We develop a new method for deriving local laws for a large class of random matrices. It is applicable to many matrix models built from sums and products of deterministic or independent random matrices. In particular, it may be used to obtain local laws for matrix ensembles that are \emph{anisotropic} in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity. For definiteness, we present the method for sample covariance matrices of the form Q:=TXX∗T∗, where T is deterministic and X is random with independent entries. We prove that with high probability the resolvent of Q is close to a deterministic matrix, with an optimal error bound and down to optimal spectral scales. As an application, we prove the edge universality of Q by establishing the Tracy-Widom-Airy statistics of the eigenvalues of Q near the soft edges. This result applies in the single-cut and multi-cut cases. Further applications include the distribution of the eigenvectors and an analysis of the outliers and BBP-type phase transitions in finite-rank deformations; they will appear elsewhere. We also apply our method to Wigner matrices whose entries have arbitrary expectation, i.e. we consider W+A where W is a Wigner matrix and a Hermitian deterministic matrix. We prove the anisotropic local law for W+A and use it to establish edge universality.

We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics

We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, $W \sim N$. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.

It is well known that the spectral measure of eigenvalues of a rescaled square non-Hermitian random matrix with independent entries satises the circular law. We consider the product $T X$, where $T$ is a deterministic $N\times M$ matrix and $X$ is a random $M\times  N$ matrix with independent entries having zero mean and variance $(N \wedge M) ^{-1}$. We prove a general local circular law for the empirical spectral distribution (ESD) of $TX$ at any point $z$ away from the unit circle under the assumptions that$ N \sim M$, and the matrix entries $X_{ij} have suciently high moments. More precisely, if $z$ satisfes $| |z|
- 1| \ge \tau$  for arbitrarily small $\tau > 0, the ESD of $TX$ converges to $\tilde{X}_D (z) d A(z)$, where $\tilde{X}_D$ is a rotation-invariant function determined by the singular valuau of $T$ and $dA$ denotes the Lebesgue measure on $\mathbb{C}$. The local circular law is valid around $z$ up to scale $(N \wedge M) ^{-1/4+\epsilon}$ for any  $\epsilon >0$. Moreover, if $|z| >1$ or the matrix entries of $X$ have vanishing third moments, the local circular law is valid around $z$ up to scale $(N \wedge M) ^{-1/2+\epsilon}$ for any  $\epsilon >0$.