## Isotropic Random Walks on Affine Buildings

### J. Parkinson

#### Abstract

Recently, Cartwright and Woess provided a detailed analysis of isotropic random walks on the vertices of thick affine buildings of type \tilde{A}n. Their results generalise results of Sawyer, where homogeneous trees are studied (these are \tilde{A}1 buildings), and Lindlbauer and Voit, where \tilde{A}2 buildings are studied. In this paper we apply techniques of spherical harmonic analysis to prove a local limit theorem, a rate of escape theorem, and a central limit theorem for isotropic random walks on arbitrary thick regular affine buildings of irreducible type, thus providing a broad generalisation of the \tilde{A}n case.

Keywords: Affine buildings, random walks, Macdonald spherical functions.

: Primary 20E42; secondary (60G50 33D52).

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 Tuesday, November 15, 2005