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Title: Universality for a global property of the eigenvectors of Wigner matrices

Let M{sub n} be an n×n real (resp. complex) Wigner matrix and U{sub n}Λ{sub n}U{sub n}{sup *} be its spectral decomposition. Set (y{sub 1},y{sub 2}⋯,y{sub n}){sup T}=U{sub n}{sup *}x, where x = (x{sub 1}, x{sub 2}, ⋅⋅⋅, x{sub n}){sup T} is a real (resp. complex) unit vector. Under the assumption that the elements of M{sub n} have 4 matching moments with those of GOE (resp. GUE), we show that the process X{sub n}(t)=√((βn)/2 )∑{sub i=1}{sup ⌊nt⌋}(|y{sub i}|{sup 2}−1/n ) converges weakly to the Brownian bridge for any x satisfying ‖x‖{sub ∞} → 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.
Authors:
 [1] ;  [2] ;  [3]
  1. Department of Mathematics, Zhejiang University (China)
  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371 (Singapore)
  3. Department of Statistics and Applied Probability, National University of Singapore, Singapore 117546 (Singapore)
Publication Date:
OSTI Identifier:
22251512
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 2; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; EIGENVECTORS; MATRICES; VECTORS