Normality of smooth statistics for planar determinantal point processes

TL;DR

This paper establishes the asymptotic normality of smooth linear statistics for planar determinantal point processes under broad conditions, extending previous results to cases lacking analyticity and radial symmetry.

Contribution

It proves asymptotic normality for smooth linear statistics of determinantal point processes without relying on analyticity, using a streamlined proof based on the reproducing kernel property.

Findings

Asymptotic normality holds in finite variance cases.

Reproducing kernel property replaces analyticity assumptions.

Method extends to broader classes of determinantal processes.

Abstract

We consider smooth linear statistics of determinantal point processes on the complex plane, and their large scale asymptotics. We prove asymptotic normality in the finite variance case, where Soshnikov's theorem is not applicable. The setting is similar to that of Rider and Vir\'ag [Electron. J. Probab., 12, no. 45, 1238--1257, (2007)] for the complex plane, but replaces analyticity conditions by the assumption that the correlation kernel is reproducing. Our proof is a streamlined version of that of Ameur, Hedenmalm and Makarov [Duke Math J., 159, 31--81, (2011)] for eigenvalues of normal random matrices. In our case, the reproducing property is brought to bear to compensate for the lack of analyticity and radial symmetries.