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

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• 作者：Paul Bourgade ; Ashkan Nikeghbali ; Alain Rouault
• 关键词：Renormalization ; Gauge theories
• 刊名：Comptes Rendus Mathematique
• 出版年：2007
• 出版时间：15 August 2007
• 年：2007
• 卷：345
• 期：4
• 页码：229-232
• 全文大小：128 K

文摘

This Note presents some equalities in law for Z_N:=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, Z_N 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).