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