Orthogonal Polynomials for a Class of Measures with Discrete Rotational Symmetries in the Complex Plane

Abstract

We obtain the strong asymptotics of polynomials p_n(lambda ), lambda in {mathbb {C}}, orthogonal with respect to measures in the complex plane of the form e-N(|λ|2s-tλs-t¯λ¯s)dA(λ),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\hbox {e}^{-N(|\\lambda |^{2s}-t\\lambda ^s-\\overline{t}\\overline{\\lambda }^s)}\\hbox {d}A(\\lambda ), \\end{aligned}$$\\end{document}where s is a positive integer, t is a complex parameter, and hbox {d}A stands for the area measure in the plane. This problem has its origin in normal matrix models. We study the asymptotic behavior of p_n(lambda ) in the limit n,Nrightarrow infty in such a way that n/Nrightarrow T constant. Such asymptotic behavior has two distinguished regimes according to the topology of the limiting support of the eigenvalues distribution of the normal matrix model. If 0<|t|^2<T/s, the eigenvalue distribution support is a simply connected compact set of the complex plane, while for |t|^2>T/s, the eigenvalue distribution support consists of s connected components. Correspondingly, the support of the limiting zero distribution of the orthogonal polynomials consists of a closed contour contained in each connected component. Our asymptotic analysis is obtained by reducing the planar orthogonality conditions of the polynomials to equivalent contour integral orthogonality conditions. The strong asymptotics for the orthogonal polynomials is obtained from the corresponding Riemann–Hilbert problem by the Deift–Zhou nonlinear steepest descent method.