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Marches aléatoires réversibles en milieu aléatoire

Abstract : We consider two models of reversible random walks in random environment. The first one is the random walk among random conductances. We prove that the environment viewed by this walk converges to equilibrium polynomially fast in the variance sense, our main hypothesis being that the conductances are bounded away from zero. The basis of our method is the establishment of a Nash inequality, followed either by a comparison with the simple random walk or by a more direct analysis based on a martingale decomposition. For the second model that we consider, we attribute a positive value \tau_x to each x in Z^d. The random walk we consider, often called "Bouchaud's model", is reversible for the measure with weights (\tau_x). We assume that these weights are independent and identically distributed random variables with polynomial tail. We give the asymptotic behaviour of the principal eigenvalue of the generator of this random walk, with Dirichlet boundary conditions. The prominent feature of the result is a phase transition, that occurs at some threshold depending on the dimension. When the (\tau_x) are non-integrable and for d > 4, we also obtain the subdiffusive scaling limit of this model. We begin our proof by expressing the random walk as a time change of a random walk among random conductances. We then focus on proving that the time change converges, under the annealed measure, to a stable subordinator. This is achieved using previous results concerning the mixing properties of the environment viewed by the time-changed random walk.
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Contributor : Jean-Christophe Mourrat <>
Submitted on : Tuesday, May 18, 2010 - 3:20:55 PM
Last modification on : Wednesday, October 10, 2018 - 1:26:52 AM
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  • HAL Id : tel-00484257, version 1



Jean-Christophe Mourrat. Marches aléatoires réversibles en milieu aléatoire. Mathématiques [math]. Université de Provence - Aix-Marseille I; PUC de Chile, 2010. Français. ⟨tel-00484257⟩



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