A localization-delocalization transition for nonhomogeneous random matrices

We consider $N\times N$ self-adjoint Gaussian random matrices defined by an arbitrary deterministic sparsity pattern with $d$ nonzero entries per row. We show that such random matrices exhibit a canonical localization-delocalization transition near the edge of the spectrum: when $d\gg\log N$ the random matrix possesses a delocalized approximate top eigenvector, while when $d\ll\log N$ any approximate top eigenvector is localized. The key feature of this phenomenon is that it is universal with respect to the sparsity pattern, in contrast to the delocalization properties of exact eigenvectors which are sensitive to the specific sparsity pattern of the random matrix.