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Publication - Professor Tamara Grava

    A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions

    Citation

    Basor, E, Bleher, P, Buckingham, R, Grava, T, Its, A, Its, E & Keating, J, 2019, ‘A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions’. Nonlinearity.

    Abstract

    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 σ-Painlev´e 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 σ-Painlev´e 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 σ-Painlev´e III0
    equation. Using the conformal block expansion of the τ -functions associated with the σ-Painlev´e V and the σ-Painlev´e III0 equations leads to general conjectures for the joint moments

    Full details in the University publications repository