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For each $n$, let $A_n=(\sigma_{ij})$ be an $n\times n$ deterministic matrixand let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centeredentries of unit variance. We study the asymptotic behavior of the empiricalspectral distribution $\mu_n^Y$ of the rescaled entry-wise product \[ Y_n =\left(\frac1{\sqrt{n}} \sigma_{ij}X_{ij}\right). \] For our main result weprovide a deterministic sequence of probability measures $\mu_n$, eachdescribed 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 ofour results is to allow some of the entries $\sigma_{ij}$ to vanish, providedthat the standard deviation profiles $A_n$ satisfy a certain quantitativeirreducibility property. An important step is to obtain quantitative bounds onthe solutions to an associate system of Schwinger--Dyson equations, which weaccomplish in the general sparse setting using a novel graphical bootstrapargument.

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