Journal of Statistical Physics

, Volume 150, Issue 2, pp 285–298

Localization for a Random Walk in Slowly Decreasing Random Potential

  • Christophe Gallesco
  • Serguei Popov
  • Gunter M. Schütz
Article

Abstract

We consider a continuous time random walk X in a random environment on ℤ+ such that its potential can be approximated by the function V:ℝ+→ℝ given by \(V(x)=\sigma W(x) -\frac {b}{1-\alpha}x^{1-\alpha}\) 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) \(\lim_{t\to\infty} \frac{X_{t}}{(C^{*}(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alpha}}}=1\) with \(C^{*}=\frac{2\alpha b}{\sigma^{2}(1-2\alpha)}\). In fact, we prove that by showing that there is a trap located around \((C^{*}(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alpha}}\) (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 ln2t.

Keywords

KMT strong coupling Brownian motion with drift Localization Random walk in random environment Reversibility 

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Christophe Gallesco
    • 1
  • Serguei Popov
    • 1
  • Gunter M. Schütz
    • 2
  1. 1.Department of Statistics, Institute of Mathematics, Statistics and Scientific ComputationUniversity of Campinas–UNICAMPCampinasBrazil
  2. 2.Theoretical Soft Matter and Biophysics, Institute of Complex SystemsForschungszentrum JülichJülichGermany

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