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James Parkinson
Isotropic random walks on affine buildings
(Marches aléatoires isotropes sur les immeubles affines)
Annales de l'institut Fourier, 57 no. 2 (2007), p. 379-419, doi: 10.5802/aif.2262
Article PDF | Reviews MR 2310945 | Zbl 1177.60046 | 1 citation in Cedram
Class. Math.: 20E42, 60G50, 33D52
Keywords: Affine buildings, random walks, Macdonald spherical functions

Résumé - 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).

Bibliography

[1] Philippe Bougerol, “Théorème central limite local sur certains groupes de Lie”, Annales Scientifiques de L’É.N.S. 14 (1981), p. 403-432 Numdam |  MR 654204 |  Zbl 0488.60013
[2] N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, Elements of Mathematics, Springer-Verlag, Berlin Heidelberg New York, 2002  MR 1890629 |  Zbl 0983.17001
[3] Kenneth Brown, Buildings, Springer-Verlag, New York, 1989  MR 969123 |  Zbl 0715.20017
[4] D. I. Cartwright, “Spherical Harmonic Analysis on Buildings of Type $\widetilde{A}_n$”, Monatsh. Math. 133 (2001) no. 2, p. 93-109 Article |  MR 1860293 |  Zbl 1008.51019
[5] D. I. Cartwright & W. Woess, “Isotropic Random Walks in a Building of Type $\widetilde{A}_d$”, Mathematische Zeitschrift 247 (2004), p. 101-135 Article |  MR 2054522 |  Zbl 1060.60070
[6] Kenneth R. Davidson, ${C}^*$-Algebras by Example, Fields Institute Monographs, American Mathematical Society, Providence, Rhode Island, U.S.A., 1996  MR 1402012 |  Zbl 0958.46029
[7] A. Figà-Talamanca & C. Nebbia, Harmonic analysis and representation theory for groups acting on homogeneous trees, London Mathematical Society Lecture Notes Series 162, C.U.P., Cambridge, 1991  MR 1152801 |  Zbl 00050232
[8] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9, Springer-Verlag, New York-Berlin, 1978  MR 499562 |  Zbl 0447.17001
[9] M. Lindlbauer & M. Voit, “Limit Theorems for Isotropic Random Walks on Triangle Buildings”, J. Aust. Math. Soc. 73 (2002) no. 3, p. 301-333 Article |  MR 1936256 |  Zbl 1028.60005
[10] I. G. Macdonald, Spherical Functions on a Group of $p$-adic type, Publications of the Ramanujan Institute, Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras, Madras, 1971  MR 435301 |  Zbl 0302.43018
[11] I. G. Macdonald, “The Poincaré Series of a Coxeter Group”, Math. Ann. 199 (1972), p. 161-174 Article |  MR 322069 |  Zbl 0286.20062
[12] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1995  MR 1354144 |  Zbl 0824.05059
[13] I. G. Macdonald, Affine Hecke Algebras and Orthogonal Polynomials, Cambridge Tracts in Mathematics 157, C.U.P., Cambridge, 2003  MR 1976581 |  Zbl 1024.33001
[14] J. Parkinson, Buildings and Hecke Algebras, Ph.D. Thesis, Sydney University, 2005 arXiv |  MR 2206366
[15] J. Parkinson, “Buildings and Hecke Algebras”, Journal of Algebra 297 (2006) no. 1, p. 1-49 Article |  MR 2206366 |  Zbl 1095.20003
[16] J. Parkinson, “Spherical Harmonic Analysis on Affine Buildings”, Mathematische Zeitschrift 253 (2006) no. 3, p. 571-606 Article |  MR 2221087 |  Zbl 05071445
[17] Mark Ronan, Lectures on Buildings, Perspectives in Mathematics, Academic Press, 1989  MR 1005533 |  Zbl 0694.51001
[18] Stanley Sawyer, “Isotropic Random Walks in a Tree”, Z. Wahrsch. Verw. Gebiete 42 (1978), p. 279-292 Article |  MR 491493 |  Zbl 0362.60075
[19] Frank Spitzer, Principles of Random Walk (second edition), Graduate Texts in Mathematics, Springer-Verlag, 1964  MR 388547 |  Zbl 0119.34304
[20] Wolfgang Woess, Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics, C.U.P., 2000  Zbl 0951.60002
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