On slowdown and speedup of transient random walks in random environment

We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^κ)$ from the origin, $κ\in(0,1)$. We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time $n$, the particle is at a distance of order $O(n^{ν_0})$ from the origin, $ν_0\in (0,κ)$), and speedup (at time $n$, the particle is at a distance of order $n^{ν_1}$ from the origin, $ν_1\in (κ,1)$), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time $n$, the particle is located around $(-n^ν)$, thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.

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