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Isotropic local laws for sample covariance and generalized Wigner matrices. (English) Zbl 1288.15044
Summary: 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 $$\operatorname{Im} z \geq N^{-1+\varepsilon}$$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.

##### MSC:
 15B52 Random matrices (algebraic aspects) 60B20 Random matrices (probabilistic aspects) 15A18 Eigenvalues, singular values, and eigenvectors
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