Universality for a global property of the eigenvectors of Wigner matrices

• Main Authors: Bao, Zhigang, Pan, Guangming, Zhou, Wang
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2014
• Subjects: DRNTU::Science::Mathematics
• Online Access: https://hdl.handle.net/10356/103709; http://hdl.handle.net/10220/20034
• Institution: Nanyang Technological University

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.