Orthogonal polynomials in the normal matrix model with two insertions

TL;DR

This paper analyzes orthogonal polynomials with a specific complex weight, deriving their asymptotics and zero distribution in a normal matrix model, using advanced Riemann-Hilbert techniques under certain parameter restrictions.

Contribution

It provides the first strong asymptotics and zero distribution results for these orthogonal polynomials in the complex plane with a novel approach via multiple orthogonal polynomials.

Findings

Asymptotic zero distribution (mother body) characterized for large degrees.

Explicit form of the equilibrium measure (droplet) for eigenvalues.

Application of Riemann-Hilbert steepest descent method to complex orthogonal polynomials.

Abstract

We consider orthogonal polynomials with respect to the weight $|z^2+a^2|^{cN}e^{-N|z|^2}$ in the whole complex plane. We obtain strong asymptotics and the limiting normalized zero counting measure (mother body) of the orthogonal polynomials of degree $n$ in the scaling limit $n,N\to \infty$ such that $\frac{n}{N}\to t$. We restrict ourselves to the case $a^2\geq 2c$, $cN$ integer, and $t<t^{*}$ where $t^{*}$ is a constant depending only on $a,c$. Due to this restriction, the mother body is supported on an interval. We also find the two dimensional equilibrium measure (droplet) associated with the eigenvalues in the corresponding normal matrix model. Our method relies on the recent result that the planar orthogonal polynomials are a part of a vector of type I multiple orthogonal polynomials, and this enables us to apply the steepest descent method to the associated Riemann-Hilbert problem.