Universality for fluctuations of counting statistics of random normal matrices

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}{2π\sqrtπ}\int_{\partial_* A} \sqrt{ΔQ(z)} \, d\mathcal H^1(z), \end{align*} where ${\partial }_{\ast }A$ 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.

arXiv comment33 pages. Minor typos corrected

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