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

Abstract

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 \[ Y_n = \left(\frac1{\sqrt{n}} σ_{ij}X_{ij}\right). \] 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 ${\mu }_n^Y-{\mu }_n$ 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.

50 pages. The original arXiv submission has been split into two parts. This is the first part and was published in the Electronic Journal of Probability. The second part is titled: Non-Hermitian random matrices with a variance profile (II): properties and examples

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• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• Non-Hermitian random matrices with a variance profile (I): Deterministic equivalents and limiting ESDs

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