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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Дата:2016
Автори: Grava, T., Its, A., Kapaev, A., Mezzadri, F.
Формат: Стаття
Мова:English
Опубліковано: Інститут математики НАН України 2016
Назва видання:Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/148536
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати: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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spelling irk-123456789-1485362019-02-19T01:24:11Z On the Tracy-Widomβ Distribution for β=6 Grava, T. Its, A. Kapaev, A. Mezzadri, F. 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. 2016 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/148536 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
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.
format Article
author Grava, T.
Its, A.
Kapaev, A.
Mezzadri, F.
spellingShingle Grava, T.
Its, A.
Kapaev, A.
Mezzadri, F.
On the Tracy-Widomβ Distribution for β=6
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Grava, T.
Its, A.
Kapaev, A.
Mezzadri, F.
author_sort Grava, T.
title On the Tracy-Widomβ Distribution for β=6
title_short 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_sort on the tracy-widomβ distribution for β=6
publisher Інститут математики НАН України
publishDate 2016
url http://dspace.nbuv.gov.ua/handle/123456789/148536
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.
series Symmetry, Integrability and Geometry: Methods and Applications
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first_indexed 2023-05-20T17:29:42Z
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