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 ∈ C is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is s ∈ iR, due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as n → ∞ have been recently studied for s ∈ iR, 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 = ±2 or some other points on the breaking curve. Contents 3.1. The Riemann-Hilbert Problem and The Modified External Field 3.2. Overview of Deift-Zhou Nonlinear Steepest Descent 3.3. Small Norm Riemann-Hilbert Problems 3.4. Unwinding the Transformations 3.5. The Global Parametrix 3.6. The Local Parametrices 4. The Global Phase Portrait -Continuation in Parameter Space 4.1. Breaking Curves 4.2. Quadratic Differentials 4.3. The Genus 0 and 1 h-functions 4.4. Proof of Theorem 2.1 4.5. Proof of Theorem 2.2 4.6. Proof of Theorem 2.4 5. Double Scaling Limit near Regular Breaking Points 5.1. Definition of the Double Scaling Limit 5.2. Opening of the Lenses 5.3. Parametrix around the Critical Point 5.4. Proof of Theorem 2.5 6. Double Scaling Limit near a Critical Breaking Point 6.1. Outline of Steepest Descent 6.2. Local parametrix at z = 1. 6.3. Proof of Theorem 2.6 References Key words and phrases. Orthogonal polynomials in the complex plane; Riemann-Hilbert problem; Continuation in parameter space; asymptotic analysis.