Rostyslav Kozhan: Relative Strong Szego Theorem

Abstract:
We establish a relative version of the Strong Szego Theorem for Toeplitz determinants. As an application we obtain the Central Limit Theorem for linear statistics of the eigenvalues of orthogonal polynomial ensembles of random unitary matrices. Our method can handle measures with essential support on the full circle or a single arc that satisfy the Lopez conditions. In particular, this allows the measure to have a singular component within or outside of the arc. In the talk I will review some basics of the theory of orthogonal polynomials on the unit circle and various forms of the classical (strong) Szego theorem: from the viewpoint of Toeplitz determinants, of random matrix theory, and of orthogonal polynomials. Joint work with M.Duits (KTH).