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

Abstract

AbstractConsider the subspace $${{{\mathscr {W}}}_{n}}$$ W n of $$L^2({{\mathbb {C}}},dA)$$ L 2 ( C , d A ) consisting of all weighted polynomials $$W(z)=P(z)\cdot e^{-\frac{1}{2}nQ(z)},$$ W ( z ) = P ( z ) · e - 1 2 n Q ( z ) , where P(z) is a holomorphic polynomial of degree at most $$n-1$$ n - 1 , $$Q(z)=Q(z,{\bar{z}})$$ Q ( z ) = Q ( z , z ¯ ) is a fixed, real-valued function called the “external potential”, and $$dA=\tfrac{1}{2\pi i}\, d{\bar{z}}\wedge dz$$ d A = 1 2 π i d z ¯ ∧ d z is normalized Lebesgue measure in the complex plane $${{\mathbb {C}}}$$ C . We study large n asymptotics for the reproducing kernel $$K_n(z,w)$$ K n ( z , w ) of $${{\mathscr {W}}}_n$$ 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 $$\hat{{{\mathbb {C}}}}\setminus S$$ C ^ \ S containing $$\infty $$ ∞ , leaving aside such cases which are at this point well-understood. For the Ginibre kernel, corresponding to $$Q=|z|^2$$ 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 $$z\ne w$$ z ≠ w when both z and w are on the boundary $${\partial }U$$ ∂ U , we obtain that up to unimportant factors (cocycles) the correlations obey the asymptotic $$\begin{aligned} K_n(z,w)\sim \sqrt{2\pi n}\,\Delta Q(z)^{\frac{1}{4}}\,\Delta Q(w)^{\frac{1}{4}}\,S(z,w) \end{aligned}$$ K n ( z , w ) ∼ 2 π n Δ Q ( z ) 1 4 Δ Q ( w ) 1 4 S ( z , w ) where S(z, w) is the Szegő kernel, i.e., the reproducing kernel for the Hardy space $$H^2_0(U)$$ H 0 2 ( U ) of analytic functions on U vanishing at infinity, equipped with the norm of $$L^2({\partial }U,|dz|)$$ L 2 ( ∂ U , | d z | ) . 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.