Delocalization of eigenvectors of random matrices with independent entries

Understand

We prove that an n by n random matrix G with independent entries is completely delocalized.

• Suppose 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(n^{-1/2}), modulo logarithmic corrections.
• This comes a consequence of a new, geometric, approach to delocalization for random matrices.