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

Author: 

Keating, J
Basor, E
Bleher, P
Buckingham, R
Grava, T
Its, A
Its, E

Publication Date: 

13 September 2019

Journal: 

Nonlinearity

Last Updated: 

2020-06-13T19:09:31.26+01:00

Issue: 

2019

Volume: 

32

DOI: 

10.1088/1361-6544/ab28c7

page: 

4033-4078

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

Symplectic id: 

1055642

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article