bolthausen. A generalization of the results on exit measures to certain anisotropic random walks in random environment is available at arXiv:1309.3169.