Painlev\'e IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

TL;DR

This paper analyzes the asymptotic behavior of complex-plane orthogonal polynomials linked to a normal matrix model, revealing a transition described by Painlevé IV equations and providing numerical insights into related Fredholm determinants.

Contribution

It establishes a connection between eigenvalue distribution transitions in a matrix model and Painlevé IV asymptotics, including numerical evaluation of associated determinants.

Findings

Eigenvalue distribution undergoes a corner-type singularity transition.

Asymptotics of orthogonal polynomials are described by Painlevé IV solutions.

Fredholm determinants are computed numerically and shown to be pole-free on a semiaxis.

Abstract

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to 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 Painlev\'e IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painlev\'e transcendent is pole-free on a semiaxis.