Random Matrices with Merging Singularities and the Painlevé V Equation
We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form 1Zn∣∣det(M²−tI)∣∣αe−nTrV(M)dM, where M is an n×n Hermitian matrix, α>−1/2 and t∈R, in double scaling limits where n→∞ and simultaneously t→0. If t is proportional to 1/n²,...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2016 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2016
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147729 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Random Matrices with Merging Singularities and the Painlevé V Equation / T. Claeys, B. Fahs // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 33 назв. — англ. |
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Claeys, T. Fahs, B. 2019-02-15T18:50:36Z 2019-02-15T18:50:36Z 2016 Random Matrices with Merging Singularities and the Painlevé V Equation / T. Claeys, B. Fahs // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 33 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 60B20; 35Q15; 33E1 DOI:10.3842/SIGMA.2016.031 https://nasplib.isofts.kiev.ua/handle/123456789/147729 We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form 1Zn∣∣det(M²−tI)∣∣αe−nTrV(M)dM, where M is an n×n Hermitian matrix, α>−1/2 and t∈R, in double scaling limits where n→∞ and simultaneously t→0. If t is proportional to 1/n², a transition takes place which can be described in terms of a family of solutions to the Painlevé V equation. These Painlevé solutions are in general transcendental functions, but for certain values of α, they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel. 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. edgements The authors are grateful to I. Krasovsky and N. Simm for useful discussions. They were supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007/2013)/ ERC Grant Agreement 307074 and by the Belgian Interuniversity Attraction Pole P07/18. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Random Matrices with Merging Singularities and the Painlevé V Equation Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Random Matrices with Merging Singularities and the Painlevé V Equation |
| spellingShingle |
Random Matrices with Merging Singularities and the Painlevé V Equation Claeys, T. Fahs, B. |
| title_short |
Random Matrices with Merging Singularities and the Painlevé V Equation |
| title_full |
Random Matrices with Merging Singularities and the Painlevé V Equation |
| title_fullStr |
Random Matrices with Merging Singularities and the Painlevé V Equation |
| title_full_unstemmed |
Random Matrices with Merging Singularities and the Painlevé V Equation |
| title_sort |
random matrices with merging singularities and the painlevé v equation |
| author |
Claeys, T. Fahs, B. |
| author_facet |
Claeys, T. Fahs, B. |
| publishDate |
2016 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form 1Zn∣∣det(M²−tI)∣∣αe−nTrV(M)dM, where M is an n×n Hermitian matrix, α>−1/2 and t∈R, in double scaling limits where n→∞ and simultaneously t→0. If t is proportional to 1/n², a transition takes place which can be described in terms of a family of solutions to the Painlevé V equation. These Painlevé solutions are in general transcendental functions, but for certain values of α, they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147729 |
| citation_txt |
Random Matrices with Merging Singularities and the Painlevé V Equation / T. Claeys, B. Fahs // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 33 назв. — англ. |
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2025-12-07T20:05:16Z |
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2025-12-07T20:05:16Z |
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