A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions. (13th September 2019)

• Record Type: Journal Article
• Authors: Basor, Estelle; Bleher, Pavel; Buckingham, Robert; Grava, Tamara; Its, Alexander; Its, Elizabeth; Keating, Jonathan P
• Is Part Of: Nonlinearity. Volume 32:Number 10(2019)
• Page Start: 4033
• Page End: 4078
• Publication Date: 2019-09-13
• Subjects: CUE ensembles -- Riemann zeta function -- Riemann–Hilbert problems -- Painleve equations; 11M50 -- 33E17 -- 35Q15 -- 60B20; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab28c7
• Languages: English
• ISSNs: 0951-7715

Abstract

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.