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arXiv · 1811.00064

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

Estelle Basor·Pavel Bleher·Robert Buckingham·Tamara Grava·Alexander Its·Elizabeth Its·Jonathan P. Keating
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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 sigma-Painleve 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 sigma-Painleve 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 sigma-Painleve III equation. Using the conformal block expansion of the tau-functions associated with the sigma-Painleve V and the sigma-Painleve III equations leads to general conjectures for the joint moments.

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Estelle Basor, Pavel Bleher, Robert Buckingham, Tamara Grava, Alexander Its, Elizabeth Its, Jonathan P. Keating. 2019-06-26. A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions. https://doi.org/10.1088/1361-6544%2Fab28c7

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