Random matrices beyond Wigner-Dyson-Mehta

Random matrix theory (RMT) was created in the 1950's by the pioneering idea of E.

Wigner predicting that the distribution of the gaps between energy levels of complex quantum systems is universal in the sense that it is independent of the physical details of the model apart from its basic symmetry type.

He proposed to study eigenvalues of large matrices with random entries as the fundamental model for his new universal statistics, later coined Wigner-Dyson-Mehta (WDM) universality.

Mathematical research in RMT has largely been inspired by the WDM universality, specifically for Hermitian matrices in the mean-field regime.

In physics, however, Wigner's vision has always been viewed from a higher perspective: random matrices are routinely used to model complex Hamilton operators for various observables on all scales. This project explores how this broader physics interpretation of RMT can be rigorously justified.

We focus on three new directions.First, we study Hermitian models beyond the conventional universality problem for eigenvalues and, among others, establish that Gaussian fluctuations prevail for most other physically relevant quantities such as multi-point Green functions or generic observables. In turn, these results help establish WDM universality for new ensembles that have previously not been accessible.

Second, we develop the universality theory of non-Hermitian random matrices on all scales, leading to Gaussian Free Field on mesoscopic scales, and universality on microscopic scales.

Third, we apply rigorous RMT to several key problems in disordered quantum systems, such as scattering theory in quantum dots and wires, fluctuation of density of states detecting the Anderson metal-insulator transition and the Sachdev-Ye-Kitaev model of fermions with random interactions.The main impact of the project will be to establish the ubiquity of Gaussianity and to develop new mathematical tools to apply RMT to realistic physical models beyond WDM universality.