Random walks in Dirichlet environment: an overview

Sabot, Christophe; Tournier, Laurent

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 463-509.

Les marches aléatoires en environnement de Dirichlet (RWDE) correspondent à des marches alétoires en environnement aléatoire dont les probabilités de transition en chaque site sont indépendantes et distribuées suivant une même loi de Dirichlet. Le modèle est donc paramétré par une famille de poids ${\left({\alpha }_i\right)}_{i=1,...,2d}$, un pour chaque direction dirigée de ${\mathbb{Z}}^d$. Dans ce cas, la loi moyennée est celle de la marche renforcée avec renforcement linéaire sur les arêtes orientées. Les RWDE ont une propriété remarquable d’invariance en loi par retournement du temps, de laquelle découlent plusieurs résultats encore inaccessibles dans le cas général, comme la propriété d’équivalence des points de vue statiques et dynamiques ou comme la caractérisation des régimes de transience directionnelle et de ballisticité. Dans cet article, nous présentons les développements récents sur ce modèle et donnons plusieurs esquisses de démonstrations mettant en relief les arguments centraux du sujet. Nous présentons aussi des calculs nouveaux sur la fonction de taux des grandes déviations dans le cas unidimensionnel.

Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in Random Environment (RWRE) on ${\mathbb{Z}}^d$ where the transition probabilities are i.i.d. at each site with a Dirichlet distribution. Hence, the model is parametrized by a family of positive weights ${\left({\alpha }_i\right)}_{i=1,...,2d}$, one for each oriented direction of ${\mathbb{Z}}^d$. In this case, the annealed law is that of a reinforced random walk, with linear reinforcement on directed edges. RWDE have a remarkable property of statistical invariance by time reversal from which can be inferred several properties that are still inaccessible for general environments, such as the equivalence of static and dynamic points of view and a description of the directionally transient and ballistic regimes. In this paper we review the recent developments on this model and give several sketches of proofs presenting the core of the arguments. We also present new computations of the large deviation rate function for one dimensional RWDE.

Classification : 60K37, 60K35
Mots clés : Random walk in random environment, Dirichlet distribution, Reinforced random walks, invariant measure viewed from the particle

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