On slowdown and speedup of transient random walks in random environment

TL;DR

This paper analyzes the probabilities of slowdown, speedup, and backtracking in one-dimensional transient random walks in random environments, focusing on the sub-ballistic regime and providing insights into large deviation behaviors.

Contribution

It offers a detailed study of moderate deviations for transient random walks, including quenched and annealed probabilities of slowdown, speedup, and backtracking, extending understanding in the sub-ballistic regime.

Findings

Quantifies probabilities of slowdown and speedup in the sub-ballistic regime.

Provides estimates for backtracking probabilities.

Extends results to the ballistic case for slowdown.

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.