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On the joint moments of the characteristic polynomials of random unitary matrices (2005.13961v2)
Published 28 May 2020 in math.PR, math-ph, and math.MP
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.