Weak quenched limiting distributions for transient one-dimensional random walk in a random environment

TL;DR

This paper investigates the asymptotic behavior of hitting times in a one-dimensional transient random walk in a random environment, revealing that while quenched distributions do not converge to a fixed distribution, they do converge weakly to a random probability measure.

Contribution

It demonstrates that quenched distributions of hitting times converge weakly to a random measure, generalizing previous results on averaged limits and non-existence of fixed quenched limits.

Findings

Quenched distributions converge weakly to a random probability measure.

Results extend understanding of hitting time distributions in random environments.

The limit measure exhibits interesting stability properties.

Abstract

We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $\kappa>0$ that determines the fluctuations of the process. When $0<\kappa<2$, the averaged distributions of the hitting times of the random walk converge to a $\kappa$-stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true that for almost every fixed environment, the distributions of the hitting times (centered and scaled in any manner) converge to a non-degenerate distribution. We show, however, that the quenched distributions do have a limit in the weak sense. That is, the quenched distributions of the hitting times -- viewed as a random probability measure -- converge in distribution to a random probability measure, which has interesting stability properties. Our results generalize both the averaged limiting distribution and the non-existence of quenched limiting distributions.