Skip to content

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

Research output: Contribution to journalArticle

  • Estelle Basor
  • Pavel Bleher
  • Robert Buckingham
  • Tamara Grava
  • Alexander Its
  • Elizabeth Its
  • Jonathan P. Keating
Original languageEnglish
Pages (from-to)4033-4078
Number of pages47
Issue number10
DateAccepted/In press - 11 Jun 2019
DatePublished (current) - 13 Sep 2019


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.

    Research areas

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

Download statistics

No data available



  • Full-text PDF (accepted author manuscript)

    Rights statement: This is the author accepted manuscript (AAM). The final published version (version of record) is available online via IOP Publishing at . Please refer to any applicable terms of use of the publisher.

    Accepted author manuscript, 512 KB, PDF document


View research connections

Related faculties, schools or groups