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

TL;DR

This paper derives asymptotic formulas for the reproducing kernel in weighted polynomial spaces, revealing boundary correlation decay in random normal matrix models, especially at the edge of the droplet.

Contribution

It provides a Szegő-type asymptotic formula for the reproducing kernel at the boundary of the droplet in weighted polynomial spaces, connecting to random matrix theory.

Findings

Asymptotic formula for the kernel at the edge of the droplet.

Description of slow decay of correlations at the boundary.

Connection to the Szegő kernel and Hardy space analysis.

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.