The infinite valley for a recurrent random walk in random environment

Z. Shi (1) , N. Gantert , Y. Peres

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys. 92 (1984) 491-506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.

Domaines: Probabilités [math.PR]

Z. Shi, N. Gantert, Y. Peres. The infinite valley for a recurrent random walk in random environment. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2010, 46 (2), pp.525-536. ⟨10.1214/09-AIHP205⟩. ⟨hal-00485283⟩