Gernot Akemann: Complex normal, symmetric or self-dual random matrices: Approximation by a 2d Coulomb gas and analytic…

Abstract:

 It has been conjectured that among all 38 classes of non-Hermitian random matrices, only 3 different local bulk statistics exist. This conjecture is based on numerically generated nearest neighbor spacing distributions. The simplest representatives for these 3 bulk statistics are complex Ginibre matrices (class A), complex symmetric (class AI\(^\dagger\)), and complex self-dual random matrices (class AII\(^\dagger\)). While class A is very well understood as a determinantal point process, we are only beginning to explore the latter two.

First, based on numerics, I will show that both nearest and next-to-nearest neighbor spacing distributions of classes AI\(^\dagger\) and AII\(^\dagger\) can be well approximated by a 2-dimensional Coulomb gas at inverse temperature \(\beta =1.4\) and \(\beta=2.6\), respectively. For class A, this map is exact with \(\beta = 2\).

Second, I will present the first analytic results for the expectation value of two characteristic polynomials in classes AI\(^\dagger\) and AII\(^\dagger\). This includes results at finite matrix size as well as global and local edge and bulk asymptotics.