Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 105, 26 pages      arXiv:1607.01351      https://doi.org/10.3842/SIGMA.2016.105
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

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

Tamara Grava ab, Alexander Its c, Andrei Kapaev d and Francesco Mezzadri a
a) School of Mathematics, University of Bristol, Bristol, BS8 1SN, UK
b) SISSA, via Bonomea 265, 34100, Trieste, Italy
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

Received July 04, 2016, in final form October 25, 2016; Published online November 01, 2016

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.

Key words: $\beta$-ensamble; $\beta$-Tracy-Widom distribution; Painlevé II equation.

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