Localization for a random walk in slowly decreasing random potential

Abstract

We consider a continuous time random walk $X$ in random environment on ${\mathbb{Z}}^{+}$ such that its potential can be approximated by the function $V:{\mathbb{R}}^{+}\to \mathbb{R}$ 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 ${\ln }^{2}t$.

14pages, 7 figures

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Cited by in corpus (1)

• Random walk in mixed random environment without uniform ellipticity