On slowdown and speedup of transient random walks in random environment

Details On slowdown and speedup of transient random walks in random environment On slowdown and speedup of transient random walks in random environment

2008-06-04

arxiv.org/abs/0806.0790v3

Probability Theory and Related Fields, v. 147, p. 43-88, 2010

Mathematics

Probability

43 pages, 4 figures; to appear in Probability Theory and Related Fields

Scientific paper

10.1007/s00440-009-0201-2

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 $O(n^\kappa)$ from the origin, $\kappa\in(0,1)$. 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 $O(n^{\nu_0})$ from the origin, $\nu_0\in (0,\kappa)$), and speedup (at time $n$, the particle is at a distance of order $n^{\nu_1}$ from the origin, $\nu_1\in (\kappa,1)$), 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 $(-n^\nu)$, 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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