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

We consider n × 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 ε-pseudospectrum in the consecutive limits n → ∞ and ε → 0. The limiting spectral measure is identified as the Brown measure of a deformed operator-valued circular element with the help of [6].