Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated with a certain normal matrix model. The model depends on a parameter, and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209866 |
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| Cite this: | Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane / M. Bertola, J.G.E. Rebelo, T. Grava // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 43 назв. — англ. |
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Bertola, M. Rebelo, J.G.E. Grava, T. 2025-11-27T18:03:44Z 2018 Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane / M. Bertola, J.G.E. Rebelo, T. Grava // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 43 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34M55; 34M56; 33C15 arXiv: 1802.01153 https://nasplib.isofts.kiev.ua/handle/123456789/209866 https://doi.org/10.3842/SIGMA.2018.091 We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated with a certain normal matrix model. The model depends on a parameter, and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition, we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlevé IV equation. We determine the Fredholm determinant associated with such a solution, and we compute it numerically on the real line, showing also that the corresponding Painlevé transcendent is pole-free on a semiaxis. The authors wish to thank the anonymous referees for their many suggestions for improving this manuscript. T.G. acknowledges the support of the H2020-MSCA-RISE-2017 PROJECT No. 778010 IPADEGAN. M.B. acknowledges the support by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN-2016-06660 and the FQRNT grant “Matrices Aléatoires, Processus Stochastiques et Systèmes Intégrables” (2013–PR–166790). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane |
| spellingShingle |
Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane Bertola, M. Rebelo, J.G.E. Grava, T. |
| title_short |
Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane |
| title_full |
Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane |
| title_fullStr |
Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane |
| title_full_unstemmed |
Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane |
| title_sort |
painlevé iv critical asymptotics for orthogonal polynomials in the complex plane |
| author |
Bertola, M. Rebelo, J.G.E. Grava, T. |
| author_facet |
Bertola, M. Rebelo, J.G.E. Grava, T. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated with a certain normal matrix model. The model depends on a parameter, and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition, we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlevé IV equation. We determine the Fredholm determinant associated with such a solution, and we compute it numerically on the real line, showing also that the corresponding Painlevé transcendent is pole-free on a semiaxis.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209866 |
| citation_txt |
Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane / M. Bertola, J.G.E. Rebelo, T. Grava // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 43 назв. — англ. |
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| first_indexed |
2025-12-07T13:42:14Z |
| last_indexed |
2025-12-07T13:42:14Z |
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