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


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.


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Ziliang Che. "Universality of random matrices with correlated entries." Electron. J. Probab. 22 1 - 38, 2017.


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

Primary: 15B52 , 60B20

Keywords: correlated random matrix , spetral statistics , Universality

Vol.22 • 2017
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