Isotropic local laws for sample covariance and generalized Wigner matrices

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 z ≥ N-1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.