A local limit theorem for random walks on the chambers of A₂ buildings

2009/12/07 by James Parkinson, Bruno Schapira, Parkinson, James +1
Mathematics · #20E42 #60G50 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Random Matrices and Applications #Representation Theory (math.RT) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0912.1301

openalex publication_date 2009/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we outline an approach for analysing random walks on the chambers of buildings. The types of walks that we consider are those which are well adapted to the structure of the building: Namely walks with transition probabilities p(c,d) depending only on the Weyl distance δ(c,d). We carry through the computations for thick locally finite affine buildings of type A₂ to prove a local limit theorem for these buildings. The technique centres around the representation theory of the associated Hecke algebra. This representation theory is particularly well developed for affine Hecke algebras, with elegant harmonic analysis developed by Opdam. We give an introductory account of this theory in the second half of this paper.

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