Global Phase Portrait and Large Degree Asymptotics for the Kissing Polynomials

Abstract

We study a family of monic orthogonal polynomials which are orthogonal with respect to the varying, complex valued weight function, $\exp(nsz)$, over the interval $[-1,1]$, where $s\in\mathbb{C}$ is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is $s\in i \mathbb{R}$, due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as $n\to\infty$ have been recently studied for $s\in i\mathbb{R}$, and our main goal is to extend these results to all $s$ in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so-called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann-Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter $s$ is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter $s$ approaches a breaking curve, by considering double scaling limits as $s$ approaches these points. We shall see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points $s=\pm 2$ or some other points on the breaking curve.