On the Joint Moments of the Characteristic Polynomials of Random Unitary Matrices

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 [ 40] 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 σ-Painlevé III’ equation.