Eigenvector distribution of Wigner matrices

Antti Knowles Harvard University Jun Yin Harvard University

Probability mathscidoc:1608.28015

Probability Theory & Related Fields, 155, (3), 543-582, 2011
We consider $N\times N$ Hermitian or symmetric random matrices with independent entries. The distribution of the $(i,j)$-th matrix element is given by a probability measure $\nu_{ij}$ whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution $\nu_{ij}$ coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector-eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.
random matrix, universality, eigenvector distribution
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  title={Eigenvector distribution of Wigner matrices},
  author={Antti Knowles, and Jun Yin},
  booktitle={Probability Theory & Related Fields},
Antti Knowles, and Jun Yin. Eigenvector distribution of Wigner matrices. 2011. Vol. 155. In Probability Theory & Related Fields. pp.543-582. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823162731354228412.
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