Random matrices: Universal properties of eigenvectors

Details Random matrices: Universal properties of eigenvectors Random matrices: Universal properties of eigenvectors

2011-03-14

arxiv.org/abs/1103.2801v2

Mathematics

Probability

25 pages, no figures, to appear, Random Matrices: Theory and applications. This is the final version, incorporating the refere

Scientific paper

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. In another application, we prove a universality result for the resolvent, up to the real axis. This implies universality of the inverse matrix.

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