Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

Abstract

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated with a certain normal matrix model. The model depends on a parameter, and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition, we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlevé IV equation. We determine the Fredholm determinant associated with such a solution, and we compute it numerically on the real line, showing also that the corresponding Painlevé transcendent is pole-free on a semiaxis.