We prove that an random matrix with independent entries is completely delocalized. Suppose that the entries of have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high probability all unit eigenvectors of have all coordinates of magnitude , modulo logarithmic corrections. This comes as a consequence of a new, geometric approach to delocalization for random matrices.
"Delocalization of eigenvectors of random matrices with independent entries." Duke Math. J. 164 (13) 2507 - 2538, 1 October 2015. https://doi.org/10.1215/00127094-3129809