Comportement asymptotique des polyn\^omes orthogonaux associ\'es \`a un poids ayant un z\'ero d'ordre fractionnaire sur le cercle. Applications aux valeurs propres d'une classe de matrices al\'eatoires unitaires

Abstract

Asymptotic behavior of orthogonal polynomials on the circle, with respect to a weight having a fractional zero on the torus. Applications to the eigenvalues of certain unitary random matrices. This paper is devoted to the orthogonal polynomial on the circle, with respect to a weight of type $ f=(1-\cos \theta )^\alpha c$ where $c$ is a sufficiently smooth function and $\alpha \in ]-{1/2}, {1/2}[$. We obtain an asymptotic expansion of the coefficients of this polynomial and of $\Phi^{(p)}_{N}(1)$ for all integer $p$. 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 resuts related with the orthogonal polynomials are essentialy based on the inversion of Toeplitz matice associated to the symbol $f$.