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Limit theorems for isotropic random walks on triangle buildings

Part of: Structure and classification of infinite or finite groups Limit theorems Basic hypergeometric functions Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  09 April 2009

Marc Lindlbauer  and
Michael Voit
Marc Lindlbauer
Affiliation:
Mathematisches Institut, Universität TübingenAuf der Morgenstelle 10, 72076 Tübingen, Germany and GSF-Forschungszentrum, fü Umwelt und Gesundheit, 85764 Neuherberg, Germany
Michael Voit
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany e-mail: [email protected]
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Abstract

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The spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.

Keywords

triangle buildingsHall-Littlewood polynomialspolynimial hypergroupsisotropic rondom walkslaw of large numberscentral limit theoremlocal limit theorem

MSC classification

Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60F05: Central limit and other weak theorems 60F15: Strong theorems 33D52: Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) 20E42: Groups with a $BN$-pair; buildings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 3 , December 2002 , pp. 301 - 334
Copyright
Copyright © Australian Mathematical Society 2002

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