Comportement asymptotique des polynômes orthogonaux associés à un poids ayant un zéro d'ordre fractionnaire sur le cercle. Applications aux valeurs propres d'une classe de matrices aléatoires unitaires

2009/04/05 by Philippe Rambour, Rambour, Philippe, Abdellatif Seghier +1
Mathematics · #47B39 #47BXX #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #math.FA #math.PR #msc:47B39 #msc:47BXX

paper · pdf · doi:10.48550/arxiv.0904.0777

arxiv created 2009/04/27 · arxiv updated 2009/12/01

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 θ)^αc where c is a sufficiently smooth function and α∈ ]-1/2, 1/2[. We obtain an asymptotic expansion of the coefficients of this polynomial and of Φ^(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.

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