Localisation in 1D random random walks

TL;DR

This paper investigates localization phenomena in one-dimensional random walks under random force fields, revealing finite site-occupation probabilities and connections to the Random Energy Model's equilibrium phase.

Contribution

It introduces a novel analysis of localization in 1D random walks using an equivalence with a directed walk model with traps, highlighting new long-term behaviors.

Findings

Average position dispersion diverges over time

Probability of two particles sharing a site approaches a finite constant

Long-term properties resemble the equilibrium phase of the Random Energy Model

Abstract

Diffusion in a one dimensional random force field leads to interesting localisation effects, which we study using the equivalence with a directed walk model with traps. We show that although the average dispersion of positions $\bar{< x^2 > - < x > ^2}$ diverges for long times, the probability that two particles occupy the same site tends to a finite constant in the small bias phase of the model. Interestingly, the long time properties of this off-equilibrium, aging phase is similar to the equilibrium phase of the Random Energy Model.