Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

TL;DR

This paper investigates aging phenomena and localization in one-dimensional random walks in random environments with zero speed, providing new insights into their long-term behavior and distribution at large times.

Contribution

It introduces a detailed analysis of aging and localization phenomena in sub-ballistic regimes, with precise distribution estimates in the quenched setting.

Findings

Aging phenomenon involving the generalized Arcsine law is established.

Localization occurs at the foot of valleys of height log t.

Distribution of the walk at time t is sharply estimated.

Abstract

We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of "valleys" of height $\log t$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.