Large deviations at the origin of random walk in random environment

TL;DR

This paper investigates the large deviation behavior of a random walk in i.i.d. random environments, focusing on the origin, and describes the atypical events leading to large deviations in both nestling and marginally nestling cases.

Contribution

It provides a detailed description of the atypical events causing large deviations at the origin, extending understanding to nestling and marginally nestling environments.

Findings

Quenched and averaged large deviation rate functions coincide at the origin.

Atypical events involve the walk seeking regions where the environment mimics elements minimizing the rate function.

Periodic environments are significant in describing these atypical events.

Abstract

We consider a random walk in an i.i.d. random environment on Zd and study properties of its large deviation rate function at the origin. It was proved by Comets, Gantert and Zeitouni in dimension d = 1 in 1999 and later by Varadhan in dimensions d >= 2 in 2003 that, for uniformly elliptic i.i.d. random environments, the quenched and the averaged large deviation rate functions coincide at the origin. Here we provide a description of an atypical event realizing the correct quenched large deviation rate in the nestling and marginally nestling setting: the random walk seeks regions of space where the environment emulates the element in the convex hull of the support of the law of the environment at a site which minimizes the rate function. Periodic environments play a natural role in this description.