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

Nathanaël Enriquez;  Christophe Sabot;  Olivier Zindy — 2009

Bulletin de la Société Mathématique de France

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 log t . In the quenched setting, we also sharply estimate the distribution of the walk at time t .

Limit laws for transient random walks in random environment on ℤ

Nathanaël Enriquez;  Christophe Sabot;  Olivier Zindy — 2009

Annales de l’institut Fourier

We consider transient random walks in random environment on ℤ with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level n converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.

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