Quenched limits for transient, zero speed one-dimensional random walk in random environment

Abstract

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 but with zero speed, so that X_n 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 {n_k} and {x_k} depending on the environment only, such that X_nk−x_k=o(log n_k)² (a localized regime). On the other hand, there exist sequences {t_m} and {s_m} depending on the environment only, such that logs_m/log t_m→s<1 and P_ω(X_tm/s_m≤x)→1/2 for all x>0 and →0 for x≤0 (a spread out regime).