A graph limit approach to the eigenvectors of non-symmetric random sign matrices

Random matrix theory is a quickly evolving area of probability theory, which has interesting connections to other fields of mathematics and physics as well. In this work we consider non-symmetric random matrices with independent, zero mean, plus-minus $n^{-1/2}$ entries. It is well known that the eigenvalues of such a matrix are distributed approximately uniformly on the unit disk of the complex plane; however, less is known about the eigenvectors. In the talk we present a result stating that the empirical distribution of the delocalized eigenvectors are in some sense close to a Gaussian distribution. We will summarize the concentration result for random matrices with respect to a metric coming from graph limit theory, which is an important element in the proofs - together with other tools from probability theory and information theory.

Joint work with Balázs Szegedy.