Abstract
We study the TracyWidom 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 Laxpair representation for the BloemendalVirag equation. The latter is a linear PDE which describes the TracyWidom functions corresponding to general values of β. Using his Lax pair, Rumanov derives an explicit formula for the TracyWidom β = 6 function in terms of the second Painlev\'e transcendent and the solution of an auxiliary ODE. Rumanov also shows that this formula allows him to derive formally the asymptotic expansion of the TracyWidom 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 HastinsMcLeod 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 TracyWidom 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.
Original language  English 

Article number  105 
Number of pages  26 
Journal  Symmetry, Integrability and Geometry: Methods and Applications 
Volume  12 
DOIs  
Publication status  Published  1 Nov 2016 
Keywords
 βensamble
 βTracy–Widom distribution
 Painlevé II equation
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Professor Francesco Mezzadri
 Probability, Analysis and Dynamics
 School of Mathematics  Professor of Mathematical Physics
 Applied Mathematics
 Mathematical Physics
Person: Academic , Member