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

摘要

The first part of this paper is devoted to the study of F_N{Phi_N} the orthogonal polynomials on the circle, with respect to a weight of type f = (1 − cos θ) α c where c is a sufficiently smooth function and ${alpha > -frac{1}{2}}${alpha > -frac{1}{2}}. We obtain an asymptotic expansion of the coefficients F*(p)_N(1){Phi^{*(p)}_{N}(1)} for all integer p where F*_N{Phi^*_N} is defined by F*_N (z) = zN [`(F)]_N(frac1z) (z ¹ 0){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.