Quenched limits for transient, ballistic, sub-Gaussian one-dimensional random walk in random environment

We consider a nearest-neighbor, one-dimensional random walk {X-n}(n >= 0) in a random i.i.d. environment, in the regime where the walk is transient with speed vp > 0 and there exists an s is an element of (1, 2) such that the annealed law of n(-1/s) (X-n - nvp) converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences (t(k)) and {t'(k)} depending on the environment only, such that a quenched central limit theorem holds along the subsequence tk, but the quenched limiting distribution along the subsequence t'(k) centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:math/0704.1778v1 [math.PR]) which handled the case when the parameter s is an element of (0, 1).