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In this paper we outline an approach for analysing random walks on thechambers of buildings. The types of walks that we consider are those which arewell adapted to the structure of the building: Namely walks with transitionprobabilities $p(c,d)$ depending only on the Weyl distance $\delta(c,d)$. Wecarry through the computations for thick locally finite affine buildings oftype $\tilde{A}_2$ to prove a local limit theorem for these buildings. Thetechnique centres around the representation theory of the associated Heckealgebra. This representation theory is particularly well developed for affineHecke algebras, with elegant harmonic analysis developed by Opdam. We give anintroductory account of this theory in the second half of this paper.

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