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 SIGMA, 2016, Volume 12, 105, 26 pages (Mi sigma1187)

On the Tracy–Widom$_\beta$ Distribution for $\beta=6$

Tamara Gravaab, Alexander Itsc, Andrei Kapaevd, Francesco Mezzadrib

a SISSA, via Bonomea 265, 34100, Trieste, Italy
b School of Mathematics, University of Bristol, Bristol, BS8 1SN, UK
c Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, Indianapolis, IN 46202-3216, USA
d Department of Mathematical Physics, St. Petersburg State University, St. Petersburg, Russia

Abstract: We study the Tracy–Widom distribution function for Dyson's $\beta$-ensemble with $\beta = 6$. 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 $\beta$. Using his Lax pair, Rumanov derives an explicit formula for the Tracy–Widom $\beta=6$ 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 $\beta$-ensembles formulated by Percy Deift during the June 2015 Montreal Conference on integrable systems.

Keywords: $\beta$-ensamble; $\beta$-Tracy–Widom distribution; Painlevé II equation.

 Funding Agency Grant Number Leverhulme Trust VP2-2014-034RF-2015-442 National Science Foundation DMS-1361856 Saint Petersburg State University 11.38.215.201411.38.215.2014 Engineering and Physical Sciences Research Council EP/L010305/1 PRIN 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.

DOI: https://doi.org/10.3842/SIGMA.2016.105

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Bibliographic databases:

ArXiv: 1607.01351
Document Type: Article
MSC: 30E20; 60B20; 34M50
Received: July 4, 2016; in final form October 25, 2016; Published online November 1, 2016
Language: English

Citation: Tamara Grava, Alexander Its, Andrei Kapaev, Francesco Mezzadri, “On the Tracy–Widom$_\beta$ Distribution for $\beta=6$”, SIGMA, 12 (2016), 105, 26 pp.

Citation in format AMSBIB
\Bibitem{GraItsKap16}
\paper On the Tracy--Widom$_\beta$ Distribution for $\beta=6$
\jour SIGMA
\yr 2016
\vol 12
\papernumber 105
\totalpages 26
\mathnet{http://mi.mathnet.ru/sigma1187}
\crossref{https://doi.org/10.3842/SIGMA.2016.105}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84996483423}

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This publication is cited in the following articles:
1. Percy Deift, “Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II”, SIGMA, 13 (2017), 016, 23 pp.
2. V. A. Pavlenko, B. I. Suleimanov, ““Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$”, Ufa Math. J., 9:4 (2017), 97–107
3. V. A. Pavlenko, B. I. Suleimanov, “Solutions to analogues of non-stationary Schrödinger equations defined by isomonodromic Hamilton system $H^{2+1+1+1}$”, Ufa Math. J., 10:4 (2018), 92–102
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