Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 3, p. 685-709

We consider a nearest-neighbor, one-dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient with speed vP>0 and there exists an s∈(1, 2) such that the annealed law of n-1/s(Xn-nvP) 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 {tk} and {tk'} depending on the environment only, such that a quenched central limit theorem holds along the subsequence tk, but the quenched limiting distribution along the subsequence tk' 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).

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 vP>0 et où il existe s∈(1, 2) tel que la loi «annealed» (i.e., moyennée) de n-1/s(Xn-nvP) 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 {tk} et {tk'}, dépendant de l'environnement, tel qu'un théorème de limite centrale quenched est valide le long de la suite tk, mais où la distribution limite suivant la suite tk' 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).

DOI : https://doi.org/10.1214/08-AIHP149
Classification:  60K37,  60F05,  82C41,  82D30
Keywords: random walk, random environment
@article{AIHPB_2009__45_3_685_0,
     author = {Peterson, Jonathon},
     title = {Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     pages = {685-709},
     doi = {10.1214/08-AIHP149},
     zbl = {1178.60071},
     mrnumber = {2548499},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_3_685_0}
}
Peterson, Jonathon. Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 3, pp. 685-709. doi : 10.1214/08-AIHP149. http://www.numdam.org/item/AIHPB_2009__45_3_685_0/

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