Tail estimates for one-dimensional random walk in random environment

Abstract

Suppose that the integers are assigned i.i.d. random variables {ω _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 atx, moves one step to the right with probability ω _x , and one step to the left with probability 1-ω _x . Solomon (1975) determined the almost-sure asymptotic speed (=rate of escape) of a RWRE. For certain environment distributions where the drifts 2ω _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.