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Ewens Measures on Compact Groups and Hypergeometric Kernels

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Séminaire de Probabilités XLIII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2006))

Abstract

On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as n tends to infinity to a limit kernel at the singularity.

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Acknowledgement

A.N.’s work is supported by the Swiss National Science Foundation (SNF) grant 200021_119970/1.

A.R’s work is partly supported by the ANR project Grandes Matrices Alatoires ANR-08-BLAN-0311-01.

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Authors and Affiliations

  1. Institut Telecom & Université Paris 6, 46 rue Barrault, 75634, Paris Cedex 13, France

    Paul Bourgade

  2. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland

    Ashkan Nikeghbali

  3. Université Versailles-Saint Quentin, LMV, Bâtiment Fermat, 45 avenue des Etats-Unis, 78035, Versailles Cedex, France

    Alain Rouault

Authors
  1. Paul Bourgade
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  2. Ashkan Nikeghbali
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  3. Alain Rouault
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Correspondence to Alain Rouault .

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Editors and Affiliations

  1. Lab. de Probabilités et Modèles Aléatoir, Université Pierre et Marie Curie, place Jussieu 4, Paris Cedex 05, 75252, France

    Catherine Donati-Martin

  2. IECN, Campus scientifique, BP 239, Nancy-Université, INRIA, Vandoeuvre-lès-Nancy, 54506, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des Etats-Unis 45, Versailles, 78035, France

    Alain Rouault

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Bourgade, P., Nikeghbali, A., Rouault, A. (2011). Ewens Measures on Compact Groups and Hypergeometric Kernels. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIII. Lecture Notes in Mathematics(), vol 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15217-7_15

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