Random matrices: Universal properties of eigenvectors

Understand

The four moment theorem asserts, roughly speaking, that the joint distribution of a small number of eigenvalues of a Wigner random matrix (when measured at the scale of the mean eigenvalue spacing) depends only on the first four moments of the entries of the matrix.

• In this paper, we extend the four moment theorem to also cover the coefficients of the \emph{eigenvectors} of a Wigner random matrix.
• A similar result (with different hypotheses) has been proved recently by Knowles and Yin, using a different method.
• As an application, we prove some central limit theorems for these eigenvectors.