Universality for a global property of the eigenvectors of Wigner matrices
Date of Issue2014
School of Physical and Mathematical Sciences
Let M n be an n×n real (resp. complex) Wigner matrix and UnΛnU∗n be its spectral decomposition. Set (y1,y2⋯,yn)T=U∗nx , where x = (x 1, x 2, ⋅⋅⋅, x n ) T is a real (resp. complex) unit vector. Under the assumption that the elements of M n have 4 matching moments with those of GOE (resp. GUE), we show that the process Xn(t)=βn2−−−√∑⌊nt⌋i=1(|yi∣∣2−1n) converges weakly to the Brownian bridge for any x satisfying ‖x‖∞ → 0 as n → ∞, where β = 1 for the real case and β = 2 for the complex case. Such a result indicates that the orthogonal (resp. unitary) matrices with columns being the eigenvectors of Wigner matrices are asymptotically Haar distributed on the orthogonal (resp. unitary) group from a certain perspective.
Journal of mathematical physics
© 2014 AIP Publishing LLC. This paper was published in Journal of Mathematical Physics and is made available as an electronic reprint (preprint) with permission of AIP Publishing LLC. The paper can be found at the following official DOI: http://dx.doi.org/10.1063/1.4864735. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.