Universality for a global property of the eigenvectors of Wigner matrices - Bao, Zhigang (2014)
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.