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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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