Delocalization of eigenvectors of random matrices with independent entries

TL;DR

This paper proves that under certain conditions, all eigenvectors of a large random matrix with independent entries are uniformly spread out, with each coordinate roughly of size 1/√n, using a new geometric approach.

Contribution

It introduces a novel geometric method to establish complete delocalization of eigenvectors in random matrices with independent entries.

Findings

Eigenvectors are uniformly delocalized with high probability.

Coordinates of eigenvectors are of order O(n^{-1/2}) with logarithmic corrections.

The approach applies to matrices with bounded variance and exponential tail decay.

Abstract

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.