Ewens Measures on Compact Groups and Hypergeometric Kernels

• Published in: Séminaire de Probabilités XLIII pp. 351 - 377
• Main Authors: Bourgade, Paul, Nikeghbali, Ashkan, Rouault, Alain
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg
• Series: Lecture Notes in Mathematics
• Subjects: Characteristic polynomials; Correlation kernel; Decomposition of Haar measure; Ewens sampling formula; Random matrices
• DOI: 10.1007/978-3-642-15217-7_15
• ISBN: 3642152163; 9783642152160
• ISSN: 0075-8434

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.