Marche aléatoire sur un immeuble affine de type $\tilde{A}_r$ et mouvement brownien de la chambre de Weyl

Bruno Schapira (1, 2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

In this paper we study a random walk on an affine building of type $\tilde{A}_r$, whose radial part, when suitably normalized, converges to the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane \cite{Bia2}. This extends also the link at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol and Jeulin in rank one \cite{BJ}. The main ingredients of the proof are a combinatorial formula on the building and the estimate of the transition density proved in \cite{AST}.

Mots clés: processus GUE; Marche aléatoire; immeuble affine; système de racines; processus GUE.

Domaines: Probabilités [math.PR]

Bruno Schapira. Marche aléatoire sur un immeuble affine de type $\tilde{A}_r$ et mouvement brownien de la chambre de Weyl. 2007. ⟨hal-00114681v2⟩