  We consider a nearest-neighbor, one dimensional random walk
$\{X_n\}_{n\geq0}$ 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 $P_{\omega}(X_{t_m}/s_m\leq x)\to1/2$ for all $x>0$ and $\to0$ for
$x\leq0$ (a spread out regime).

Probability