Isotropic local laws for sample covariance and generalized Wigner matrices

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Isotropic local laws for sample covariance and generalized Wigner matrices

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Title: Isotropic local laws for sample covariance and generalized Wigner matrices
Author: Alex, Bloemendal; Erdos, Laszlo; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun

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Citation: Alex, Bloemendal, László Erdős, Antti Knowles, Horng-Tzer Yau, and Jun Yin. 2014. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” Electronic Journal of Probability 19 (0). doi:10.1214/ejp.v19-3054.
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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 MarchenkoPastur 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 hv,(X ∗X − z) −1wi − hv, wim(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v, w ∈ C N . We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Im z > N −1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.
Published Version: doi:10.1214/EJP.v19-3054
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:34253795
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