We prove an operator level limit for the circular Jacobi $\beta$-ensemble. As a result, we characterize the counting function of the limit point process via coupled systems of stochastic differential equations. We also show that the normalized characteristic polynomials converge to a random analytic function, which we characterize via the joint distribution of its Taylor coefficients at zero and as the solution of a stochastic differential equation system. We also provide analogous results for the real orthogonal $\beta$-ensemble.

中文翻译：

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圆形 Jacobi β-系综的算子级限制

我们证明了循环雅可比的算子级别限制$\beta$-合奏。因此，我们通过随机微分方程的耦合系统来描述极限点过程的计数函数。我们还表明，归一化特征多项式收敛到一个随机解析函数，我们通过其泰勒系数在零处的联合分布和随机微分方程组的解来表征该函数。我们还为实正交提供了类似的结果$\beta$-合奏。