On the joint moments of the characteristic polynomials of random unitary matrices

TL;DR

This paper derives the asymptotic behavior of joint moments of characteristic polynomials and their derivatives for random unitary matrices, confirming a 2001 conjecture and linking results to Painlevé equations.

Contribution

It proves a conjecture on joint moments of characteristic polynomials and derivatives, providing a probabilistic representation and connecting to Painlevé equations.

Findings

Asymptotics of joint moments established for general exponents

Probabilistic representation of leading order coefficient provided

Connections made between characteristic function and Painlevé III' equation

Abstract

We establish the asymptotics of the joint moments of the characteristic polynomial of a random unitary matrix and its derivative for general real values of the exponents, proving a conjecture made by Hughes in 2001. Moreover, we give a probabilistic representation for the leading order coefficient in the asymptotic in terms of a real-valued random variable that plays an important role in the ergodic decomposition of the Hua-Pickrell measures. This enables us to establish connections between the characteristic function of this random variable and the $\sigma$-Painlev\'{e} III' equation.