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.

A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions / Basor, Estelle; Bleher, Pavel; Buckingham, Robert; Grava, Tamara; Its, Alexander; Its, Elizabeth; Keating, Jonathan P. - In: NONLINEARITY. - ISSN 0951-7715. - 32:10(2019), pp. 4033-4078. [10.1088/1361-6544/ab28c7]

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

Grava, Tamara
Membro del Collaboration group
;
2019

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 -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.
32
10
4033
4078
https://doi.org/10.1088/1361-6544/ab28c7
https://arxiv.org/abs/1811.00064
Basor, Estelle; Bleher, Pavel; Buckingham, Robert; Grava, Tamara; Its, Alexander; Its, Elizabeth; Keating, Jonathan P
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/103996
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