Leslie Molag : Fluctuations of random normal matrices in various regimes of non-Hermiticity

Speaker
Leslie Molag , University of Madrid

Abstract
We study linear statistics associated to the eigenvalues of random normal matrices in various regimes of non-Hermiticity. Ameur, Hedenmalm, and Makarov (2015) proved that, for smooth test functions, the linear statistics satisfy a Central Limit Theorem. However, less is known for test functions that are indicator functions of some set $A$, and most known results require radial symmetry. When $A$ is strictly inside the droplet, we prove in generality that the variance is of order $\sqrt n |\partial A|$. We explain how this implies a holographic principle.
The remainder of the talk focuses on the elliptic Ginibre ensemble with non-Hermiticity parameter $\tau=1-\kappa n^{-\alpha}$, where $0<\alpha<1$. We demonstrate that the corresponding rescaled smooth linear statistics satisfy a Central Limit Theorem that interpolates between dimensions 1 and 2, and we provide an exact formula for the limiting variance. This result has an interpretation in terms of log-correlated fields. Our approach is an adaptation of the method of Ward identities, well-known in theoretical physics, applied to Hermitian random matrices by Johansson (1998) and later extended to random normal matrices by Ameur, Hedenmalm, and Makarov (2015).