Universality for fluctuations of counting statistics of random normal matrices

by Prof. Leslie Molag (Carlos III University of Madrid)

Thursday 9 Oct 2025, 14:00 → 15:00 Europe/Brussels

B/2nd floor-B.203 - Seminar room (Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve))

We consider the fluctuations of the number of eigenvalues of n×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 contained the droplet, we show that the variance of the number of eigenvalues in A has a limiting behavior given by an integral with respect to the one-dimensional Hausdorff measure over the measure theoretic boundary of A, with integrand depending on Q. 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 the Hausdorff measure is replaced by the harmonic measure at infinity 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.