The beta-Wishart ensemble
We introduce a “broken-arrow” matrix model for the β-Wishart ensemble, which improves on the traditional bidiagonal model by generalizing to non-identity covariance parameters. We prove that its joint eigenvalue density involves the correct hypergeometric function of two matrix arguments, and a cont...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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American Institute of Physics (AIP)
2015
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| Online Access: | http://hdl.handle.net/1721.1/92863 https://orcid.org/0000-0001-7676-3133 |
| _version_ | 1826206640738664448 |
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| author | Dubbs, Alexander Joseph Edelman, Alan Koev, Plamen Venkataramana, Praveen |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Dubbs, Alexander Joseph Edelman, Alan Koev, Plamen Venkataramana, Praveen |
| author_sort | Dubbs, Alexander Joseph |
| collection | MIT |
| description | We introduce a “broken-arrow” matrix model for the β-Wishart ensemble, which improves on the traditional bidiagonal model by generalizing to non-identity covariance parameters. We prove that its joint eigenvalue density involves the correct hypergeometric function of two matrix arguments, and a continuous parameter β > 0. If we choose β = 1, 2, 4, we recover the classical Wishart ensembles of general covariance over the reals, complexes, and quaternions. Jack polynomials are often defined as the eigenfunctions of the Laplace-Beltrami operator. We prove that Jack polynomials are in addition eigenfunctions of an integral operator defined as an average over a β-dependent measure on the sphere. When combined with an identity due to Stanley, we derive a definition of Jack polynomials. An efficient numerical algorithm is also presented for simulations. The algorithm makes use of secular equation software for broken arrow matrices currently unavailable in the popular technical computing languages. The simulations are matched against the cdfs for the extreme eigenvalues. The techniques here suggest that arrow and broken arrow matrices can play an important role in theoretical and computational random matrix theory including the study of corners processes. We provide a number of simulations illustrating the extreme eigenvalue distributions that are likely to be useful for applications. We also compare the n → ∞ answer for all β with the free-probability prediction. |
| first_indexed | 2024-09-23T13:35:48Z |
| format | Article |
| id | mit-1721.1/92863 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T13:35:48Z |
| publishDate | 2015 |
| publisher | American Institute of Physics (AIP) |
| record_format | dspace |
| spelling | mit-1721.1/928632022-09-28T14:54:47Z The beta-Wishart ensemble Dubbs, Alexander Joseph Edelman, Alan Koev, Plamen Venkataramana, Praveen Massachusetts Institute of Technology. Department of Mathematics Dubbs, Alexander Joseph Edelman, Alan Venkataramana, Praveen We introduce a “broken-arrow” matrix model for the β-Wishart ensemble, which improves on the traditional bidiagonal model by generalizing to non-identity covariance parameters. We prove that its joint eigenvalue density involves the correct hypergeometric function of two matrix arguments, and a continuous parameter β > 0. If we choose β = 1, 2, 4, we recover the classical Wishart ensembles of general covariance over the reals, complexes, and quaternions. Jack polynomials are often defined as the eigenfunctions of the Laplace-Beltrami operator. We prove that Jack polynomials are in addition eigenfunctions of an integral operator defined as an average over a β-dependent measure on the sphere. When combined with an identity due to Stanley, we derive a definition of Jack polynomials. An efficient numerical algorithm is also presented for simulations. The algorithm makes use of secular equation software for broken arrow matrices currently unavailable in the popular technical computing languages. The simulations are matched against the cdfs for the extreme eigenvalues. The techniques here suggest that arrow and broken arrow matrices can play an important role in theoretical and computational random matrix theory including the study of corners processes. We provide a number of simulations illustrating the extreme eigenvalue distributions that are likely to be useful for applications. We also compare the n → ∞ answer for all β with the free-probability prediction. San Jose State University (Woodward Fund for Applied Mathematics) National Science Foundation (U.S.) (SOLAR Grant No. 1035400) National Science Foundation (U.S.) (SOLAR Grant No. DMS-1035400) National Science Foundation (U.S.) (SOLAR Grant No. DMS-1016086) National Science Foundation (U.S.) (NSF GRFP) 2015-01-14T18:45:15Z 2015-01-14T18:45:15Z 2013-08 2013-04 Article http://purl.org/eprint/type/JournalArticle 00222488 http://hdl.handle.net/1721.1/92863 Dubbs, Alexander, Alan Edelman, Plamen Koev, and Praveen Venkataramana. “The Beta-Wishart Ensemble.” Journal of Mathematical Physics 54, no. 8 (2013): 083507. © 2013 AIP. https://orcid.org/0000-0001-7676-3133 en_US http://dx.doi.org/10.1063/1.4818304 Journal of Mathematical Physics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Institute of Physics (AIP) MIT web domain |
| spellingShingle | Dubbs, Alexander Joseph Edelman, Alan Koev, Plamen Venkataramana, Praveen The beta-Wishart ensemble |
| title | The beta-Wishart ensemble |
| title_full | The beta-Wishart ensemble |
| title_fullStr | The beta-Wishart ensemble |
| title_full_unstemmed | The beta-Wishart ensemble |
| title_short | The beta-Wishart ensemble |
| title_sort | beta wishart ensemble |
| url | http://hdl.handle.net/1721.1/92863 https://orcid.org/0000-0001-7676-3133 |
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