Tail estimates for one-dimensional random walk in random environment

Suppose that the integers are assigned i.i.d. random variables {omega(x)} (taking values in the unit interval), 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. For certain environment distributions where the drifts 2 omega(x) - 1 can take both positive and negative values, we show that the chance of the RWRE deviating below this speed has a polynomial rate of decay, and determine the exponent in this power law; for environments which allow only positive and zero drifts, we show that these large-deviation probabilities decay like exp(-Cn(1/3)). This differs sharply from the rates derived by Greven and den-Hollander (1994) for large deviation probabilities conditioned on the environment. As a by product we also provide precise tail and moment estimates for the total population size in a Branching Process with Random Environment.