  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.
