# Supercritical Regime for the Kissing Polynomials

## TL;DR

This paper analyzes the asymptotic behavior of a family of orthogonal polynomials with complex oscillatory weights in the supercritical regime, using quadratic differentials and Riemann-Hilbert techniques, revealing new phenomena at certain parameter values.

## Contribution

It extends the asymptotic analysis of kissing polynomials to the supercritical regime, explicitly constructing the global parametrix and identifying parameter values where polynomials do not exist.

## Key findings

- Asymptotics are obtained for ment yond the critical lue.
- Explicit construction of the global parametrix using elliptic integrals.
- Identification of a discrete set of lues where polynomials fail to exist.

## Abstract

We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function $e^{ni\lambda z}$ on $[-1,1]$, where $\lambda$ is a positive parameter. This family of polynomials has appeared in the literature recently in connection with complex quadrature rules, and their asymptotics have been previously studied when $\lambda$ is smaller than a certain critical value, $\lambda_c$. Our main goal is to compute their asymptotics when $\lambda>\lambda_c$. We first provide a geometric description, based on the theory of quadratic differentials, of the curves in the complex plane which will eventually support the asymptotic zero distribution of these polynomials. Next, using the powerful Riemann-Hilbert formulation of the orthogonal polynomials due to Fokas, Its, and Kitaev, along with its method of asymptotic solution via Deift-Zhou nonlinear steepest descent, we provide uniform asymptotics of the polynomials throughout the complex plane. Although much of this asymptotic analysis follows along the lines of previous works in the literature, the main obstacle appears in the construction of the so-called global parametrix. This construction is carried out in an explicit way with the help of certain integrals of elliptic type. In stark contrast to the situation one typically encounters in the presence of real orthogonality, an interesting byproduct of this construction is that there is a discrete set of values of $\lambda$ for which one cannot solve the model Riemann-Hilbert problem, and as such the corresponding polynomials fail to exist.