Orthogonal polynomials for a class of measures with discrete rotational symmetries in the complex plane

TL;DR

This paper derives the strong asymptotics of orthogonal polynomials with respect to complex measures exhibiting discrete rotational symmetries, revealing how their zero distributions and support topology change with parameters, using Riemann-Hilbert analysis.

Contribution

It introduces a method to analyze orthogonal polynomials with complex measures related to normal matrix models, especially addressing different topological regimes of eigenvalue support.

Findings

Asymptotics depend on the parameter regime, with support topology changing at critical values.

Zero distribution support consists of a contour in each connected component of the eigenvalue support.

Method reduces planar orthogonality to contour integral conditions and applies Riemann-Hilbert analysis.

Abstract

We obtain the strong asymptotics of polynomials $p_n(\lambda)$, $\lambda\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|\lambda|^{2s}-t\lambda^s-\overline{t\lambda}^s)}dA(\lambda), $$ where $s$ is a positive integer, $t$ is a complex parameter and $dA$ stands for the area measure in the plane. Such problem has its origin from normal matrix models. We study the asymptotic behaviour of $p_n(\lambda)$ in the limit $n,N\to\infty$ in such a way that $n/N\to T$ constant. Such asymptotic behaviour has two distinguished regimes according to the topology of the limiting support of the eigenvalue distribution of the normal matrix model. If $0<|t|^2<T/s$, the eigenvalue distribution support is a simply connected compact set of the complex plane, while for $|t|^2>T/s$ the eigenvalue distribution support consists of $s$ connected components. Correspondingly the support of the limiting zero distribution of the orthogonal polynomials consists of a closed contour contained in each connected component. Our asymptotic analysis is obtained by reducing the planar orthogonality conditions of the polynomials to an equivalent system of contour integral orthogonality conditions. The strong asymptotics for the orthogonal polynomials is obtained from the corresponding Riemann--Hilbert problem by the Deift--Zhou nonlinear steepest descent method.