Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices

Abstract

We consider sample covariance matrices of the form $X^{\ast }X$, 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 $(X^{\ast }X-z{)}^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $⟨v,(X^{\ast }X-z{)}^{-1}w⟩-⟨v,w⟩m(z)$, 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 $Imz\ge N^{-1+\epsilon }$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.

References in corpus (2)

• Local semicircle law and complete delocalization for Wigner random matrices
• On the principal components of sample covariance matrices

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