Browsing by Author "Prokhorov, A."

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    Mixed moments of characteristic polynomials of random unitary matrices editors-pick
    (AIP, 2019) Bailey, E. C.; Bettin, S.; Blower, G.; Conrey, J. B.; Prokhorov, A.; Rubinstein, M. O.; Snaith, N. C.; Mathematical Sciences, School of Science
    Following the work of Conrey, Rubinstein, and Snaith [Commun. Math. Phys. 267, 611 (2006)] and Forrester and Witte [J. Phys. A: Math. Gen. 39, 8983 (2006)], we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group U(N) (also known as the CUE) and relate the moment to the solution of a Painlevé differential equation. We also calculate a simple form for the asymptotic behavior of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle.
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    On Some Hamiltonian Properties of the Isomonodromic Tau Functions
    (World Scientific, 2018-08) Its, Alexander R.; Prokhorov, A.; Mathematical Sciences, School of Science
    We discuss some new aspects of the theory of the Jimbo–Miwa–Ueno tau function which have come to light within the recent developments in the global asymptotic analysis of the tau functions related to the Painlevé equations. Specifically, we show that up to the total differentials the logarithmic derivatives of the Painlevé tau functions coincide with the corresponding classical action differential. This fact simplifies considerably the evaluation of the constant factors in the asymptotics of tau functions, which has been a long-standing problem of the asymptotic theory of Painlevé equations. Furthermore, we believe that this observation is yet another manifestation of L. D. Faddeev’s emphasis of the key role which the Hamiltonian aspects play in the theory of integrable system.
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    On β=6 Tracy-Widom distribution and the second Calogero-Painlevé system
    (arXiv, 2020) Its, A.; Prokhorov, A.; Mathematical Sciences, School of Science
    The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of theclassical Painlevé equations to the case of the several Painlevé “particles” coupled via the Calogero type interactions. In 2014, I. Rumanov discovered a remarkable fact that a particular case of the second Calogero–Painlevé IIequation describes the Tracy-Widom distribution function for the general beta-ensembles with the even valuesof parameter beta. Most recently, in 2017 work of M. Bertola, M. Cafasso, and V. Rubtsov, it was proven that all Calogero-Painlevé systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous asymptotic analysis of the Calogero Painlevé equations which in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of beta beyond the classical β = 1,2,4. In this work we shall start an asymptotic analysis of the Calogero-Painlevé system with a special focus on the Calogero-Painlevé system corresponding to β = 6 Tracy-Widom distribution function.