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## 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 language | English |
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Pages (from-to) | 4033-4078 |

Number of pages | 47 |

Journal | Nonlinearity |

Volume | 32 |

Issue number | 10 |

DOIs | |

Publication status | Published - 13 Sep 2019 |

## Keywords

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