Isotropic random walks on affine buildings

James Parkinson[1]

  • [1] University of Sydney School of Mathematics and Statistics F07 Sydney NSW 2006 (Australia)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 2, page 379-419
  • ISSN: 0373-0956

Abstract

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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 A ˜ n buildings are studied, Lindlbauer and Voit where A ˜ 2 buildings are studied, and Sawyer where homogeneous trees are studied (these are A ˜ 1 buildings).

How to cite

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Parkinson, James. "Isotropic random walks on affine buildings." Annales de l’institut Fourier 57.2 (2007): 379-419. <http://eudml.org/doc/10226>.

@article{Parkinson2007,
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).},
affiliation = {University of Sydney School of Mathematics and Statistics F07 Sydney NSW 2006 (Australia)},
author = {Parkinson, James},
journal = {Annales de l’institut Fourier},
keywords = {Affine buildings; random walks; Macdonald spherical functions; affine buildings; MacDonald spherical functions},
language = {eng},
number = {2},
pages = {379-419},
publisher = {Association des Annales de l’institut Fourier},
title = {Isotropic random walks on affine buildings},
url = {http://eudml.org/doc/10226},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Parkinson, James
TI - Isotropic random walks on affine buildings
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 2
SP - 379
EP - 419
AB - 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).
LA - eng
KW - Affine buildings; random walks; Macdonald spherical functions; affine buildings; MacDonald spherical functions
UR - http://eudml.org/doc/10226
ER -

References

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  1. Philippe Bougerol, Théorème central limite local sur certains groupes de Lie, Annales Scientifiques de L’É.N.S. 14 (1981), 403-432 Zbl0488.60013MR654204
  2. N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, (2002), Springer-Verlag, Berlin Heidelberg New York Zbl0983.17001MR1890629
  3. Kenneth Brown, Buildings, (1989), Springer-Verlag, New York Zbl0715.20017MR969123
  4. D. I. Cartwright, Spherical Harmonic Analysis on Buildings of Type A ˜ n , Monatsh. Math. 133 (2001), 93-109 Zbl1008.51019MR1860293
  5. D. I. Cartwright, W. Woess, Isotropic Random Walks in a Building of Type A ˜ d , Mathematische Zeitschrift 247 (2004), 101-135 Zbl1060.60070MR2054522
  6. Kenneth R. Davidson, C * -Algebras by Example, (1996), American Mathematical Society, Providence, Rhode Island, U.S.A. Zbl0958.46029MR1402012
  7. A. Figà-Talamanca, C. Nebbia, Harmonic analysis and representation theory for groups acting on homogeneous trees, 162 (1991), C.U.P., Cambridge Zbl1154.22301MR1152801
  8. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, 9 (1978), Springer-Verlag, New York-Berlin Zbl0447.17001MR499562
  9. M. Lindlbauer, M. Voit, Limit Theorems for Isotropic Random Walks on Triangle Buildings, J. Aust. Math. Soc. 73 (2002), 301-333 Zbl1028.60005MR1936256
  10. I. G. Macdonald, Spherical Functions on a Group of p -adic type, (1971), Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras, Madras Zbl0302.43018MR435301
  11. I. G. Macdonald, The Poincaré Series of a Coxeter Group, Math. Ann. 199 (1972), 161-174 Zbl0286.20062MR322069
  12. I. G. Macdonald, Symmetric Functions and Hall Polynomials, (1995), Clarendon Press, Oxford Zbl0824.05059MR1354144
  13. I. G. Macdonald, Affine Hecke Algebras and Orthogonal Polynomials, 157 (2003), C.U.P., Cambridge Zbl1024.33001MR1976581
  14. J. Parkinson, Buildings and Hecke Algebras, (2005), Sydney University Zbl1095.20003MR2206366
  15. J. Parkinson, Buildings and Hecke Algebras, Journal of Algebra 297 (2006), 1-49 Zbl1095.20003MR2206366
  16. J. Parkinson, Spherical Harmonic Analysis on Affine Buildings, Mathematische Zeitschrift 253 (2006), 571-606 Zbl1171.43009MR2221087
  17. Mark Ronan, Lectures on Buildings, (1989), Academic Press Zbl0694.51001MR1005533
  18. Stanley Sawyer, Isotropic Random Walks in a Tree, Z. Wahrsch. Verw. Gebiete 42 (1978), 279-292 Zbl0362.60075MR491493
  19. Frank Spitzer, Principles of Random Walk (second edition), (1964), Springer-Verlag Zbl0119.34304MR388547
  20. Wolfgang Woess, Random Walks on Infinite Graphs and Groups, (2000), C.U.P. Zbl0951.60002

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