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Ewens Measures on Compact Groups and Hypergeometric Kernels

Part of the Lecture Notes in Mathematics book series (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.

  • Decomposition of Haar measure
  • Random matrices
  • Characteristic polynomials
  • Ewens sampling formula
  • Correlation kernel

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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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Correspondence to 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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