Random walk on a building of type A_r and Brownian motion of the Weyl chamber

2006/11/17 by Bruno Schapira, Schapira, Bruno
Mathematics · #05C25 #60B10 #60B15 #60C05 #60J10 #60J25 #60J35 #60J60 #Advanced Algebra and Geometry #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.math/0611529

openalex publication_date 2006/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study a random walk on an affine building of type 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 \citeBia2. 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 \citeBJ. The main ingredients of the proof are a combinatorial formula on the building and the estimate of the transition density proved in \citeAST.

Related

• A local limit theorem for random walks on the chambers of A₂ buildings
• Random walks conditioned to stay in Weyl chambers of type C and D
• Heat kernel and Green function estimates on affine buildings of type A_r
• Sharp estimates for distinguished random walks on affine buildings of type Ar
• Random walks in Weyl chambers