Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Jonathon Peterson

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Abstract

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(XnnvP) 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).

Résumé

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(XnnvP) 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).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 3 (2009), 685-709.

Dates
First available in Project Euclid: 4 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1249391380

Digital Object Identifier
doi:10.1214/08-AIHP149

Mathematical Reviews number (MathSciNet)
MR2548499

Zentralblatt MATH identifier
1178.60071

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60F05: Central limit and other weak theorems 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50] 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
Random walk Random environment

Citation

Peterson, Jonathon. Quenched limits for transient, ballistic, sub-Gaussian one-dimensional random walk in random environment. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 3, 685--709. doi:10.1214/08-AIHP149. https://projecteuclid.org/euclid.aihp/1249391380


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