Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment

Christophe Sabot and Laurent Tournier

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Abstract

We consider random walks in a random environment given by i.i.d. Dirichlet distributions at each vertex of ℤd or, equivalently, oriented edge reinforced random walks on ℤd. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of ℤd. We prove that, as soon as these weights are nonsymmetric, the random walk is transient in a direction (i.e., it satisfies Xn ⋅ n +∞ for some ) with positive probability. In dimension 2, this result is strenghened to an almost sure directional transience thanks to the 0–1 law from [Ann. Probab. 29 (2001) 1716–1732]. Our proof relies on the property of stability of Dirichlet environment by time reversal proved in [Random walks in random Dirichlet environment are transient in dimension d ≥ 3 (2009), Preprint]. In a first part of this paper, we also give a probabilistic proof of this property as an alternative to the change of variable computation used initially.

Résumé

On s’intéresse aux marches aléatoires dans un environnement défini par des variables de Dirichlet i.i.d. en chaque sommet de ℤd ou, de façon équivalente, aux marches aléatoires renforcées par arêtes orientées sur ℤd. Les paramètres de ce modèle sont un 2d-uplet de réels positifs indexé par les vecteurs unitaires de ℤd. On démontre que, dès que ces poids ne sont pas symétriques, la marche aléatoire est transiente dans une direction (c’est-à-dire qu’elle satisfait Xn ⋅ n +∞ pour un certain ) avec probabilité positive. En dimension 2, la loi du 0–1 de [Ann. Probab. 29 (2001) 1716–1732] permet de renforcer ce résultat en transience directionnelle presque-sûre. La preuve repose sur la propriété de stabilité des environnements de Dirichlet par renversement temporel introduite dans [Random walks in random Dirichlet environment are transient in dimension d≥3 (2009), Preprint] et dont on donne une nouvelle démonstration, de nature plus probabiliste, en première partie du présent article.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 1 (2011), 1-8.

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1294170226

Digital Object Identifier
doi:10.1214/09-AIHP344

Mathematical Reviews number (MathSciNet)
MR2779393

Zentralblatt MATH identifier
1209.60055

Subjects
Primary: 60K37: Processes in random environments 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Random walk Random environment Dirichlet distribution Directional transience Time reversal

Citation

Sabot, Christophe; Tournier, Laurent. Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 1, 1--8. doi:10.1214/09-AIHP344. https://projecteuclid.org/euclid.aihp/1294170226


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