Leonid Pastur(Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine)

Asymptotics of Orthogonal Polynomials, Quasiperiodic Jacobi Matrices, and Random Matrices

We present a review of results on asymptotics of orthogonal polynomials, stressing their spectral aspects and similarity in two cases considered. They are polynomials orthonormal on a finite union of disjoint intervals with respect to the Szegö weight and polynomials orthonormal on the whole line with respect to varying weights and having the same union of intervals as the set of oscillations of asymptotics. In both cases we construct double infinite Jacobi matrices with generically quasiperiodic coefficients and show that each of them is an isospectral deformation of another. Related results on asymptotic eigenvalue distribution of a class of random matrices of large size are also discussed.