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

Estelle Basor, Pavel Bleher, Robert Buckingham, Tamara Grava, Alexander Its, Elizabeth Its, Jonathan P. Keating

Research output: Contribution to journalArticle (Academic Journal)peer-review

3 Citations (Scopus)
160 Downloads (Pure)

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-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 σ-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 σ-Painlevé III' equation. Using the conformal block expansion of the τ-functions associated with the σ-Painlevé V and the σ-Painlevé III equations leads to general conjectures for the joint moments.

Original languageEnglish
Pages (from-to)4033-4078
Number of pages47
JournalNonlinearity
Volume32
Issue number10
DOIs
Publication statusPublished - 13 Sep 2019

Keywords

  • CUE ensembles
  • Riemann zeta function
  • Riemann-Hilbert problems
  • Painleve equations

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