Göttingen 2025 – wissenschaftliches Programm

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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 4: Dynamics and Chaotic Behaviour

MP 4.3: Vortrag

Mittwoch, 2. April 2025, 11:50–12:10, ZHG001

Complex symmetric, self-dual, and Ginibre random matrices: Analytical results for three classes of bulk and edge statistics — •Noah Ayguen — Bielefeld University, Bielefeld, Germany

The energy eigenvalues of chaotic quantum systems are expected to follow random matrix statistics, where closed systems relate to Hermitian random matrices while open systems with complex eigenvalues relate to non-Hermitian matrices. The random matrix model depends on the corresponding symmetry class of the physical systems under consideration. Recently, based on numerics, it has been conjectured that among such classes of non-Hermitian random matrices only three different local bulk statistics of complex eigenvalues exist. Motivated by these new insights, we find new analytic results for expectation values of characteristic polynomials, using the technique of Grassmann variables. The simplest representatives of these 3 bulk statistics are the Gaussian ensembles of well-known complex Ginibre matrices, complex symmetric, and complex self-dual random matrices. In the Cartan classification scheme of non-Hermitian random matrices they are labelled as class A, AI^† and AII^†, respectively. (Based on joint work with G. Akemann, M. Kieburg, P. Päßler arXiv:2410.21032)

Keywords: Random matrix theory; Universality; Open quantum systems