Ewens measures on compact groups and hypergeometric kernels

Paul Bourgade, Ashkan Nikeghbali, Alain Rouault

Research output: Chapter in Book/Report/Conference proceedingChapter

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.

Original languageEnglish (US)
Title of host publicationSeminaire de Probabilites XLIII
Pages351-377
Number of pages27
Volume2006
DOIs
StatePublished - 2011

Publication series

NameLecture Notes in Mathematics
Volume2006
ISSN (Print)00758434

Fingerprint

Unitary group
Compact Group
Ewens Sampling Formula
kernel
Decompose
Haar Measure
Associated Polynomials
Orthogonal Group
Characteristic polynomial
Independent Random Variables
Unit circle
Spectral Properties
Symmetric group
Orthogonal Polynomials
Probability Measure
Infinity
Singularity
Tend
Family

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Bourgade, P., Nikeghbali, A., & Rouault, A. (2011). Ewens measures on compact groups and hypergeometric kernels. In Seminaire de Probabilites XLIII (Vol. 2006, pp. 351-377). (Lecture Notes in Mathematics; Vol. 2006). https://doi.org/10.1007/978-3-642-15217-7_15

Ewens measures on compact groups and hypergeometric kernels. / Bourgade, Paul; Nikeghbali, Ashkan; Rouault, Alain.

Seminaire de Probabilites XLIII. Vol. 2006 2011. p. 351-377 (Lecture Notes in Mathematics; Vol. 2006).

Research output: Chapter in Book/Report/Conference proceedingChapter

Bourgade, P, Nikeghbali, A & Rouault, A 2011, Ewens measures on compact groups and hypergeometric kernels. in Seminaire de Probabilites XLIII. vol. 2006, Lecture Notes in Mathematics, vol. 2006, pp. 351-377. https://doi.org/10.1007/978-3-642-15217-7_15
Bourgade P, Nikeghbali A, Rouault A. Ewens measures on compact groups and hypergeometric kernels. In Seminaire de Probabilites XLIII. Vol. 2006. 2011. p. 351-377. (Lecture Notes in Mathematics). https://doi.org/10.1007/978-3-642-15217-7_15
Bourgade, Paul ; Nikeghbali, Ashkan ; Rouault, Alain. / Ewens measures on compact groups and hypergeometric kernels. Seminaire de Probabilites XLIII. Vol. 2006 2011. pp. 351-377 (Lecture Notes in Mathematics).
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