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Random Matrices with Merging Singularities and the Painlevé V Equation
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2016, том 12
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Random Matrices with Merging Singularities and the Painlevé V Equation
Інший ID:
2010 Mathematics Subject Classification: 60B20; 35Q15; 33E1
DOI:10.3842/SIGMA.2016.031
DOI:10.3842/SIGMA.2016.031
Посилання:
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.
Підтримка:
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.
Дата:
2016
Переглядів:
637
Завантажень:
293
Анотація:
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.
