The Generalised Dyson Circular Unitary Ensemble: Asymptotic Distribution of the Eigenvalues at the Origin of the Spectrum

Abstract

The first part of this paper is devoted to the study of \({\Phi_N}\) the orthogonal polynomials on the circle, with respect to a weight of type f = (1 − cos θ)αc where c is a sufficiently smooth function and \({\alpha > -\frac{1}{2}}\). We obtain an asymptotic expansion of the coefficients \({\Phi^{*(p)}_{N}(1)}\) for all integer p where \({\Phi^*_N}\) is defined by \({\Phi^*_N (z) = z^N \bar \Phi_N(\frac{1}{z})\ (z \not=0)}\). These results allow us to obtain an asymptotic expansion of the associated Christofel–Darboux kernel, and to compute the distribution of the eigenvalues of a family of random unitary matrices. The proof of the results related to the orthogonal polynomials are essentially based on the inversion of the Toeplitz matrix associated to the symbol f.