On $β=6$ Tracy-Widom distribution and the second Calogero-Painlevé system
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arXiv
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2020
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| author | Its, Alexander Prokhorov, Andrei |
| author_facet | Its, Alexander Prokhorov, Andrei |
| contents | The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlevé equations to the case of the several Painlevé ``particles'' coupled via the Calogero type interactions. In 2014, I. Rumanov discovered the remarkable fact that a particular case of the second Calogero-Painlevé II equation describes the Tracy-Widom distribution function for the general beta-ensembles with even values of the parameter beta. Most recently, in 2017 work of M. Bertola, M. Cafasso, and V. Rubtsov, it was proven that all Calogero-Painlevé systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous, based on the Deift-Zhou nonlinear steepest descent method, asymptotic analysis of the Calogero-Painlevé equations. This in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of beta beyond the classical $β=1, 2, 4.$ In this work we shall start an asymptotic analysis of the Calogero-Painlevé system with a special focus on the Calogero-Painlevé system corresponding to $β= 6$ Tracy-Widom distribution function. The principle technical challenge is the implementation of the nonlinear steepest descent approach beyond the $2\times 2$ matrix dimension of the corresponding Riemann-Hilbert problem; in our case, it is $6\times 6$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_06733 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On $β=6$ Tracy-Widom distribution and the second Calogero-Painlevé system Its, Alexander Prokhorov, Andrei Exactly Solvable and Integrable Systems Classical Analysis and ODEs Probability 33E17, 34E05, 34M55, 34M55, 34M56, 37J35, 41A60, 60B20, 82B44 The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlevé equations to the case of the several Painlevé ``particles'' coupled via the Calogero type interactions. In 2014, I. Rumanov discovered the remarkable fact that a particular case of the second Calogero-Painlevé II equation describes the Tracy-Widom distribution function for the general beta-ensembles with even values of the parameter beta. Most recently, in 2017 work of M. Bertola, M. Cafasso, and V. Rubtsov, it was proven that all Calogero-Painlevé systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous, based on the Deift-Zhou nonlinear steepest descent method, asymptotic analysis of the Calogero-Painlevé equations. This in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of beta beyond the classical $β=1, 2, 4.$ In this work we shall start an asymptotic analysis of the Calogero-Painlevé system with a special focus on the Calogero-Painlevé system corresponding to $β= 6$ Tracy-Widom distribution function. The principle technical challenge is the implementation of the nonlinear steepest descent approach beyond the $2\times 2$ matrix dimension of the corresponding Riemann-Hilbert problem; in our case, it is $6\times 6$. |
| title | On $β=6$ Tracy-Widom distribution and the second Calogero-Painlevé system |
| topic | Exactly Solvable and Integrable Systems Classical Analysis and ODEs Probability 33E17, 34E05, 34M55, 34M55, 34M56, 37J35, 41A60, 60B20, 82B44 |
| url | https://arxiv.org/abs/2010.06733 |