Perturbed Toeplitz operators and radial determinantal processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 934-960.

Nous étudions une classe d'ensembles déterminantaux dans le plan complexe invariants par rotation; cette classe comprend les cas des valeurs propres de matrices gaussiennes aléatoires et des zéros de certaines familles de polynomes aléatoires. Le résultat principal est un critère pour l'existence d'un théorème de la limite centrale pour la statistique des angles entre les points. La preuve utilise une formule exacte reliant la fonction génératrice de telles statistiques au déterminant d'une matrice de Toeplitz perturbée.

We study a class of rotation invariant determinantal ensembles in the complex plane; examples include the eigenvalues of Gaussian random matrices and the roots of certain families of random polynomials. The main result is a criterion for a central limit theorem to hold for angular statistics of the points. The proof exploits an exact formula relating the generating function of such statistics to the determinant of a perturbed Toeplitz matrix.

DOI : 10.1214/12-AIHP501
Classification : 60B20, 60F05, 47B35
Mots-clés : random matrices, determinantal processes, Toeplitz operators, Szegö-Widom limit theorem 
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Ehrhardt, Torsten; Rider, Brian. Perturbed Toeplitz operators and radial determinantal processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 934-960. doi : 10.1214/12-AIHP501. http://archive.numdam.org/articles/10.1214/12-AIHP501/

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