From random matrices to quasi-periodic Jacobi matrices via orthogonal polynomials

We present an informal 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 R 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 quasi-periodic 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 shortly discussed.