Isotropic random walks on affine buildings

Parkinson, James

Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 379-419.

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 ${\tilde{A}}_1$).

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).

Classification : 20E42, 60G50, 33D52
Keywords: Affine buildings, random walks, Macdonald spherical functions
Mot clés : immeubles affines, marche aléatoire, fonctions sphériques de Macdonald

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

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

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

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

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

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

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

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

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

[10] Macdonald, I. G. 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 no. 2 | MR | Zbl

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

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

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

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

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

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

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

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

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

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