Universality for fluctuations of counting statistics of random normal matrices

Abstract We consider the fluctuations of the number of eigenvalues of random normal matrices depending on a potential in a given set . 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 . When is a Borel set strictly inside the droplet, we show that the variance of the number of eigenvalues in has a limiting behavior given by where is the measure theoretic boundary of , denotes the one‐dimensional Hausdorff measure, and . We also consider the case where is a microscopic dilation of the droplet and fully generalize a result by Akemann, Byun, and Ebke for arbitrary potentials. In this result is replaced by the harmonic measure at 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.