On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators
By applying an idea of Borodin and Olshanski [J. Algebra 313 (2007), 40-60], we study various scaling limits of determinantal point processes with trace class projection kernels given by spectral projections of selfadjoint Sturm-Liouville operators. Instead of studying the convergence of the kernels...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2016 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2016
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147850 |
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| Cite this: | On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators / F. Bornemann // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862734350545584128 |
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| author | Bornemann, F. |
| author_facet | Bornemann, F. |
| citation_txt | On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators / F. Bornemann // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | By applying an idea of Borodin and Olshanski [J. Algebra 313 (2007), 40-60], we study various scaling limits of determinantal point processes with trace class projection kernels given by spectral projections of selfadjoint Sturm-Liouville operators. Instead of studying the convergence of the kernels as functions, the method directly addresses the strong convergence of the induced integral operators. We show that, for this notion of convergence, the Dyson, Airy, and Bessel kernels are universal in the bulk, soft-edge, and hard-edge scaling limits. This result allows us to give a short and unified derivation of the known formulae for the scaling limits of the classical random matrix ensembles with unitary invariance, that is, the Gaussian unitary ensemble (GUE), the Wishart or Laguerre unitary ensemble (LUE), and the MANOVA (multivariate analysis of variance) or Jacobi unitary ensemble (JUE).
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| first_indexed | 2025-12-07T19:42:46Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147850 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:42:46Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bornemann, F. 2019-02-16T09:16:49Z 2019-02-16T09:16:49Z 2016 On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators / F. Bornemann // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 15B52; 34B24; 33C45 DOI:10.3842/SIGMA.2016.083 https://nasplib.isofts.kiev.ua/handle/123456789/147850 By applying an idea of Borodin and Olshanski [J. Algebra 313 (2007), 40-60], we study various scaling limits of determinantal point processes with trace class projection kernels given by spectral projections of selfadjoint Sturm-Liouville operators. Instead of studying the convergence of the kernels as functions, the method directly addresses the strong convergence of the induced integral operators. We show that, for this notion of convergence, the Dyson, Airy, and Bessel kernels are universal in the bulk, soft-edge, and hard-edge scaling limits. This result allows us to give a short and unified derivation of the known formulae for the scaling limits of the classical random matrix ensembles with unitary invariance, that is, the Gaussian unitary ensemble (GUE), the Wishart or Laguerre unitary ensemble (LUE), and the MANOVA (multivariate analysis of variance) or Jacobi unitary ensemble (JUE). This paper is a contribution to the Special Issue on Asymptotics and Universality in Random Matrices,
 Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy.
 The full collection is available at http://www.emis.de/journals/SIGMA/Deift-Tracy.html. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators Article published earlier |
| spellingShingle | On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators Bornemann, F. |
| title | On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators |
| title_full | On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators |
| title_fullStr | On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators |
| title_full_unstemmed | On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators |
| title_short | On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators |
| title_sort | on the scaling limits of determinantal point processes with kernels induced by sturm-liouville operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147850 |
| work_keys_str_mv | AT bornemannf onthescalinglimitsofdeterminantalpointprocesseswithkernelsinducedbysturmliouvilleoperators |