On the Tracy-Widomβ Distribution for β=6
We study the Tracy-Widom distribution function for Dyson's β-ensemble with β=6. The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Virág equation. The latter is a linear PDE which describes the Tracy-Widom functions...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2016 |
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Інститут математики НАН України
2016
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148536 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On the Tracy-Widomβ Distribution for β=6 / T. Grava, A. Its, A. Kapaev, F. Mezzadri // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 26 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862726246128943104 |
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| author | Grava, T. Its, A. Kapaev, A. Mezzadri, F. |
| author_facet | Grava, T. Its, A. Kapaev, A. Mezzadri, F. |
| citation_txt | On the Tracy-Widomβ Distribution for β=6 / T. Grava, A. Its, A. Kapaev, F. Mezzadri // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 26 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study the Tracy-Widom distribution function for Dyson's β-ensemble with β=6. The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Virág equation. The latter is a linear PDE which describes the Tracy-Widom functions corresponding to general values of β. Using his Lax pair, Rumanov derives an explicit formula for the Tracy-Widom β=6 function in terms of the second Painlevé transcendent and the solution of an auxiliary ODE. Rumanov also shows that this formula allows him to derive formally the asymptotic expansion of the Tracy-Widom function. Our goal is to make Rumanov's approach and hence the asymptotic analysis it provides rigorous. In this paper, the first one in a sequel, we show that Rumanov's Lax-pair can be interpreted as a certain gauge transformation of the standard Lax pair for the second Painlevé equation. This gauge transformation though contains functional parameters which are defined via some auxiliary nonlinear ODE which is equivalent to the auxiliary ODE of Rumanov's formula. The gauge-interpretation of Rumanov's Lax-pair allows us to highlight the steps of the original Rumanov's method which needs rigorous justifications in order to make the method complete. We provide a rigorous justification of one of these steps. Namely, we prove that the Painlevé function involved in Rumanov's formula is indeed, as it has been suggested by Rumanov, the Hastings-McLeod solution of the second Painlevé equation. The key issue which we also discuss and which is still open is the question of integrability of the auxiliary ODE in Rumanov's formula. We note that this question is crucial for the rigorous asymptotic analysis of the Tracy-Widom function. We also notice that our work is a partial answer to one of the problems related to the β-ensembles formulated by Percy Deift during the June 2015 Montreal Conference on integrable systems.
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| first_indexed | 2025-12-07T18:56:31Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148536 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:56:31Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
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| spelling | Grava, T. Its, A. Kapaev, A. Mezzadri, F. 2019-02-18T14:51:20Z 2019-02-18T14:51:20Z 2016 On the Tracy-Widomβ Distribution for β=6 / T. Grava, A. Its, A. Kapaev, F. Mezzadri // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 26 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 30E20; 60B20; 34M50 DOI:10.3842/SIGMA.2016.105 https://nasplib.isofts.kiev.ua/handle/123456789/148536 We study the Tracy-Widom distribution function for Dyson's β-ensemble with β=6. The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Virág equation. The latter is a linear PDE which describes the Tracy-Widom functions corresponding to general values of β. Using his Lax pair, Rumanov derives an explicit formula for the Tracy-Widom β=6 function in terms of the second Painlevé transcendent and the solution of an auxiliary ODE. Rumanov also shows that this formula allows him to derive formally the asymptotic expansion of the Tracy-Widom function. Our goal is to make Rumanov's approach and hence the asymptotic analysis it provides rigorous. In this paper, the first one in a sequel, we show that Rumanov's Lax-pair can be interpreted as a certain gauge transformation of the standard Lax pair for the second Painlevé equation. This gauge transformation though contains functional parameters which are defined via some auxiliary nonlinear ODE which is equivalent to the auxiliary ODE of Rumanov's formula. The gauge-interpretation of Rumanov's Lax-pair allows us to highlight the steps of the original Rumanov's method which needs rigorous justifications in order to make the method complete. We provide a rigorous justification of one of these steps. Namely, we prove that the Painlevé function involved in Rumanov's formula is indeed, as it has been suggested by Rumanov, the Hastings-McLeod solution of the second Painlevé equation. The key issue which we also discuss and which is still open is the question of integrability of the auxiliary ODE in Rumanov's formula. We note that this question is crucial for the rigorous asymptotic analysis of the Tracy-Widom function. We also notice that our work is a partial answer to one of the problems related to the β-ensembles formulated by Percy Deift during the June 2015 Montreal Conference on integrable systems. 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. A. Its and T. Grava acknowledge the support of the Leverhulme Trust visiting Professorship
 grant VP2-2014-034. A. Its acknowledges the support by the NSF grant DMS-1361856 and by
 the SPbGU grant N 11.38.215.2014. A. Kapaev acknowledges the support by the SPbGU grant
 N 11.38.215.2014. F. Mezzadri was partially supported by the EPSRC grant no. EP/L010305/1.
 T. Grava acknowledges the support by the Leverhulme Trust Research Fellowship RF-2015-442
 from UK and PRIN Grant “Geometric and analytic theory of Hamiltonian systems in finite and
 infinite dimensions” of Italian Ministry of Universities and Researches. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Tracy-Widomβ Distribution for β=6 Article published earlier |
| spellingShingle | On the Tracy-Widomβ Distribution for β=6 Grava, T. Its, A. Kapaev, A. Mezzadri, F. |
| title | On the Tracy-Widomβ Distribution for β=6 |
| title_full | On the Tracy-Widomβ Distribution for β=6 |
| title_fullStr | On the Tracy-Widomβ Distribution for β=6 |
| title_full_unstemmed | On the Tracy-Widomβ Distribution for β=6 |
| title_short | On the Tracy-Widomβ Distribution for β=6 |
| title_sort | on the tracy-widomβ distribution for β=6 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148536 |
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