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

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) = z^N `(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.