Universality for a global property of the eigenvectors of Wigner matrices
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 sho...
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| Format: | Journal Article |
| Language: | English |
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2014
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| Online Access: | https://hdl.handle.net/10356/103709 http://hdl.handle.net/10220/20034 |
| _version_ | 1811680738822586368 |
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| author | Bao, Zhigang Pan, Guangming Zhou, Wang |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Bao, Zhigang Pan, Guangming Zhou, Wang |
| author_sort | Bao, Zhigang |
| collection | NTU |
| description | 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. |
| first_indexed | 2024-10-01T03:29:50Z |
| format | Journal Article |
| id | ntu-10356/103709 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T03:29:50Z |
| publishDate | 2014 |
| record_format | dspace |
| spelling | ntu-10356/1037092023-02-28T19:30:05Z Universality for a global property of the eigenvectors of Wigner matrices Bao, Zhigang Pan, Guangming Zhou, Wang School of Physical and Mathematical Sciences DRNTU::Science::Mathematics 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. Published version 2014-07-03T03:10:26Z 2019-12-06T21:18:30Z 2014-07-03T03:10:26Z 2019-12-06T21:18:30Z 2014 2014 Journal Article Bao, Z., Pan, G., & Zhou, W. (2014). Universality for a global property of the eigenvectors of Wigner matrices. Journal of Mathematical Physics, 55(2), 023303-. 0022-2488 https://hdl.handle.net/10356/103709 http://hdl.handle.net/10220/20034 10.1063/1.4864735 en 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. application/pdf |
| spellingShingle | DRNTU::Science::Mathematics Bao, Zhigang Pan, Guangming Zhou, Wang Universality for a global property of the eigenvectors of Wigner matrices |
| title | Universality for a global property of the eigenvectors of Wigner matrices |
| title_full | Universality for a global property of the eigenvectors of Wigner matrices |
| title_fullStr | Universality for a global property of the eigenvectors of Wigner matrices |
| title_full_unstemmed | Universality for a global property of the eigenvectors of Wigner matrices |
| title_short | Universality for a global property of the eigenvectors of Wigner matrices |
| title_sort | universality for a global property of the eigenvectors of wigner matrices |
| topic | DRNTU::Science::Mathematics |
| url | https://hdl.handle.net/10356/103709 http://hdl.handle.net/10220/20034 |
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