Ewens Measures on Compact Groups and Hypergeometric Kernels

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 explic...

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Bibliographic Details
Published inSéminaire de Probabilités XLIII pp. 351 - 377
Main Authors Bourgade, Paul, Nikeghbali, Ashkan, Rouault, Alain
Format Book Chapter
LanguageEnglish
Published Berlin, Heidelberg Springer Berlin Heidelberg
SeriesLecture Notes in Mathematics
Subjects
Characteristic polynomials
Correlation kernel
Decomposition of Haar measure
Ewens sampling formula
Random matrices
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Summary: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.
ISBN:3642152163
9783642152160
ISSN:0075-8434
1617-9692
DOI:10.1007/978-3-642-15217-7_15