Random Walks in a Dirichlet Environment

Sabot, Christophe; Enriquez, Nathanaël

doi:10.1214/ejp.v11-350

Cited by 11 publications

References 14 publications

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Random-walk in Beta-distributed random environment

Barraquand

Corwin

2016

Probab. Theory Relat. Fields

108

230

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We introduce an exactly-solvable model of random walk in random environment that we call the Beta RWRE. This is a random walk in $\mathbb{Z}$ which performs nearest neighbour jumps with transition probabilities drawn according to the Beta distribution. We also describe a related directed polymer model, which is a limit of the $q$-Hahn interacting particle system. Using a Fredholm determinant representation for the quenched probability distribution function of the walker's position, we are able to prove second order cube-root scale corrections to the large deviation principle satisfied by the walker's position, with convergence to the Tracy-Widom distribution. We also show that this limit theorem can be interpreted in terms of the maximum of strongly correlated random variables: the positions of independent walkers in the same environment. The zero-temperature counterpart of the Beta RWRE can be studied in a parallel way. We also prove a Tracy-Widom limit theorem for this model.Comment: 54 pages, 9 figures. v3: extended revised version. To appear in Probab. Theory Related Field

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Random-walk in Beta-distributed random environment

Barraquand

Corwin

2016

Probab. Theory Relat. Fields

108

230

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Conduction and Diffusion in Percolating Systems

Hughes¹

SpringerReference

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

Sabot

Tournier

2011

Ann. Inst. H. Poincaré Probab. Statist.

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We consider random walks in a random environment given by i.i.d. Dirichlet distributions at each vertex of Z d or, equivalently, oriented edge reinforced random walks on Z d . The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of Z d . We prove that, as soon as these weights are nonsymmetric, the random walk is transient in a direction (i.e., it satisfies X n • → 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 Z d ou, de façon équivalente, aux marches aléatoires renforcées par arêtes orientées sur Z d . Les paramètres de ce modèle sont un 2d-uplet de réels positifs indexé par les vecteurs unitaires de Z d . 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 X n • → 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.

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Random Walks in a Dirichlet Environment

Cited by 11 publications

References 14 publications

Random-walk in Beta-distributed random environment

Random-walk in Beta-distributed random environment

Conduction and Diffusion in Percolating Systems

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

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