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

We consider a nearest-neighbor, one dimensional random walk $\{X_n{\}}_{n\ge 0}$ 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 $X_{n_k}-x_k=o(\log n_k{)}^2$ (a localized regime). On the other hand, there exist sequences $\{t_m\}$ and $\{s_m\}$ depending on the environment only, such that $\log s_m/\log t_m\to s<1$and$P_{\omega}(X_{t_m}/s_m\leq x)\to1/2$forall$x>0$and$\to0$for$x\leq0$ (a spread out regime).