Universality for fluctuations of counting statistics of random normal matrices

J. Marzo; L. D. Molag; J. Ortega-Cerdà

Paper

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

. 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

. When

is a Borel set strictly inside the droplet, we show that the variance of the number of eigenvalues

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

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.