Delocalization of eigenvectors of random matrices

Abstract.Heuristically, delocalization for a random matrix means that its normalized eigenvectors look like the vectors uniformly distributed over the unit sphere. This can be made precise in a number of different ways. We will consider two complimentary approaches to delocalization. For a matrix with independent entries, we show that with high probability, the largest coordinate of a normalized eigenvector is of the same order as for a uniform random unit vector. This means that the Euclidean norm of an eigenvector cannot be concentrated on a few coordinates.
 On the other hand, we show that with high probability, any sufficiently large set of coordinates of an eigenvector carries a non-negligible portion of its Euclidean norm. The latter result pertains to a large class of random matrices including the ones  with independent entries, symmetric, skew-symmetric matrices, as well as more general ensembles.
Joint work with Roman Vershynin.