Szeg\H{o} type asymptotics for the reproducing kernel in spaces of full-plane weighted polynomials

Abstract

In this work we find and discuss an asymptotic formula, as $n\to\infty$, for the reproducing kernel $K_n(z,w)$ in spaces of full-plane weighted polynomials $W(z)=P(z)\cdot e^{-\frac 12nQ(z)},$ where $P(z)$ is a holomorphic polynomial of degree at most $n-1$ and $Q(z)$ is a fixed, real-valued function termed "external potential". The kernel $K_n$ corresponds precisely to the canonical correlation kernel in the theory of random normal matrices. As is well-known, the large $n$ behaviour of $K_n(z,w)$ must depend 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 $Q$. In the particular case when $z$ and $w$ are at the edge and $z\ne w$, we prove the formula $K_n(z,w)\sim\sqrt{2\pi n}\,\Delta Q(z)^{\frac 1 4}\Delta Q(w)^{\frac 14}\,S(z,w)$ where $S(z,w)$ is the Szeg\H{o} kernel associated with the Hardy space $H^2_0(U)$ of analytic functions on unbounded component $U$ of $\hat{\mathbb{C}}\setminus S$ which vanish at infinity. 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.