Isotropic local laws for sample covariance and generalized Wigner matrices

A. Bloemendal, L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of Probability 19 (2014) 33.

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Journal Article | Published | English
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Abstract
We consider sample covariance matrices of the form X∗X, where X is an M×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 ⟨v,(X∗X−z)−1w⟩−⟨v,w⟩m(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v,w∈CN. We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Iz≥N−1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.
Publishing Year
Date Published
2014-03-15
Journal Title
Electronic Journal of Probability
Volume
19
Article Number
33
ISSN
IST-REx-ID

Cite this

Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. Isotropic local laws for sample covariance and generalized Wigner matrices. Electronic Journal of Probability. 2014;19:33. doi:10.1214/EJP.v19-3054
Bloemendal, A., Erdös, L., Knowles, A., Yau, H., & Yin, J. (2014). Isotropic local laws for sample covariance and generalized Wigner matrices. Electronic Journal of Probability, 19, 33. https://doi.org/10.1214/EJP.v19-3054
Bloemendal, Alex, László Erdös, Antti Knowles, Horng Yau, and Jun Yin. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” Electronic Journal of Probability 19 (2014): 33. https://doi.org/10.1214/EJP.v19-3054.
A. Bloemendal, L. Erdös, A. Knowles, H. Yau, and J. Yin, “Isotropic local laws for sample covariance and generalized Wigner matrices,” Electronic Journal of Probability, vol. 19, p. 33, 2014.
Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. 2014. Isotropic local laws for sample covariance and generalized Wigner matrices. Electronic Journal of Probability. 19, 33.
Bloemendal, Alex, et al. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” Electronic Journal of Probability, vol. 19, Institute of Mathematical Statistics, 2014, p. 33, doi:10.1214/EJP.v19-3054.
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