Density of eigenvalues and its perturbation invariance in unitary ensembles of random matrices

We generally study the density of eigenvalues in unitary ensembles of random matrices from the recurrence coefficients with regularly varying conditions for the orthogonal polynomials. By using a new method, we calculate directly the moments of the density (which has been obtained in the work of Nevai and Dehesa, Van Assche and others on asymptotic zero distribution), and prove that scaling eigenvalues converge weakly, in probability and almost surely to the Nevai–Ullmann measure. Furthermore, we can prove that the density is invariant when the weight function is perturbed by a polynomial.