Delocalization of eigenvectors of random matrices with independent entries

Abstract

We prove that an $n\times n$ random matrix $G$ with independent entries is completely delocalized. Suppose that the entries of $G$ have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high probability all unit eigenvectors of $G$ have all coordinates of magnitude $O\left(n^{-1/2}\right)$, modulo logarithmic corrections. This comes as a consequence of a new, geometric approach to delocalization for random matrices.