Stable fluctuations for ballistic random walks in random environment on Z

TL;DR

This paper studies the asymptotic behavior of hitting times for ballistic random walks in random environments on Z, providing a complete description of the limit law in the zero speed regime and explicit results for Dirichlet environments.

Contribution

It offers a new proof for the convergence of hitting times to a stable distribution in the zero speed case, including explicit results for Dirichlet environments.

Findings

Hitting times converge to a stable distribution in the zero speed regime.

The limit law's scale parameter is explicitly characterized.

Dirichlet environments yield particularly explicit results.

Abstract

We consider transient random walks in random environment on Z in the positive speed (ballistic) and critical zero speed regimes. A classical result of Kesten, Kozlov and Spitzer proves that the hitting time of level $n$, after proper centering and normalization, converges to a completely asymmetric stable distribution, but does not describe its scale parameter. Following a previous article by three of the authors, where the (non-critical) zero speed case was dealt with, we give a new proof of this result in the subdiffusive case that provides a complete description of the limit law. The case of Dirichlet environment turns out to be remarkably explicit.