Random matrices: Universal properties of eigenvectors
arXiv:1103.2801
Abstract
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.
25 pages, no figures, to appear, Random Matrices: Theory and applications. This is the final version, incorporating the referee's suggestions
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- Eigenvector Distribution of Wigner Matrices
- Convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix
- Eigenvectors of Wigner matrices: universality of global fluctuations
- Euclidean distance between Haar orthogonal and gaussian matrices
- On Finite Rank Deformations of Wigner Matrices II: Delocalized Perturbations