Optimal delocalization for generalized Wigner matrices

We consider eigenvector statistics of large random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, it can be shown that extremal coordinates are no larger than $C\sqrt{\log N/N}$ with high probability.

There has been an extensive amount of work on generalizing such a result, known as delocalization, to a more general entry distribution. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdös–Yau, the analysis of high moments as well as new level repulsion estimates which will be presented during the talk.

This is based on a joint work with P. Lopatto.