Isotropic random walks on affine buildings

[Marches aléatoires isotropes sur les immeubles affines]

Parkinson, James

Annales de l'Institut Fourier, Tome 57 (2007), p. 379-419 / Harvested from Numdam

Résumé

Dans cet article, nous utilisons les techniques de l’analyse harmonique sphérique pour démontrer un théorème local limite, un théorème sur la vitesse de fuite et un théorème central limite pour les marches aléatoires isotropes sur des immeubles affines épais arbitraires de type irréductible. Cela généralise des résultats de Cartwright et Woess sur les immeubles de type $\tilde{A}_n$, de Lindlbauer et Voit sur les immeubles de type $\tilde{A}_2$ et de Sawyer sur les arbres homogènes (qui sont des immeubles de type $\tilde{A}_1$).

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 $\tilde{A}_n$ buildings are studied, Lindlbauer and Voit where $\tilde{A}_2$ buildings are studied, and Sawyer where homogeneous trees are studied (these are $\tilde{A}_1$ buildings).

Détail

Publié le : 2007-01-01

MR 2310945 | Zbl 1177.60046 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.2262
Classification:  20E42,  60G50,  33D52
Mots clés: immeubles affines, marche aléatoire, fonctions sphériques de Macdonald

Bibliographie

[1] Bougerol, Philippe Théorème central limite local sur certains groupes de Lie, Annales Scientifiques de L’É.N.S., Tome 14 (1981), pp. 403-432 | Numdam | MR 654204 | Zbl 0488.60013

[2] Bourbaki, N. Lie Groups and Lie Algebras, Chapters 4–6, Springer-Verlag, Berlin Heidelberg New York, Elements of Mathematics (2002) | MR 1890629 | Zbl 0983.17001

[3] Brown, Kenneth Buildings, Springer-Verlag, New York (1989) | MR 969123 | Zbl 0715.20017

[4] Cartwright, D. I. Spherical Harmonic Analysis on Buildings of Type $\widetilde{A}_n$, Monatsh. Math., Tome 133 (2001) no. 2, pp. 93-109 | Article | MR 1860293 | Zbl 1008.51019

[5] Cartwright, D. I.; Woess, W. Isotropic Random Walks in a Building of Type $\widetilde{A}_d$, Mathematische Zeitschrift, Tome 247 (2004), pp. 101-135 | Article | MR 2054522 | Zbl 1060.60070

[6] Davidson, Kenneth R. ${C}^*$-Algebras by Example, American Mathematical Society, Providence, Rhode Island, U.S.A., Fields Institute Monographs (1996) | MR 1402012 | Zbl 0958.46029

[7] Figà-Talamanca, A.; Nebbia, C. Harmonic analysis and representation theory for groups acting on homogeneous trees, C.U.P., Cambridge, London Mathematical Society Lecture Notes Series, Tome 162 (1991) | MR 1152801 | Zbl 00050232

[8] Humphreys, J. E. Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York-Berlin, Graduate Texts in Mathematics, Tome 9 (1978) | MR 499562 | Zbl 0447.17001

[9] Lindlbauer, M.; Voit, M. Limit Theorems for Isotropic Random Walks on Triangle Buildings, J. Aust. Math. Soc., Tome 73 (2002) no. 3, pp. 301-333 | Article | MR 1936256 | Zbl 1028.60005

[10] Macdonald, I. G. Spherical Functions on a Group of $p$-adic type, Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras, Madras, Publications of the Ramanujan Institute (1971) no. 2 | MR 435301 | Zbl 0302.43018

[11] Macdonald, I. G. The Poincaré Series of a Coxeter Group, Math. Ann., Tome 199 (1972), pp. 161-174 | Article | MR 322069 | Zbl 0286.20062

[12] Macdonald, I. G. Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, Oxford Mathematical Monographs (1995) | MR 1354144 | Zbl 0824.05059

[13] Macdonald, I. G. Affine Hecke Algebras and Orthogonal Polynomials, C.U.P., Cambridge, Cambridge Tracts in Mathematics, Tome 157 (2003) | MR 1976581 | Zbl 1024.33001

[14] Parkinson, J. Buildings and Hecke Algebras, Sydney University, Ph.D. Thesis (2005) | MR 2206366

[15] Parkinson, J. Buildings and Hecke Algebras, Journal of Algebra, Tome 297 (2006) no. 1, pp. 1-49 | Article | MR 2206366 | Zbl 1095.20003

[16] Parkinson, J. Spherical Harmonic Analysis on Affine Buildings, Mathematische Zeitschrift, Tome 253 (2006) no. 3, pp. 571-606 | Article | MR 2221087 | Zbl 05071445

[17] Ronan, Mark Lectures on Buildings, Academic Press, Perspectives in Mathematics (1989) | MR 1005533 | Zbl 0694.51001

[18] Sawyer, Stanley Isotropic Random Walks in a Tree, Z. Wahrsch. Verw. Gebiete, Tome 42 (1978), pp. 279-292 | Article | MR 491493 | Zbl 0362.60075

[19] Spitzer, Frank Principles of Random Walk (second edition), Springer-Verlag, Graduate Texts in Mathematics (1964) | MR 388547 | Zbl 0119.34304

[20] Woess, Wolfgang Random Walks on Infinite Graphs and Groups, C.U.P., Cambridge Tracts in Mathematics (2000) | Zbl 0951.60002