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 the support of the asymptotic eigenvalue distribution in the complex plane exactly coincides with the ε-pseudospectrum in the consecutive limits n → ∞ and ε → 0. Furthermore, we provide a description of this support in terms of a single real-valued function on the complex plane. As a level set of this locally real analytic function, the spectral edge is a real analytic variety of dimension at most one.