Quenched Sub-Exponential 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 _n } (called a RWRE) which, when at x, 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 v _α (=rate of escape) of a RWRE. Greven and den Hollander (1994) have proved a large deviation principle for X _n /n, conditional upon the environment, with deterministic rate function. For certain environment distributions where the drifts 2 ω _x -1 can take both positive and negative values, their rate function vanisheson an interval (0,v _α). We find the rate of decay on this interval and prove it is a stretched exponential of appropriate exponent, that is the absolute value of the log of the probability that the empirical mean X _n /n is smaller than v, v∈ (0,v _α), behaves roughly like a fractional power of n. The annealed estimates of Dembo, Peres and Zeitouni (1996) play a crucial role in the proof. We also deal with the case of positive and zero drifts, and prove there a quenched decay of the form .