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