Browsing by Author "Its, Elizabeth"

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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.
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    Riemann–Hilbert approach to the elastodynamic equation: half plane
    (Springer, 2021-05) Its, Alexander; Its, Elizabeth; Mathematical Sciences, School of Science
    We show, how the Riemann–Hilbert approach to the elastodynamic equations, which have been suggested in our preceding papers, works in the half plane case. We pay a special attention to the emergence of the Rayleigh waves within the scheme.