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

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).