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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.

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