Precise large deviations for random walk in random environment

TL;DR

This paper investigates the probabilities of rare events for one-dimensional random walks in random environments, providing precise large deviation estimates in both ballistic and sub-ballistic regimes.

Contribution

It introduces sharp large deviation results for random walks in random environments, detailing behavior in both ballistic and sub-ballistic regimes.

Findings

Established precise large deviations in the ballistic regime.

Described the probability of slowdown in the sub-ballistic regime.

Provided a comprehensive analysis of the walk's behavior in different regimes.

Abstract

We study one-dimensional nearest neighbour random walk in site-random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown.