Browsing by Author "Grava, Tamara"

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    On the Tracy-Widomββ Distribution for β=6
    (2016) Grava, Tamara; Its, Alexander; Kapaev, Andrei; Mezzadri, Francesco; Department of Mathematical Sciences, School of Science
    We study the Tracy-Widom distribution function for Dyson's ββ-ensemble with β=6β=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 ββ. Using his Lax pair, Rumanov derives an explicit formula for the Tracy-Widom β=6β=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 ββ-ensembles formulated by Percy Deift during the June 2015 Montreal Conference on integrable systems.
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    A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions
    (IOP, 2019-10) Basor, Estelle; Bleher, Pavel; Buckingham, Robert; Grava, Tamara; Its, Alexander; Its, Elizabeth; Keating, Jonathan P.; Mathematical Sciences, School of Science
    We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the -Painlevé V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann–Hilbert method. We use this connection with the -Painlevé V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the -Painlevé III equation. Using the conformal block expansion of the -functions associated with the -Painlevé V and the -Painlevé III equations leads to general conjectures for the joint moments.