We consider $N\times N$ Hermitian or symmetric random matrices with independent entries. The distribution of the $(i,j)$-th matrix element is given by a probability measure $\nu_{ij}$ whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution $\nu_{ij}$ coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector-eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.

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.