Universality for fluctuations of counting statistics of random normal matrices

TL;DR

This paper studies the fluctuations in the number of eigenvalues of random normal matrices within a set, revealing universal limiting behaviors and extending previous results to more general potentials.

Contribution

It establishes a universal limit for the variance of eigenvalue counts and generalizes boundary fluctuation results for arbitrary potentials.

Findings

Variance of eigenvalue count scales as 1/√n with a specific limit involving the potential's Laplacian.

Derived a boundary fluctuation result involving harmonic measure for general potentials.

Extended previous results to microscopic dilations of the droplet and arbitrary potentials.

Abstract

We consider the fluctuations of the number of eigenvalues of $n\times n$ random normal matrices depending on a potential $Q$ in a given set $A$. These eigenvalues are known to form a determinantal point process, and are known to accumulate on a compact set called the droplet under mild conditions on $Q$. When $A$ is a Borel set strictly inside the droplet, we show that the variance of the number of eigenvalues $N_A^{(n)}$ in $A$ has a limiting behavior given by \begin{align*} \lim_{n\to\infty} \frac1{\sqrt n}\operatorname{Var } N_A^{(n)} = \frac{1}{2\pi\sqrt\pi}\int_{\partial_* A} \sqrt{\Delta Q(z)} \, d\mathcal H^1(z), \end{align*} where $\partial_* A$ is the measure theoretic boundary of $A$, $d\mathcal H^1(z)$ denotes the one-dimensional Hausdorff measure, and $\Delta = \partial_z \overline{\partial_z}$. We also consider the case where $A$ is a microscopic dilation of the droplet and fully generalize a result by Akemann, Byun and Ebke for arbitrary potentials. In this result $d\mathcal H^1(z)$ is replaced by the harmonic measure at $\infty$ associated with the exterior of the droplet. This second result is proved by strengthening results due to Hedenmalm-Wennman and Ameur-Cronvall on the asymptotic behavior of the associated correlation kernel near the droplet boundary.