A note on quenched moderate deviations for Sinai's random walk in random environment

¹ Université Paris 7, UFR de Mathématiques, case 7012, 2, place Jussieu, 75251 Paris Cedex 05, France; e-mail: Partially supported by CNRS (UMR 7599 “Probabilités et Modèles Aléatoires”) and by the “Réseau Mathématiques France-Brésil”.
² Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, CEP 05508–090, São Paulo SP, Brasil; e-mail: Partially supported by CNPq (302981/02–0), and by the “Rede Matemática Brasil-França".

Received: 23 May 2003
Revised: 13 October 2003

Abstract

We consider the continuous time, one-dimensional random walk in random environment in Sinai's regime. We show that the probability for the particle to be, at time t and in a typical environment, at a distance larger than t^a (0<a<1) from its initial position, is exp{-Const ⋅ t^a/[(1 - a)lnt](1 + o(1))}.