Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The infinite valley for a recurrent random walk in random environment

Nina Gantert, Yuval Peres, and Zhan Shi

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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.


Nous prouvons que les mesures empiriques d’une marche aléatoire unidimensionnelle en environnement aléatoire convergent étroitement vers la loi stationnaire d’une marche aléatoire dans une vallée infinie. La construction de cette vallée infinie revient à Golosov, voir Comm. Math. Phys. 92 (1984) 491–506. En applications, nous obtenons la convergence étroite du maximum des temps locaux et du temps local d’intersections de la marche aléatoire en environnement aléatoire; de plus, nous identifions la constante représentant la “limsup” presque sûre du maximum des temps locaux.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 2 (2010), 525-536.

First available in Project Euclid: 11 May 2010

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Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60J50: Boundary theory 60J55: Local time and additive functionals 60F10: Large deviations

Random walk in random environment Empirical distribution Local time Self-intersection local time


Gantert, Nina; Peres, Yuval; Shi, Zhan. The infinite valley for a recurrent random walk in random environment. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 2, 525--536. doi:10.1214/09-AIHP205.

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