Central limit theorem for eigenvectors of heavy tailed matrices
We consider the eigenvectors of symmetric matrices with independent heavy tailed entries, such as matrices with entries in the domain of attraction of α-stable laws, or adjacencymatrices of Erdos-Renyi graphs. We denote by U=[uij] the eigenvectors matrix (corresponding to increasing eigenvalues) and...
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| Format: | Article |
| Language: | en_US |
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Institute of Mathematical Statistics
2014
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| Online Access: | http://hdl.handle.net/1721.1/89525 https://orcid.org/0000-0003-4524-8627 |
| _version_ | 1826199907698999296 |
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| author | Benaych-Georges, Florent Guionnet, Alice |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Benaych-Georges, Florent Guionnet, Alice |
| author_sort | Benaych-Georges, Florent |
| collection | MIT |
| description | We consider the eigenvectors of symmetric matrices with independent heavy tailed entries, such as matrices with entries in the domain of attraction of α-stable laws, or adjacencymatrices of Erdos-Renyi graphs. We denote by U=[uij] the eigenvectors matrix (corresponding to increasing eigenvalues) and prove that the bivariate process [formula] indexed by s,t∈[0,1], converges in law to a non trivial Gaussian process. An interesting part of this result is the n−1/2 rescaling, proving that from this point of view, the eigenvectors matrix U behaves more like a permutation matrix (as it was proved by Chapuy that for U a permutation matrix, n−1/2 is the right scaling) than like a Haar-distributed orthogonal or unitary matrix (as it was proved by Rouault and Donati-Martin that for U such a matrix, the right scaling is 1). |
| first_indexed | 2024-09-23T11:27:43Z |
| format | Article |
| id | mit-1721.1/89525 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:27:43Z |
| publishDate | 2014 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/895252022-10-01T03:47:01Z Central limit theorem for eigenvectors of heavy tailed matrices Benaych-Georges, Florent Guionnet, Alice Massachusetts Institute of Technology. Department of Mathematics Guionnet, Alice We consider the eigenvectors of symmetric matrices with independent heavy tailed entries, such as matrices with entries in the domain of attraction of α-stable laws, or adjacencymatrices of Erdos-Renyi graphs. We denote by U=[uij] the eigenvectors matrix (corresponding to increasing eigenvalues) and prove that the bivariate process [formula] indexed by s,t∈[0,1], converges in law to a non trivial Gaussian process. An interesting part of this result is the n−1/2 rescaling, proving that from this point of view, the eigenvectors matrix U behaves more like a permutation matrix (as it was proved by Chapuy that for U a permutation matrix, n−1/2 is the right scaling) than like a Haar-distributed orthogonal or unitary matrix (as it was proved by Rouault and Donati-Martin that for U such a matrix, the right scaling is 1). Simons Foundation National Science Foundation (U.S.) (Grant DMS-1307704) 2014-09-15T15:31:57Z 2014-09-15T15:31:57Z 2014-06 2013-10 Article http://purl.org/eprint/type/JournalArticle 1083-6489 http://hdl.handle.net/1721.1/89525 Benaych-Georges, Florent, and Alice Guionnet. “Central Limit Theorem for Eigenvectors of Heavy Tailed Matrices.” Electronic Journal of Probability 19, no. 0 (January 2, 2014). https://orcid.org/0000-0003-4524-8627 en_US http://dx.doi.org/10.1214/EJP.v19-3093 Electronic Journal of Probability Creative Commons Attribution http://creativecommons.org/licenses/by/2.5/ application/pdf Institute of Mathematical Statistics Institute of Mathematical Statistics |
| spellingShingle | Benaych-Georges, Florent Guionnet, Alice Central limit theorem for eigenvectors of heavy tailed matrices |
| title | Central limit theorem for eigenvectors of heavy tailed matrices |
| title_full | Central limit theorem for eigenvectors of heavy tailed matrices |
| title_fullStr | Central limit theorem for eigenvectors of heavy tailed matrices |
| title_full_unstemmed | Central limit theorem for eigenvectors of heavy tailed matrices |
| title_short | Central limit theorem for eigenvectors of heavy tailed matrices |
| title_sort | central limit theorem for eigenvectors of heavy tailed matrices |
| url | http://hdl.handle.net/1721.1/89525 https://orcid.org/0000-0003-4524-8627 |
| work_keys_str_mv | AT benaychgeorgesflorent centrallimittheoremforeigenvectorsofheavytailedmatrices AT guionnetalice centrallimittheoremforeigenvectorsofheavytailedmatrices |