We study the power spectrum of eigen-angles of random matrices drawn from the circular unitary ensemble ${\rm CUE}(N)$ and show that it can be evaluated in terms of either a Fredholm determinant, or a Toeplitz determinant, or a sixth Painlev\'e function. In the limit of infinite-dimensional matrices, $N\rightarrow\infty$, we derive a ${\it\, concise\,}$ parameter-free formula for the power spectrum which involves a fifth Painlev\'e transcendent and interpret it in terms of the ${\rm Sine}_2$ determinantal random point field. Further, we discuss a universality of the predicted power spectrum law and tabulate it for easy use by random-matrix-theory and quantum chaos practitioners.

中文翻译：

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圆形酉系综的功率谱

我们研究了从圆形酉系综 ${\rm CUE}(N)$ 中提取的随机矩阵的特征角的功率谱，并表明它可以根据 Fredholm 行列式或 Toeplitz 行列式或第六个 Painlev\'e 函数。在无限维矩阵 $N\rightarrow\infty$ 的极限下，我们推导出了一个 ${\it\, concise\,}$ 功率谱的无参数公式，它涉及到第五个 Painlev\'e 超越和解释它根据 ${\rm Sine}_2$ 行列式随机点场。此外，我们讨论了预测功率谱定律的普遍性并将其制成表格，以便随机矩阵理论和量子混沌实践者使用。