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2018 Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs
Nicholas Cook, Walid Hachem, Jamal Najim, David Renfrew
Electron. J. Probab. 23: 1-61 (2018). DOI: 10.1214/18-EJP230

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 \[ Y_n = \left (\frac 1{\sqrt{n} } \sigma _{ij}X_{ij}\right ). \] 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.

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Nicholas Cook. Walid Hachem. Jamal Najim. David Renfrew. "Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs." Electron. J. Probab. 23 1 - 61, 2018. https://doi.org/10.1214/18-EJP230

Information

Received: 2 March 2018; Accepted: 3 October 2018; Published: 2018
First available in Project Euclid: 30 October 2018

zbMATH: 1401.60008
MathSciNet: MR3878135
Digital Object Identifier: 10.1214/18-EJP230

Subjects:
Primary: 15B52
Secondary: 15A18‎, 60B20

JOURNAL ARTICLE
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Vol.23 • 2018
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