Localization for a random walk in slowly decreasing random potential

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+ → R given by V (x) = σW (x) − b1−αx1−α where σW a Brownian motion with diffusion coefficient σ> 0 and parameters b, α are such that b> 0 and 0 < α < 1/2. We show that P-a.s. (where P is the averaged law) limt→ ∞ Xt (C∗(ln ln t)−1 ln t) 1 α = 1 with C ∗ = 2αbσ2(1−2α). In fact, we prove that by showing that there is a trap located around (C∗(ln ln t)−1 ln t) 1 α (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 ln2 t