Large Deviation Principle for Random Walk in a Quenched Random Environment in the Low Speed Regime

Abstract

We consider a one-dimensional random walk $(X_n)_{n \times \mathbb{N}}$ in a random environment of zero or strictly positive drifts. We establish a full large deviation principle for $X_n/n$ of the correct order $n/(\log n)^2$ in the low speed regime, valid for almost every environment. This completes the large deviation picture obtained earlier by Greven and den Hollander and Gantert and Zeitouni in the case of zero and positive drifts. The proof uses coarse graining along with concentration of measure techniques.