Poplavskyi M. Spectral properties of ensembles of random unitary matrices

Purpose of the work is the analysis of the local eigenvalue distribution inside and on the boundary of the spectrum for unitary matrix models and asymptotic properties of polynomials orthogonal on the unit circle with a varying weight. The objects of study are unitary matrix models and polynomials orthogonal on the unit circle with a varying weight. The obtained results are the following. For system of polynomials orthogonal on the unit circle with a varying weight, in assumption of one interval support of the equilibrium measure of the system, the asymptotic formulas for Verblunsky coefficients are proved. Using this result the universality conjecture at the edge of the spectrum for unitary matrix models is proved in assumption of the one interval spectrum. For interior points of the spectrum uniform convergence of the first martingale density to the density of the normalized counting measure is obtained and the universaliry conjecture for interior points is proved. Also for unitary matrix models the universality conjecture for the gap probability in the bulk and at the edge of the spectrum is proved.