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




This article is cited in 3 scientific papers (total in 3 papers)
On the Tracy–Widom$_\beta$ Distribution for $\beta=6$
Tamara Grava^{ab}, Alexander Its^{c}, Andrei Kapaev^{d}, Francesco Mezzadri^{b} ^{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 462023216, 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 Laxpair 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 Laxpair 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 gaugeinterpretation of Rumanov's Laxpair 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 
VP22014034 RF2015442 
National Science Foundation 
DMS1361856 
Saint Petersburg State University 
11.38.215.2014 11.38.215.2014 
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 VP22014034. A. Its acknowledges the support by the NSF grant DMS1361856 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 RF2015442
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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ArXiv:
1607.01351
MSC: 30E20; 60B20; 34M50 Received: July 4, 2016; in final form October 25, 2016; Published online November 1, 2016
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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
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\paper On the TracyWidom$_\beta$ Distribution for $\beta=6$
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\papernumber 105
\totalpages 26
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This publication is cited in the following articles:

Percy Deift, “Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II”, SIGMA, 13 (2017), 016, 23 pp.

V. A. Pavlenko, B. I. Suleimanov, ““Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$”, Ufa Math. J., 9:4 (2017), 97–107

V. A. Pavlenko, B. I. Suleimanov, “Solutions to analogues of nonstationary Schrödinger equations defined by isomonodromic Hamilton system $H^{2+1+1+1}$”, Ufa Math. J., 10:4 (2018), 92–102

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