The Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights

Abstract

We obtain Plancherel–Rotach-type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form on the real line, assuming that V has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application, we extend the universality class for bulk and edge asymptotics of eigenvalue statistics in unitary invariant Hermitian random matrix theory. We develop a new technique of asymptotic analysis for matrix Riemann–Hilbert problems with nonanalytic jump matrices suitable for analyzing such problems even near transition points where the solution changes from oscillatory to exponential behavior.