Isotropic random walks on affine buildings

Abstract

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. This generalises results of Cartwright and Woess where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover accent="true"> <mml:mi>A</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mi>n</mml:mi> </mml:msub> </mml:math> buildings are studied, Lindlbauer and Voit where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover accent="true"> <mml:mi>A</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mn>2</mml:mn> </mml:msub> </mml:math> buildings are studied, and Sawyer where homogeneous trees are studied (these are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover accent="true"> <mml:mi>A</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> </mml:math> buildings).