The Dirichlet random walk
[La marche aléatoire de Dirichlet]
Boulanger, Adrien1 ; Glorieux, Olivier2
1 Technopôle Château-Gombert 39, rue Frédéric Joliot-Curie 13453 MARSEILLE Cedex 13 (France)
2 Université Rennes 1, IRMAR / UFR Math, Campus de Beaulieu, 35042 Rennes Cedex (France)
Annales Henri Lebesgue, Tome 5 (2022), pp. 1295-1328.

Dans cet article on définit un processus stochastique sur les revêtements galoisiens d’une variété compacte. Les positions successives du processus sont données en tirant uniformément par rapport à la mesure Riemannienne un point dans le domaine de Dirichlet associé à la position précédente. On démontre un théorème à la Kesten  : la vitesse de fuite de la marche est positive si et seulement si le revêtement est moyennable. On étudie plus particulièrement le cas où le groupe de revêtement est hyperbolique. On démontre alors la convergence presque sûre au bord ainsi qu’un théorème central limite.

In this article we define and study a stochastic process on Galoisian covers of compact manifolds. The successive positions of the process are defined recursively by picking a point uniformly in the Dirichlet domain of the previous one. We prove a theorem à la Kesten for such a process: the escape rate of the random walk is positive if and only if the cover is non amenable. We also investigate more in details the case where the deck group is Gromov hyperbolic, showing the almost sure convergence to the boundary of the trajectory as well as a central limit theorem for the escape rate.

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DOI : 10.5802/ahl.152
Classification : 53C20, 58D19, 60J35
Mots clés : stochastic processes, Riemannian geometry, group actions
Boulanger, Adrien 1 ; Glorieux, Olivier 2

1 Technopôle Château-Gombert 39, rue Frédéric Joliot-Curie 13453 MARSEILLE Cedex 13 (France)
2 Université Rennes 1, IRMAR / UFR Math, Campus de Beaulieu, 35042 Rennes Cedex (France) 
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Boulanger, Adrien; Glorieux, Olivier. The Dirichlet random walk. Annales Henri Lebesgue, Tome 5 (2022), pp. 1295-1328. doi : 10.5802/ahl.152. http://archive.numdam.org/articles/10.5802/ahl.152/

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