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

Abstract : For each n, let An = (σij) be an n × n deterministic matrix and let Xn = (Xij) be an n × n random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral distribution µ Y n of the rescaled entry-wise product Yn = 1 √ n σijXij. For our main result we provide a deterministic sequence of probability measures µn, each described by a family of Master Equations, such that the difference µ Y n − µn converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries σij to vanish, provided that the standard deviation profiles An 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. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2018, 23, pp.1 - 61. ⟨10.1214/18-EJP230⟩. ⟨hal-02385528⟩

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