Statistical properties of eigenvectors and eigenvalues of structured random matrices

We study the eigenvalues and the eigenvectors of $N\times N$ structured random matrices of the form $H = W\tilde{H}W+D$ with diagonal matrices $D$ and $W$ and $\tilde{H}$ from the Gaussian Unitary Ensemble. Using the supersymmetry technique we derive general asymptotic expressions for the density of states and the moments of the eigenvectors. We find that the eigenvectors remain ergodic under very general assumptions, but a degree of their ergodicity depends strongly on a particular choice of $W$ and $D$. For a special case of $D=0$ and random $W$, we show that the eigenvectors can become critical and are characterized by non-trivial fractal dimensions.