Asymptotics of Orthogonal Polynomials with Complex Varying Quartic Weight: Global Structure, Critical Point Behavior and the First Painlevé Equation

Abstract

We study the asymptotics of recurrence coefficients for monic orthogonal polynomials \(\pi _n(z)\) with the quartic exponential weight \(\exp \left[-N\left(\frac{1}{2}z^2 + \frac{1}{4}t z^4\right)\right]\), where \(t\in {\mathbb C}\) and \(N\rightarrow \infty \). We consider in detail the points \(t = -\frac{1}{12}\) and \(t = \frac{1}{15}\), where the recurrence coefficients of the orthogonal polynomial exhibit a behavior that involves special solutions of the first Painleve Riemann–Hilbert problem (RHP). Our principal concern is the description of their behavior in a neighborhood of special points in the \(t\)-plane (accumulating at the indicated values) where the corresponding Painleve function has poles. The nonlinear steepest descent method for the RHP is the main technique used in the paper. We note that the RHP near the critical points is very similar to the RHP describing the semiclassical limit of the focusing nonlinear Schrodinger equation near the point of gradient catastrophe that the present authors solved in 2013. Our approach is based on the technique developed in that earlier work. We also provide a numerical investigation of the “phase diagrams” in the \(t\)-plane where the recurrence coefficients exhibit different asymptotic behaviors (nonlinear Stokes’ phenomenon).