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

TL;DR

This paper investigates the spectral distribution of non-Hermitian random matrices with a variance profile, providing deterministic equivalents and analyzing the effects of sparsity and irreducibility.

Contribution

It introduces a framework for deterministic approximation of spectral distributions for variance-profile matrices, including sparse and reducible cases, using Master Equations and Schwinger--Dyson equations.

Findings

Established weak convergence of empirical spectral distributions to deterministic measures.

Developed bounds for solutions to Schwinger--Dyson equations in sparse matrix settings.

Extended analysis to matrices with vanishing entries under irreducibility conditions.

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