On slowdown and speedup of transient random walks in random environment

Abstract

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.

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

References in corpus (3)

• Quenched limits for transient, zero speed one-dimensional random walk in random environment
• Moderate deviations for diffusions with Brownian potentials
• Limit laws for transient random walks in random environment on $\z$

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