Circular \(\beta \) ensembles, CMV representation, characteristic polynomials
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.
scientific article; zbMATH DE number 5643466
scientific article