Location of the spectrum of Kronecker random matrices

Abstract

For a general class of large non-Hermitian random block matrices $\mathbf{X}$ we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of $\mathbf{X}$ as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from [arXiv:1604.08188v4] offers a unified treatment of many structured matrix ensembles.

arXiv comment33 pages, 4 figures. Some assumptions in Section 3.1 and 3.2 relaxed. Some typos corrected and references updated