Spectrum occupies pseudospectrum for random matrices with diagonal deformation and variance profile

Abstract: We consider $n\times n$ non-Hermitian random matrices with independent entries and a variance profile, as well as an additive deterministic diagonal deformation. We show that their empirical eigenvalue distribution converges to a limiting density as $n$ tends to infinity and that the support of this density in the complex plane exactly coincides with the $\varepsilon$-pseudospectrum in the consecutive limits $n \to \infty$ and $\varepsilon \to 0$. The limiting spectral measure is identified as the Brown measure of a deformed operator-valued circular element with the help of [arXiv:2409.15405].