Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials

Abstract

Consider the subspace {{{mathscr {W}}}_{n}} of L^2({{mathbb {C}}},dA) consisting of all weighted polynomials W(z)=P(z)cdot e^{-frac{1}{2}nQ(z)}, where P(z) is a holomorphic polynomial of degree at most n-1, Q(z)=Q(z,{bar{z}}) is a fixed, real-valued function called the “external potential”, and dA=tfrac{1}{2pi i}, d{bar{z}}wedge dz is normalized Lebesgue measure in the complex plane {{mathbb {C}}}. We study large n asymptotics for the reproducing kernel K_n(z,w) of {{mathscr {W}}}_n; this depends crucially on the position of the points z and w relative to the droplet S, i.e., the support of Frostman’s equilibrium measure in external potential Q. We mainly focus on the case when both z and w are in or near the component U of hat{{{mathbb {C}}}}setminus S containing infty , leaving aside such cases which are at this point well-understood. For the Ginibre kernel, corresponding to Q=|z|^2, we find an asymptotic formula after examination of classical work due to G. Szegő. Properly interpreted, the formula turns out to generalize to a large class of potentials Q(z); this is what we call “Szegő type asymptotics”. Our derivation in the general case uses the theory of approximate full-plane orthogonal polynomials instigated by Hedenmalm and Wennman, but with nontrivial additions, notably a technique involving “tail-kernel approximation” and summing by parts. In the off-diagonal case zne w when both z and w are on the boundary {partial }U, we obtain that up to unimportant factors (cocycles) the correlations obey the asymptotic Kn(z,w)∼2πnΔQ(z)14ΔQ(w)14S(z,w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} K_n(z,w)\\sim \\sqrt{2\\pi n}\\,\\Delta Q(z)^{\\frac{1}{4}}\\,\\Delta Q(w)^{\\frac{1}{4}}\\,S(z,w) \\end{aligned}$$\\end{document}where S(z, w) is the Szegő kernel, i.e., the reproducing kernel for the Hardy space H^2_0(U) of analytic functions on U vanishing at infinity, equipped with the norm of L^2({partial }U,|dz|). Among other things, this gives a rigorous description of the slow decay of correlations at the boundary, which was predicted by Forrester and Jancovici in 1996, in the context of elliptic Ginibre ensembles.