Translator Disclaimer
2017 Universality of random matrices with correlated entries
Ziliang Che
Electron. J. Probab. 22: 1-38 (2017). DOI: 10.1214/17-EJP46

Abstract

We consider an $N$ by $N$ real symmetric random matrix $X=(x_{ij})$ where $\mathbb{E} [x_{ij}x_{kl}]=\xi _{ijkl}$. Under the assumption that $(\xi _{ijkl})$ is the discretization of a piecewise Lipschitz function and that the correlation is short-ranged we prove that the empirical spectral measure of $X$ converges to a probability measure. The Stieltjes transform of the limiting measure can be obtained by solving a functional equation. Under the slightly stronger assumption that $(x_{ij})$ has a strictly positive definite covariance matrix, we prove a local law for the empirical measure down to the optimal scale $\operatorname{Im} z \gtrsim N^{-1}$. The local law implies delocalization of eigenvectors. As another consequence we prove that the eigenvalue statistics in the bulk agrees with that of the GOE.

Citation

Download Citation

Ziliang Che. "Universality of random matrices with correlated entries." Electron. J. Probab. 22 1 - 38, 2017. https://doi.org/10.1214/17-EJP46

Information

Received: 11 June 2016; Accepted: 7 March 2017; Published: 2017
First available in Project Euclid: 24 March 2017

zbMATH: 1361.60008
MathSciNet: MR3629874
Digital Object Identifier: 10.1214/17-EJP46

Subjects:
Primary: 15B52, 60B20

JOURNAL ARTICLE
38 PAGES


SHARE
Vol.22 • 2017
Back to Top