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

For each $n$, let $A_n=(σ_{ij})$ 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 $μ_n^Y$ of the rescaled entry-wise product $Y_n = \left(\frac1{\sqrt{n}} σ_{ij}X_{ij}\right).$ For our main result we provide a deterministic sequence of probability measures $μ_n$, each described by a family of Master Equations, such that the difference $μ^Y_n - μ_n$ converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries $σ_{ij}$ 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.

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