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2014 Isotropic local laws for sample covariance and generalized Wigner matrices
Bloemendal Alex, László Erdős, Antti Knowles, Horng-Tzer Yau, Jun Yin
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Electron. J. Probab. 19: 1-53 (2014). DOI: 10.1214/EJP.v19-3054


We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $\langle v , (X^* X - z)^{-1}w\rangle - \langle v , w\rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v , w \in \mathbb{C}^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $\Im z \geq N^{-1+\varepsilon}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.


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Bloemendal Alex. László Erdős. Antti Knowles. Horng-Tzer Yau. Jun Yin. "Isotropic local laws for sample covariance and generalized Wigner matrices." Electron. J. Probab. 19 1 - 53, 2014.


Accepted: 15 March 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1288.15044
MathSciNet: MR3183577
Digital Object Identifier: 10.1214/EJP.v19-3054

Vol.19 • 2014
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