### 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.

pdf (502 kb)   tex (29 kb)

References

1. Baik J., Buckingham R., DiFranco J., Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function, Comm. Math. Phys. 280 (2008), 463-497, arXiv:0704.3636.
2. Bloemendal A., Virág B., Limits of spiked random matrices I, Probab. Theory Related Fields 156 (2013), 795-825, arXiv:1011.1877.
3. Borot G., Eynard B., Majumdar S.N., Nadal C., Large deviations of the maximal eigenvalue of random matrices, J. Stat. Mech. Theory Exp. 2011 (2011), P11024, 56 pages, arXiv:1009.1945.
4. Chen Y., Manning S.M., Asymptotic level spacing of the Laguerre ensemble: a Coulomb fluid approach, J. Phys. A: Math. Gen. 27 (1994), 3615-3620, cond-mat/9309010.
5. Deift P., Its A., Krasovsky I., Asymptotics of the Airy-kernel determinant, Comm. Math. Phys. 278 (2008), 643-678, math.FA/0609451.
6. Deift P., Zhou X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993), 295-368.
7. Dumitriu I., Edelman A., Matrix models for beta ensembles, J. Math. Phys. 43 (2002), 5830-5847, math-ph/0206043.
8. Flaschka H., Newell A.C., Monodromy- and spectrum-preserving deformations. I, Comm. Math. Phys. 76 (1980), 65-116.
9. Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents: the Riemann-Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, Amer. Math. Soc., Providence, RI, 2006.
10. Forrester P.J., Exact results and universal asymptotics in the Laguerre random matrix ensemble, J. Math. Phys. 35 (1994), 2539-2551.
11. Forrester P.J., Asymptotics of spacing distributions 50 years later, in Random matrix theory, interacting particle systems, and integrable systems, Math. Sci. Res. Inst. Publ., Vol. 65, Cambridge University Press, New York, 2014, 199-222, arXiv:1204.3225.
12. Hastings S.P., McLeod J.B., A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation, Arch. Rational Mech. Anal. 73 (1980), 31-51.
13. Ince E.L., Ordinary differential equations, Dover Publications, New York, 1944.
14. Nagoya H., Hypergeometric solutions to Schrödinger equations for the quantum Painlevé equations, J. Math. Phys. 52 (2011), 083509, 16 pages, arXiv:1109.1645.
15. Okamoto K., Polynomial Hamiltonians associated with Painlevé equations. II. Differential equations satisfied by polynomial Hamiltonians, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 367-371.
16. Ramírez J.A., Rider B., Virág B., Beta ensembles, stochastic Airy spectrum, and a diffusion, J. Amer. Math. Soc. 24 (2011), 919-944, math.PR/0607331.
17. Rumanov I., Beta ensembles, quantum Painlevé equations and isomonodromy systems, in Algebraic and Analytic Aspects of Integrable Systems and Painlevé Equations, Contemp. Math., Vol. 651, Amer. Math. Soc., Providence, RI, 2015, 125-155, arXiv:1408.3847.
18. Rumanov I., Classical integrability for beta-ensembles and general Fokker-Planck equations, J. Math. Phys. 56 (2015), 013508, 16 pages, arXiv:1306.2117.
19. Rumanov I., Painlevé representation of Tracy-Widom$_\beta$ distribution for $\beta=6$, Comm. Math. Phys. 342 (2016), 843-868, arXiv:1408.3779.
20. Slavyanov S.Y., Painlevé equations as classical analogues of Heun equations, J. Phys. A: Math. Gen. 29 (1996), 7329-7335.
21. Suleimanov B.I., ''Quantizations'' of the second Painlevé equation and the problem of the equivalence of its $L$-$A$ pairs, Theoret. Math. Phys. 156 (2008), 1280-1291.
22. Tracy C.A., Widom H., Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 151-174, hep-th/9211141.
23. Tracy C.A., Widom H., On orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177 (1996), 727-754, solv-int/9509007.
24. Valkó B., Virág B., Continuum limits of random matrices and the Brownian carousel, Invent. Math. 177 (2009), 463-508, arXiv:0712.2000.
25. Wasow W., Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. 14, Interscience Publishers John Wiley & Sons, Inc., New York - London - Sydney, 1965.
26. Zabrodin A., Zotov A., Classical-quantum correspondence and functional relations for Painlevé equations, Constr. Approx. 41 (2015), 385-423, arXiv:1212.5813.