Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment

Peterson, Jonathon

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 685-709.

On examine des marches aléatoires unidimensionnelles en milieu aléatoire avec un environnement i.i.d., dans le régime où la marche est transiente avec vitesse v_P>0 et où il existe s∈(1, 2) tel que la loi «annealed» (i.e., moyennée) de n^-1/s(X_n-nv_P) converge vers une loi stable de paramètre s. Sous la loi «quenched» (i.e. conditionnelement à l'environnement) on montre qu'il n'existe pas de loi limite. En particulier on prouve qu'il existe des suites {t_k} et {t_k'}, dépendant de l'environnement, tel qu'un théorème de limite centrale quenched est valide le long de la suite t_k, mais où la distribution limite suivant la suite t_k' est une distribution centrée exponentielle inverse. Ceci complète les résultats d'un article récent de Peterson et Zeitouni (arXiv:math/0704.1778v1 [math.PR]) qui traitait le case de paramètre s∈(0, 1).

We consider a nearest-neighbor, one-dimensional random walk {X_n}_n≥0 in a random i.i.d. environment, in the regime where the walk is transient with speed v_P>0 and there exists an s∈(1, 2) such that the annealed law of n^-1/s(X_n-nv_P) converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {t_k} and {t_k'} depending on the environment only, such that a quenched central limit theorem holds along the subsequence t_k, but the quenched limiting distribution along the subsequence t_k' is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:math/0704.1778v1 [math.PR]) which handled the case when the parameter s∈(0, 1).

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