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arXiv · 2005.13961

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

Theodoros Assiotis·Jonathan P. Keating·Jon Warren
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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.

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Theodoros Assiotis, Jonathan P. Keating, Jon Warren. 2020-05-28. On the joint moments of the characteristic polynomials of random unitary matrices. https://arxiv.org/abs/2005.13961

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