Random matrix models provide a phenomenological description of a vast variety of physical phenomena. Prominent examples include the eigenvalue statistics of quantum (chaotic) systems, which are characterized by the spectral form factor (SFF). Here, we calculate the SFF of unitary matrix ensembles of infinite order with the weight function satisfying the assumptions of Szegö’s limit theorem. We then consider a parameter-dependent critical ensemble which has intermediate statistics characteristic of ergodic-to-nonergodic transitions such as the Anderson localization transition. This same ensemble is the matrix model of $U(N)$ Chern-Simons theory on $S^3$ , and the SFF of this ensemble is proportional to the HOMFLY invariant of (2n,2)-torus links with one component in the fundamental and one in the antifundamental representation. This is one example of a large class of ensembles with intermediate statistics arising from topological field and string theories. Indeed, the absence of a local order parameter suggests that it is natural to characterize ergodic-to-nonergodic transitions using topological tools, such as we have done here.

中文翻译：

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中间统计的拓扑场论方法

随机矩阵模型提供了大量物理现象的现象学描述。突出的例子包括量子（混沌）系统的特征值统计，其特征在于光谱形状因子 (SFF)。在这里，我们使用满足 Szegö 极限定理假设的权重函数计算无限阶酉矩阵系综的 SFF。然后我们考虑一个依赖于参数的临界集成，它具有遍历到非遍历转换的中间统计特征，例如安德森定位转换。这个同一个集合是 $S^3$ 上的 $U(N)$ Chern-Simons 理论的矩阵模型，并且这个集合的 SFF 与 (2n,2)-torus 链接的 HOMFLY 不变量成正比，其中一个组件在基本面和反基本面的代表之一。这是一大类具有来自拓扑场和弦理论的中间统计数据的集合的一个例子。事实上，局部顺序参数的缺失表明使用拓扑工具来表征遍历到非遍历的转换是很自然的，就像我们在这里所做的那样。