Skip to Main content Skip to Navigation
Journal articles

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.
Document type :
Journal articles
Complete list of metadatas

Cited literature [72 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-02385528
Contributor : Walid Hachem <>
Submitted on : Thursday, November 28, 2019 - 7:09:03 PM
Last modification on : Wednesday, February 26, 2020 - 7:06:07 PM
Long-term archiving on: : Saturday, February 29, 2020 - 8:07:40 PM

File

non_hermitian_varprofile.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

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⟩

Share

Metrics

Record views

79

Files downloads

162