AI 中文总结
该研究建立圆雅可比系综中特征多项式联合矩与σ-Painlevé V、III'方程解的联系，解决相关问题并将Hua-Pickrell测度联合矩结果从实参数扩展到复参数。
AI 中文摘要
本文建立了圆雅可比系综（Circular Jacobi Ensemble，圆酉系综的推广）中特征多项式及其导数的联合矩与非线性Painlevé方程解之间的联系。对于有限的N，我们证明这些联合矩由σ-Painlevé V方程的解刻画，适用于所有实矩指数在其容许范围内。在适当的大N标度极限下，我们进一步证明，对于某一范围的矩指数，极限联合矩可通过σ-Painlevé III'方程的解表示。作为应用，我们回答了Assiotis等人在[Math. Physics. Anal. Geom. 25 (2022), no.2, Paper No. 15, 24pp, Remark 1.7]中提出的问题，证明了某一特殊随机变量的特征函数与复参数下的σ-Painlevé III'方程相关。此外，我们研究了圆雅可比系综中涉及特征多项式高阶导数的联合矩，进而将Assiotis等人在[Comm. Pure Appl. Math. 79 (2026), no. 7, 1771-1827, Theorem 1.11]中关于由Hua-Pickrell测度的遍历分解产生的随机变量序列联合矩的结果，从实参数扩展到复参数。
英文摘要
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 Painlevé equations. For finite $N$, we show that these joint moments are characterised by a solution of the $σ$-Painlevé 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 $σ$-Painlevé 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 $σ$-Painlevé 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.
Comments40 pages, 4 figures
