A local limit theorem for random walks on the chambers of ${\tilde{A}}_2$ buildings

James Parkinson and Bruno Schapira

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 $delta(c,d)$. We carry through the computations for thick locally finite affine buildings of type ${\tilde{A}}_2$ 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.

Keywords: Random walks, affine buildings, Hecke algebras, Plancherel Theorem.

AMS Subject Classification: Primary 20E42;; secondary 60G50.