Isotropic local laws for sample covariance and generalized Wigner matrices

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Journal Article | Published | English

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Author
Bloemendal, Alex; Erdös, LászlóISTA ; Knowles, Antti; Yau, Horng; Yin, Jun
Department
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
Publisher
Institute of Mathematical Statistics
Volume
19
Article Number
33
ISSN
IST-REx-ID
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Creative Commons Attribution 4.0 International Public License (CC-BY 4.0):
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OA Open Access
Date Uploaded
2018-12-12
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