Circular \(\beta \) ensembles, CMV representation, characteristic polynomials
type: Item
instance of: scholarly article
title: Circular \(\beta \) ensembles, CMV representation, characteristic polynomials
zbMATH Open document ID: 1179.15038
published in: Science in China. Series A
publication date: 2009-12-07T00:00:00Z
review text: The circular \(\beta\) ensembles distributed on the unit circle in the complex plane with the joint probability density characterized by a model parameter \(\beta > 0\) are considered. The special cases \(\beta =\) 1, 2 and 4 correspond to random orthogonal, unitary and symplectic matrix models with the normalized Haar measure, respectively. Recently, \textit{R. Killip} and \textit{L. Nenciu} [Int. Math. Res. Not. 2004, No.~50, 2665--2701 (2004; Zbl 1255.82004)] introduced the matrix model for circular \(\beta\) ensembles with general \(\beta\) incorporating the theory of orthogonal polynomials on the unit circle and using the CMV matrices, a class of unitary matrices investigated by \textit{M. J. Cantero, L. Moral} and \textit{L. Velázquez} [Linear Algebra Appl. 362, 29--56 (2003; Zbl 1022.42013)]. The CMV matrices play the same role among unitary matrices as Jacobi matrices in Hermitian matrices. Introducing CMV matrices provides the totally new matrix representation even for the classical circular unitary ensembles with \(\beta =\) 2. On the basis of this discovery the moment identities of characteristic polynomials are derived and a link between the characteristic polynomial and the eigenvalue counting function is given.
Mathematics Subject Classification ID: 15B52
zbMATH DE Number: 5643466
zbMATH Keywords: characteristic polynomials
zbMATH Keywords: circular \(\beta\) ensembles
zbMATH Keywords: CMV representation
zbMATH Keywords: moment identity
zbMATH Keywords: random matrix ensembles
author: Zhonggen Su
MaRDI profile type: Publication
full work available at URL: s11425-009-0099-2
OpenAlex ID: W1996463265
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DOI: 10.1007/S11425-009-0099-2
description: scientific article; zbMATH DE number 5643466