# Location of the spectrum of Kronecker random matrices

## TL;DR

This paper characterizes the eigenvalue distribution of large non-Hermitian Kronecker random matrices, showing they concentrate near a deterministic set derived from the Dyson equation, unifying analysis across various structured ensembles.

## Contribution

It provides a rigorous description of the spectral location for a broad class of non-Hermitian random matrices using Dyson equation techniques.

## Key findings

- Eigenvalues concentrate near a deterministic set
- The set is derived from the Dyson equation of the matrix's hermitization
- Analysis applies to many structured matrix ensembles

## 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.