Universality for fluctuations of counting statistics of random normal matrices

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 $A$. The eigenvalues of random normal matrices 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 $$ \lim _{n \rightarrow \infty} \frac{1}{\sqrt{n}} \operatorname{Var} N_A^{(n)}=\frac{1}{2 \pi \sqrt{\pi}} \int_{\partial_* A} \sqrt{\Delta Q(z)} d \mathcal{H}^1(z), $$ where $\partial_* A$ is the measure theoretic boundary of $A$, $d H^1(z)$ 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 $d \boldsymbol{H}^1(z)$ 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.