Large deviation principle for random walk in a quenched random environment in the low speed regime

Ž.We consider a one-dimensional random walk X in a randomn n environment of zero or strictly positive drifts. We establish a full large Ž.2deviation principle for X n of the correct order n log n in the lown 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. 1. Introduction. The model and the main result. In this paper, we continue the study, initiated by Greven and den Hollander 6, of the large deviation behavior of a Ž.nearest neighbor random walk on with quenched frozen-in site-dependent transition probabilities, which are themselves generated randomly. In 6 a