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

Abstract Consider the subspace $${{{\mathscr {W}}}_{n}}$$ W n of $$L^2({{\mathbb {C}}},dA)$$ L 2 ( C , d A ) consisting all weighted polynomials $$W(z)=P(z)\cdot e^{-\frac{1}{2}nQ(z)},$$ z = P · e - 1 Q where P ( z ) is a holomorphic polynomial degree at most $$n-1$$ , $$Q(z)=Q(z,{\bar{z}})$$ ¯ fixed, real-valued function called “external potential”, and $$dA=\tfrac{1}{2\pi i}\, d{\bar{z}}\wedge dz$$ ? i ? normalized Lebesgue measure in complex plane $${{\mathbb {C}}}$$ . We study large n asymptotics for reproducing kernel $$K_n(z,w)$$ K w $${{\mathscr {W}}}_n$$ ; this depends crucially on position points w relative to droplet S i.e., support Frostman’s equilibrium external potential Q mainly focus case when both are or near component U $$\hat{{{\mathbb {C}}}}\setminus S$$ ^ \ S containing $$\infty $$ ? leaving aside such cases which point well-understood. For Ginibre kernel, corresponding $$Q=|z|^2$$ | we find an asymptotic formula after examination classical work due G. Szeg?. Properly interpreted, turns out generalize class potentials ); what call “Szeg? type asymptotics”. Our derivation general uses theory approximate full-plane orthogonal instigated by Hedenmalm Wennman, but with nontrivial additions, notably technique involving “tail-kernel approximation” summing parts. In off-diagonal $$z\ne w$$ ? boundary $${\partial }U$$ ? U obtain that up unimportant factors (cocycles) correlations obey $$\begin{aligned} K_n(z,w)\sim \sqrt{2\pi n}\,\Delta Q(z)^{\frac{1}{4}}\,\Delta Q(w)^{\frac{1}{4}}\,S(z,w) \end{aligned}$$ ? ? 4 Szeg? Hardy space $$H^2_0(U)$$ H 0 analytic functions vanishing infinity, equipped norm $$L^2({\partial }U,|dz|)$$ Among other things, gives rigorous description slow decay boundary, was predicted Forrester Jancovici 1996, context elliptic ensembles.