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 |
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Інститут математики НАН України
2016
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Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/148536 |
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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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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 |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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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 |
work_keys_str_mv |
AT gravat onthetracywidombdistributionforb6 AT itsa onthetracywidombdistributionforb6 AT kapaeva onthetracywidombdistributionforb6 AT mezzadrif onthetracywidombdistributionforb6 |
first_indexed |
2023-05-20T17:29:42Z |
last_indexed |
2023-05-20T17:29:42Z |
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1796153419708235776 |