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Limit theorems for isotropic random walks on triangle buildings

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

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: michael.voit@uni-tuebingen.de
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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

References

[AW]Askey, R. and Wilson, J. A., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (Amer. Math. Soc., Providence, RI, 1985).Google Scholar
[BH]Bloom, W. R. and Heyer, H., Harmonic analysis of probability measures on hypergroups (de Gruyter, Berlin, 1995).CrossRefGoogle Scholar
[BG]Bouhaik, M. and Gallardo, L., ‘Un théorème limit central dans un hypergroup bidimensionnel’, Ann. Inst. H. Poincaré Probab. Statist. 28 (1992), 47–61.Google Scholar
[B]Brown, K. S., Buildings (Springer, New York, 1989).CrossRefGoogle Scholar
[C]Cartwright, D. I., ‘Spherical harmonic analysis on buildings of type Ãn’, Monatsh. Math. 133 (2001), 93–109.CrossRefGoogle Scholar
[CMSZ]Cartwright, D. I., Mantero, A. M., Steger, T. and Zappa, A., ‘Groups acting simply transitively on the vertices of a building of type Ã2. I, II’, Geom. Dedicata 47 (1993), 143–166 and 167–223.CrossRefGoogle Scholar
[CM]Cartwright, D. I. and Mlotkowski, W., ‘Harmonic analysis for groups acting on triangle buildings’, J. Austral. Math. Soc. Ser. A 56 (1994), 345–383.CrossRefGoogle Scholar
[CMS]Cartwright, D. I., Mlotkowski, W. and Steger, T., ‘Property (T) and Ã2 groups’, Ann. inst. Fourier (Grenoble) 44 (1994), 213–248.Google Scholar
[CW]Cartwright, D. I. and Woess, W., ‘Isotropic random walks in a building of type ãs’, Preprint, 2001.Google Scholar
[F]Feller, W., An introduction to probability theory and its applications, volume II, 2nd edition (Wiley, New York, 1971).Google Scholar
[Kw]Koornwinder, T. H., ‘Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators III, IV’, Indag. Math. 36 (1974), 357–369 and 370–381.CrossRefGoogle Scholar
[Ky]Korànyi, A. (editor), Harmonic functions on trees and buildings, Contemp. Math. 206, (Amer. Math. Soc., Providence, RI, 1997).CrossRefGoogle Scholar
[LR]Lasser, R. and Rosier, M., ‘A note on property (T) of orthogonal polynomials’, Arch. Math. 60 (1993), 459–463.CrossRefGoogle Scholar
[L1]Lindlbauer, M., Grenzwertsätze für Irrfahrten auf Dreiecksgebäuden und den assoziierten polynomiellen Hypergruppen (Ph.D. Thesis, Universitat Tübingen, 1998).Google Scholar
[L2]Lindlbauer, M., ‘On the rate of convergence of the laws of Markov chains associated with orthogonal polynomials’, J. Comp. Appl. Math. 99 (1998), 287–297.CrossRefGoogle Scholar
[Mac1]Macdonald, I. G., Sphericalfuncrions on a group of p-adic type, Publications of the Ramanujan Institute, No. 2 (Ramanujan Institute, University of Madras, Madras, 1971).Google Scholar
[Mac2]Macdonald, I. G., Symmetric functions and Hall polynomials, 2nd edition (Oxford University Press, New York, 1995).CrossRefGoogle Scholar
[M]Maley, F. Miller, ‘The Hall polynomial revisited’, J. Algebra 184 (1996), 363–371.CrossRefGoogle Scholar
[MZ]Mantero, A. M. and Zappa, A., ‘Spherical functions and spectrum of the Laplace operators on buildings of rank 2’, Boll. Un. Mat. ltal. B(7) 8 (1994), 419–475.Google Scholar
[P]Picardello, M. A., ‘Spherical functions and local limit theorems on free groups’. Ann. Math. Pura Appl. (4) 33 (1983), 177–191.CrossRefGoogle Scholar
[R]Ronan, M., Lectures on buildings (Academic Press, Boston, 1989).Google Scholar
[Sa]Sawyer, S., ‘Isotropic random walks in a tree’, Z. Wahrsch. Verw. Gebiete 42 (1978), 279–292.CrossRefGoogle Scholar
[Si]Simon, B., Representations of finite and compact groups (Amer. Math. Soc., 1996).Google Scholar
[T]Tits, J., ‘Spheres of radius 2 in triangle buildings I’, in: Finite geometries, buildings and related topics (ed. Kantor, W. et al. ) (Clarendon Press, Oxford, 1990) pp. 17–28.CrossRefGoogle Scholar
[VI]Voit, M., ‘Central limit theorems for random walks on that are associated with orthogonal polynomials’, J. Multivar. Anal. 34 (1990), 290–322.CrossRefGoogle Scholar
[V2]Voit, M., ‘Laws of large numbers for polynomial hypergroups and some applications’, J. Theor. Probab. 3 (1990), 245–266.CrossRefGoogle Scholar
[V3]Voit, M., ‘A law of the iterated logarithm for Markov chains on associated with orthogonal polynomials’, J. Theor. Probab. 6 (1993), 653–669.CrossRefGoogle Scholar
[Z1]Zeuner, H., ‘Laws of large numbers for hypergroups on +’, Math. Ann. 283 (1989). 657–678.CrossRefGoogle Scholar
[Z2]Zeuner, H., ‘Moment functions and laws of large numbers on hypergroups’, Math. Z. 211 (1992), 369–407.CrossRefGoogle Scholar
[Z3]Zeuner, H., ‘Polynomial hypergroups in several variables’, Arch. Math. 58 (1992), 425–434.CrossRefGoogle Scholar