We study the ensembles of direct product of $m$ random unitary matrices of size $N$ drawn from a given circular ensemble. We calculate the statistical measures, viz. number variance and spacing distribution to investigate the level correlations and fluctuation properties of the eigenangle spectrum. Similar to the random unitary matrices, the level statistics is stationary for the ensemble constructed by their direct product. We find that the eigenangles are uncorrelated in the small spectral intervals. While, in large spectral intervals, the spectrum is rigid due to strong long-range correlations between the eigenangles. The analytical and numerical results are in good agreement. We also test our findings on the multipartite system of quantum kicked rotors.

中文翻译：

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随机酉矩阵的直接乘积：两点相关性和波动

我们研究直接积的系综 $米$ 大小的随机酉矩阵 $N$从给定的圆形集合中提取。我们计算统计措施，即。数方差和间距分布来研究本征角谱的水平相关性和波动特性。与随机酉矩阵类似，对于由它们的直接乘积构造的集成，水平统计量是平稳的。我们发现特征角在小光谱间隔中是不相关的。而在大的光谱间隔中，由于特征角之间的强长程相关性，光谱是刚性的。解析结果和数值结果吻合良好。我们还在量子踢动转子的多方系统上测试了我们的发现。