We consider Hermitian and symmetric random band matrices $H = (h_{xy})$ in $d \geq 1$ dimensions. The matrix entries $h_{xy}$, indexed by $x,y \in (\bZ/L\bZ)^d$, are independent, centred random variables with variances $s_{xy} = \E |h_{xy}|^2$. We assume that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In one dimension we prove that the eigenvectors of $H$ are delocalized if $W\gg L^{4/5}$. We also show that the magnitude of the matrix entries $\abs{G_{xy}}^2$ of the resolvent $G=G(z)=(H-z)^{-1}$ is self-averaging and we compute $\E \abs{G_{xy}}^2$. We show that, as $L\to\infty$ and $W\gg L^{4/5}$, the behaviour of $\E |G_{xy}|^2$ is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.

We analyse the spectrum of additive finite-rank deformations of $N \times N$ Wigner matrices $H$. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue $d_i$ of the deformation crosses a critical value $\pm 1$. This transition happens on the scale $|d_i| - 1 \sim N^{-1/3}$. We allow the eigenvalues $d_i$ of the deformation to depend on $N$ under the condition $|\abs{d_i} - 1| \geq (\log N)^{C \log \log N} N^{-1/3}$. We make no assumptions on the eigenvectors of the deformation. In the limit $N \to \infty$, we identify the law of the outliers and prove that the non-outliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a Gaussian matrix ensemble. A key ingredient in our proof is the \emph{isotropic local semicircle law}, which establishes optimal high-probability bounds on the quantity $< v,[(H - z)^{-1} - m(z) 1] w >$, where $m(z)$ is the Stieltjes transform of Wigner's semicircle law and $v, w$ are arbitrary deterministic vectors.

In this paper we prove the universality of covariance matrices of the form $H_{Ntimes N}={X}^{dagger}X$ where $X$ is an ${Mtimes N}$ rectangular matrix with independent real valued entries $x_{ij}$ satisfying $mathbb{E}x_{ij}=0$ and $mathbb{E}x^2_{ij}={frac{1}{M}}$, $N$, $Mto infty$. Furthermore it is assumed that these entries have sub-exponential tails or sufficiently high number of moments. We will study the asymptotics in the regime $N/M=d_Nin(0,infty),lim_{Ntoinfty}d_Nneq0,infty$. Our main result is the edge universality of the sample covariance matrix at both edges of the spectrum. In the case $lim_{Ntoinfty}d_N=1$, we only focus on the largest eigenvalue. Our proof is based on a novel version of the Green function comparison theorem for data matrices with dependent entries. En route to proving edge universality, we establish that the Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by the Marcenko-Pastur law uniformly up to the edges of the spectrum with an error of order $(Neta)^{-1}$ where $eta$ is the imaginary part of the spectral parameter in the Stieltjes transform. Combining these results with existing techniques we also show bulk universality of covariance matrices. All our results hold for both real and complex valued entries.

The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point z away from the unit circle. More precisely, if ||z|−1|≥τ for arbitrarily small τ>0, the circular law is valid around z up to scale $N^{-1/2+ \e}$ for any $\e > 0$ under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.

In the first part of this article, we proved a local version of the circular law up to the finest scale $N^{-1/2+ \e}$ for non-Hermitian random matrices at any point $z \in \C$ with ||z|−1|>c for any c>0 independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case $|z|-1=\oo(1)$. Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge $|z|-1=\oo(1)$ up to scale $N^{-1/4+ \e}$.