UZH-Logo

Maintenance Infos

Ewens measures on compact groups and hypergeometric kernels


Bourgade, P; Nikeghbali, A; Rouault, A (2011). Ewens measures on compact groups and hypergeometric kernels. In: Donati-Martin, C; Lejay, A; Rouault, A. Séminaire de Probabilités XLIII. Berlin, DE: Springer, 351-377.

Abstract

The general topic of the article is the study of probability measures on the unitary group U(N) and its closed subgroups. The authors start from the Haar measure on a compact subgroup on U(N) and show that such distributed random matrix is a product of independent random reflections. The authors also decompose certain determinants of random matrices into products of independent simply-distributed random variables. More precisely, they discuss the decomposition of det(Id-U), where U is either orthogonal, or unitary, or symplectic random matrix sampled from the Haar measure. In the second part of the article the deformations of the Haar measure on U(N) (and on some of its subgroups) are discussed. The authors consider the measure assigning to a matrix U the density proportional to det(Id-U) z ¯ det(Id-U ¯) z (where z is a complex parameter) and explain that this measure is a natural analogue of the Ewens measure on the symmetric group S(n). Recall that the Ewens measure is a probability distribution on S(n) with weight of a permutation g being proportional to θ c(g) , where c(g) stays for the number of cycles in g and θ is a real parameter. The similarity between the Ewens measures and the above measures on U(N) has been known before, especially in the context of harmonic analysis on the infinite symmetric group and the infinite-dimensional unitary group [see S. Kerov, G. Olshanski and A. Vershik, Invent. Math 158, No. 3, 551–642 (2004; Zbl 1057.43005) and G. Olshanski, Funct. Anal. 205, No. 2, 464–524 (2003; Zbl 1036.43002)]. The authors give some new explanations for the relations between these two families of measures. Finally, the distribution of the eigenvalues of the random matrices sampled from the above deformations of the Haar measure is discussed in the last part of the article.

Abstract

The general topic of the article is the study of probability measures on the unitary group U(N) and its closed subgroups. The authors start from the Haar measure on a compact subgroup on U(N) and show that such distributed random matrix is a product of independent random reflections. The authors also decompose certain determinants of random matrices into products of independent simply-distributed random variables. More precisely, they discuss the decomposition of det(Id-U), where U is either orthogonal, or unitary, or symplectic random matrix sampled from the Haar measure. In the second part of the article the deformations of the Haar measure on U(N) (and on some of its subgroups) are discussed. The authors consider the measure assigning to a matrix U the density proportional to det(Id-U) z ¯ det(Id-U ¯) z (where z is a complex parameter) and explain that this measure is a natural analogue of the Ewens measure on the symmetric group S(n). Recall that the Ewens measure is a probability distribution on S(n) with weight of a permutation g being proportional to θ c(g) , where c(g) stays for the number of cycles in g and θ is a real parameter. The similarity between the Ewens measures and the above measures on U(N) has been known before, especially in the context of harmonic analysis on the infinite symmetric group and the infinite-dimensional unitary group [see S. Kerov, G. Olshanski and A. Vershik, Invent. Math 158, No. 3, 551–642 (2004; Zbl 1057.43005) and G. Olshanski, Funct. Anal. 205, No. 2, 464–524 (2003; Zbl 1036.43002)]. The authors give some new explanations for the relations between these two families of measures. Finally, the distribution of the eigenvalues of the random matrices sampled from the above deformations of the Haar measure is discussed in the last part of the article.

Citations

2 citations in Web of Science®
1 citation in Scopus®
Google Scholar™

Altmetrics

Additional indexing

Item Type:Book Section, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2011
Deposited On:08 Jan 2012 20:11
Last Modified:05 Apr 2016 15:16
Publisher:Springer
Series Name:Lecture Notes in Mathematics
Number:2006
ISSN:0075-8434 (P)
ISBN:978-3-642-15216-0 (P) 978-3-642-15217-7 (E)
Additional Information:Kongress: Séminaire de Probabilités XLIII, 2009, Poitiers
Publisher DOI:https://doi.org/10.1007/978-3-642-15217-7_15
Related URLs:http://opac.nebis.ch/F/?local_base=EBI01&con_lng=GER&func=find-b&find_code=090&request=001948597
http://opac.nebis.ch/F/?local_base=EBI01&con_lng=GER&func=find-b&find_code=090&request=001931892
http://opac.nebis.ch/F/?local_base=NEBIS&CON_LNG=GER&func=find-b&find_code=SYS&request=006245391

Download

Full text not available from this repository.
View at publisher

TrendTerms

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.
You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.

Author Collaborations