Scaling limit of the random walk among random traps on ${\mathbb{Z}}^d$

Mourrat, Jean-Christophe

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 813-849.

Après avoir attribué une valeur positive τx à chaque x de ℤd, nous nous intéressons à une marche aléatoire au plus proche voisin et réversible pour la mesure de poids (τx), souvent appelée ≪ modèle de Bouchaud ≫. Nous supposons que ces poids sont des variables aléatoires indépendantes, de même loi non-intégrable (à queue polynomiale), et que d≥5. Nous identifions, pour presque toute réalisation des (τx), la limite sous-diffusive de ce modèle. Nous commençons la preuve en exprimant la marche aléatoire comme le changement de temps d'une marche aléatoire en conductances aléatoires. Nous nous consacrons ensuite à montrer que ce changement de temps converge, sous la loi moyennée, vers un subordinateur stable. Nous y parvenons en utilisant un résultat antérieur concernant les propriétés de mélange de l'environnement vu par la marche changée de temps.

Attributing a positive value τx to each x∈ℤd, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (τx), often known as “Bouchaud's trap model.” We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that d≥5. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing the random walk as the 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.

Classification : 60K37, 60G52, 60F17, 82D30
Mots clés : random walk in random environment, trap model, stable process, fractional kinetics

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