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

Consider the subspace Wn of L-2( C, dA) consisting of all weighted polynomials W(z) = P(z) center dot e(- 1/2) nQ(z), where P(z) is a holomorphic polynomial of degree at most n - 1, Q( z) = Q(z,(z) over bar) is a fixed, real-valued function called the "external potential", and dA = 1 2 pi i d (z) over bar boolean AND dz is normalized Lebesgue measure in the complex plane C. We study large n asymptotics for the reproducing kernel Kn(z, w) of Wn; 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) over cap \ S containing infinity, 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. Szego. Properly interpreted, the formula turns out to generalize to a large class of potentials Q(z); this is what we call "Szego 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 not equal w when both z and w are on the boundary partial derivative U, we obtain that up to unimportant factors (cocycles) the correlations obey the asymptotic Kn(z, w) similar to root 2 pi n Delta Q(z) (1/ 4) Delta Q( w) (1 /4) S(z, w) where S(z, w) is the Szegyo kernel, i.e., the reproducing kernel for the Hardy space H2 0 (U) of analytic functions onU vanishing at infinity, equipped with the norm of L2(partial derivative 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.