Localization for a random walk in slowly decreasing random potential

TL;DR

This paper analyzes the asymptotic localization of a continuous-time random walk in a slowly decreasing random potential, showing it concentrates around a predictable trap location that scales with time, contrasting with Sinai's regime.

Contribution

It provides a precise asymptotic description of the walk's localization in a slowly decreasing potential, introducing a new scaling law for trap location.

Findings

The walk localizes around a deterministic trap location.

The trap location scales with $(rac{ ext{ln } t}{ ext{ln } ext{ln } t})^{1/eta}$.

Contrasts with Sinai's regime where trap location is random on $ ext{ln}^2 t$ scale.

Abstract

We consider a continuous time random walk $X$ in random environment on $\Z^+$ such that its potential can be approximated by the function $V: \R^+\to \R$ given by $V(x)=\sig W(x) -\frac{b}{1-\alf}x^{1-\alf}$ where $\sig W$ a Brownian motion with diffusion coefficient $\sig>0$ and parameters $b$, $\alf$ are such that $b>0$ and $0<\alf<1/2$. We show that $\P$-a.s.\ (where $\P$ is the averaged law) $\lim_{t\to \infty} \frac{X_t}{(C^*(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alf}}}=1$ with $C^*=\frac{2\alf b}{\sig^2(1-2\alf)}$. In fact, we prove that by showing that there is a trap located around $(C^*(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alf}}$ (with corrections of smaller order) where the particle typically stays up to time $t$. This is in sharp contrast to what happens in the "pure" Sinai's regime, where the location of this trap is random on the scale $\ln^2 t$.