Quenched limits for the fluctuations of transient random walks in random environment on Z

TL;DR

This paper analyzes the fluctuations of hitting times for transient one-dimensional random walks in random environments, providing explicit descriptions and limiting laws in the quenched setting, extending classical results.

Contribution

It introduces a detailed description of hitting time fluctuations and their limiting Poisson process law for transient random walks in random environments, in the quenched framework.

Findings

Explicit fluctuation description in quenched environment

Limiting law characterized by a Poisson point process

Extension of classical results to quenched setting

Abstract

We consider transient nearest-neighbor random walks in random environment on Z. For a set of environments whose probability is converging to 1 as time goes to infinity, we describe the fluctuations of the hitting time of a level n, around its mean, in terms of an explicit function of the environment. Moreover, their limiting law is described using a Poisson point process whose intensity is computed. This result can be considered as the quenched analog of the classical result of Kesten, Kozlov and Spitzer [Compositio Math. 30 (1975) 145-168].