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arXiv · 2608.24423

Joint moments of characteristic polynomials in the circular Jacobi ensemble and Painlev\'e equations

Thomas Bothner·Fei Wei
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Abstract

In this paper, we establish a connection between joint moments of characteristic polynomials and their derivatives in the Circular Jacobi Ensemble, a generalisation of the Circular Unitary Ensemble, and solutions of nonlinear Painlev\'e equations. For finite $N$, we show that these joint moments are characterised by a solution of the $\sigma$-Painlev\'e V equation for all real moment exponents in their admissible range. Under an appropriate large-$N$ scaling limit, we further prove that the limiting joint moments admit a representation in terms of a solution of the $\sigma$-Painlev\'e III$'$ equation for a certain range of moment exponents. As applications, we answer a question posed by Assiotis et al. in [Math. Physics. Anal. Geom. 25 (2022), no.2, Paper No. 15, 24pp, Remark 1.7] by showing that the characteristic function of a distinguished random variable is connected with the $\sigma$-Painlev\'e III$'$ equation for complex parameters. Furthermore, we investigate joint moments involving higher-order derivatives of characteristic polynomials in the Circular Jacobi Ensemble. As a consequence, we extend a result of Assiotis et al. in [Comm. Pure Appl. Math. 79 (2026), no. 7, 1771-1827, Theorem 1.11] concerning the joint moments of a sequence of random variables arising from the ergodic decomposition of Hua-Pickrell measures, from real parameters to complex parameters.

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Thomas Bothner, Fei Wei. 2026-08-25. Joint moments of characteristic polynomials in the circular Jacobi ensemble and Painlev\'e equations. https://arxiv.org/abs/2608.24423

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