Non-Hermitian random matrices with a variance profile (I): Deterministic equivalents and limiting ESDs

For each $n$, let $A_n=({\sigma }_{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral distribution ${\mu }_n^Y$ of the rescaled entry-wise product $$Y_n=\left(\frac{1}{\sqrt{n}}{\sigma }_{ij}{X}_{ij}\right)\mathrm{.}$$ For our main result we provide a deterministic sequence of probability measures ${\mu }_n$, each described by a family of Master Equations, such that the difference $\mu^Y_n - \mu_n$ converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries ${\sigma }_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. An important step is to obtain quantitative bounds on the solutions to an associate system of Schwinger–Dyson equations, which we accomplish in the general sparse setting using a novel graphical bootstrap argument.