Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

Enriquez, Nathanaël; Sabot, Christophe; Zindy, Olivier

Bulletin de la Société Mathématique de France, Tome 137 (2009) no. 3, pp. 423-452.

Nous considérons les marches aléatoires en milieu aléatoire uni-dimensionnelles, transientes et de vitesse nulle. Un phénomène de vieillissement exprimé en fonction de la loi de l’Arcsinus généralisée est prouvé en utilisant la localisation de la marche au pied de vallées de hauteur $logt$. Dans le cas où l’environnement est fixé, nous estimons précisément la loi de la position de la marche au temps $t$.

We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height $logt$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.

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