Precise large deviation estimates for a one-dimensional random walk in a random environment

Suppose that the integers are assigned i.i.d. random variables {w(x)} (taking values in the interval [1/2, 1)), which serve as an environment. This environment defines a random walk {X-k} (called a RWRE) which, when at x, moves one step to the right with probability omega(x), and one step to the left with probability 1 - omega(x). Solomon (1975) determined the almost-sure asymptotic speed (= rate of escape) of a RWRE, in a more general set-up. Dembo, Peres and Zeitouni (1996), following earlier work by Greven and den Hollander (1994) on the quenched case, have computed rough tail asymptotics for the empirical mean of the annealed RWRE. They conjectured the form of the rate function in a full LDP. We prove in this paper their conjecture. The proof is based on a "coarse graining scheme" together with comparison techniques.