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

TL;DR

This paper studies the asymptotic behavior of orthogonal polynomials on the circle with fractional zero weights and applies these results to analyze eigenvalue distributions of certain random unitary matrices.

Contribution

It provides new asymptotic expansions for orthogonal polynomial coefficients and Christoffel-Darboux kernels with fractional zero weights, linking them to eigenvalue distributions.

Findings

Derived asymptotic expansions for polynomial coefficients.

Computed eigenvalue distributions of specific random unitary matrices.

Linked Toeplitz matrix inversion to orthogonal polynomial asymptotics.

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 $f$.