Exact spectral densities of complex noise-plus-structure random matrices

TL;DR

This paper derives exact spectral densities for complex structured random matrices using supersymmetry, revealing how normality conditions affect eigenvalue distributions.

Contribution

It provides two-fold integral formulas for spectral densities of structured random matrices, a novel analytical approach in the external field model context.

Findings

Spectral densities depend on the normality of the structure matrix S.

Exact formulas are obtained for arbitrary structural matrices.

Numerical simulations confirm theoretical results.

Abstract

We use supersymmetry to calculate exact spectral densities for a class of complex random matrix models having the form $M=S+LXR$, where $X$ is a random noise part $X$ and $S,L,R$ are fixed structure parts. This is a certain version of the "external field" random matrix models. We found two-fold integral formulas for arbitrary structural matrices. We investigate some special cases in detail and carry out numerical simulations. The presence or absence of a normality condition on $S$ leads to a qualitatively different behavior of the eigenvalue densities.