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

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.

12 pages, 4 figures

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• Statistical properties of eigenvectors and eigenvalues of structured random matrices
• Statistical analysis of chiral structured ensembles: role of matrix constraints