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

TL;DR

This paper proves a local limit theorem for specific random walks on the chambers of affine buildings of type A_2, utilizing Hecke algebra representation theory and harmonic analysis.

Contribution

It introduces a method to analyze random walks on A_2 buildings using Hecke algebra representations, providing the first local limit theorem for this setting.

Findings

Established a local limit theorem for A_2 buildings

Applied Hecke algebra representation theory to analyze random walks

Demonstrated the effectiveness of harmonic analysis in this context

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.