QUENCHED SUB-EXPONENTIAL 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 environmen t. This environment defines a random walk {X-n} (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) de termined the almost-sure asymptotic speed upsilon(alpha) (=rate of esc ape) of a RWRE. Greven and den Hollander (1994) have proved a large de viation principle for X-n/n, conditional upon the environment, with de terministic rate function. For certain environment distributions where the drifts 2 omega(x) - 1 can take both positive and negative values, their rate function vanishes on an interval (0, upsilon(alpha)). We f ind the rate of decay on this interval and prove it is a stretched exp onential of appropriate exponent, that is the absolute value of the lo g of the probability that the empirical mean X-n/n is smaller than ups ilon, upsilon is an element of (0, upsilon(alpha)), behaves roughly li ke 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 t he case of positive and zero drifts, and prove there a quenched decay of the form exp(-cn/(log n)(2)).