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

Consider the subspace W_n of L²(C, dA) consisting of all weighted polynomials W(z)=P(z)·e-12nQ(z), where P(z) is a holomorphic polynomial of degree at most n- 1 , Q(z) = Q(z, z¯) is a fixed, real-valued function called the “external potential”, and dA=12πidz¯∧dz is normalized Lebesgue measure in the complex plane C. We study large n asymptotics for the reproducing kernel K_n(z, w) of 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 C^ \ S containing ∞, 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 z≠ w when both z and w are on the boundary ∂U, we obtain that up to unimportant factors (cocycles) the correlations obey the asymptotic Kn(z,w)∼2πnΔQ(z)14ΔQ(w)14S(z,w)where S(z, w) is the Szegő kernel, i.e., the reproducing kernel for the Hardy space H02(U) of analytic functions on U vanishing at infinity, equipped with the norm of L²(∂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.