Asymptotics of discrete orthogonal polynomials and the continuum limit of the Toda lattice

Abstract

In a recent monograph, Deift and McLaughlin considered the continuum limit of the Toda lattice and solved the problem using perturbation techniques, such as the Wentzel-Kramers-Brillouin (WKB) method. We present an alternative approach based on the analytic theory of discrete orthogonal polynomials. The method is based on some extremal problems in the theory of logarithmic potentials and relies on some results for the asymptotic theory of orthogonal polynomials with a discrete orthogonality measure.