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 ∈ (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, ν₀ ∈ (0, κ)), and speedup (at time n, the particle is at a distance of order n^ν ₁ from the origin, ν₁ ∈ (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.