Spectral properties of non-Hermitian random matrices and related questions

The project focuses on the study of spectral properties of non-Hermitian families of random matrices, i.e. of particular types of rectangular arrays of randomly chosen numbers. Random matrices naturally appear in probability and statistics, computer science, and many research areas of mathematical physics. Non-Hermitian random matrices have been utilized in such diverse fields as dissipative and stochastic processes, scattering chaotic systems, and nuclear physics, and have applications ranging from neural networks and computational algorithms to communication theory and ecosystem dynamics. The main aim of the project is to achieve a better understanding of global and local (individual) behavior of complex eigenvalues of different kinds of non-Hermitian random matrices as the size of matrices becomes large. The project will also provide research training opportunities for graduate students. Since E. Wigner's and F. Dyson's works, the local behavior of the eigenvalues of random matrices has been expected to be universal, i.e. independent of the distribution of matrix entries, and, to a large extent, of the structure of random matrices. This project will focus on three important classes of non-Hermitian random matrices with two-dimensional spectral distribution: anti-Hermitian deformations of Hermitian matrices, which play an important role in the scattering theory; fully non-Hermitian matrices with independent entries with or without deformation; and sparse non-Hermitian random matrices. Universal spectral properties of these non-Hermitian random matrices will be studied by extending and combining classical random matrix techniques with a new approach based on a rigorous application of supersymmetry techniques for classical (symmetric) random matrices with real eigenvalues. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.