Quenched Sub-Exponential Tail Estimates for One-Dimensional Random Walk in Random Environment

Suppose that the integers are assigned i.i.d. random variables {ωx} (taking values in the unit interval), which serve as an environment. This environment defines a random walk {Xn} (called a RWRE) which, when at x, moves one step to the right with probability ωx, and one step to the left with probability 1- ωx. Solomon (1975) determined the almost-sure asymptotic speed vα (=rate of escape) of a RWRE. Greven and den Hollander (1994) have proved a large deviation principle for Xn /n, conditional upon the environment, with deterministic rate function. For certain environment distributions where the drifts 2 ωx-1 can take both positive and negative values, their rate function vanisheson an interval (0,vα). 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 Xn /n is smaller than v, v∈ (0,vα), 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}.