The characteristic polynomial on compact groups with Haar measure: some equalities in law

Paul Bourgade, Ashkan Nikeghbali, Alain Rouault

Research output: Contribution to journalArticlepeer-review

Abstract

This Note presents some equalities in law for ZN : = det (Id - G), where G is an element of a subgroup of the set of unitary matrices of size N, endowed with its unique probability Haar measure. Indeed, under some general conditions, ZN can be decomposed as a product of independent random variables, whose laws are explicitly known. Our results can be obtained in two ways: either by a recursive decomposition of the Haar measure (Section 1) or by previous results by Killip and Nenciu (2004) on orthogonal polynomials with respect to some measure on the unit circle (Section 2). This latter method leads naturally to a study of determinants of a class of principal submatrices (Section 3). To cite this article: P. Bourgade et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Original languageEnglish (US)
Pages (from-to)229-232
Number of pages4
JournalComptes Rendus Mathematique
Volume345
Issue number4
DOIs
StatePublished - Aug 15 2007

ASJC Scopus subject areas

  • Mathematics(all)

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