Quenched Limits for Transient, Zero Speed One-Dimensional Random Walk in Random Environment

We consider a nearest-neighbor, one dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient but with zero speed, so that Xn is of order n s for some s < 1. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible: there exist sequences {nk} and {xk} depending on the environment only, such that Xnk − xk = o(log nk) 2 (a localized regime). On the other hand, there exist sequences {tm} and {sm} depending on the environment only, such that log sm / log tm → s < 1 and Pω(Xtm/sm ≤ x) → 1/2 for all x> 0 and → 0 for x ≤ 0 (a spread out regime)