用拓扑方法揭示随机矩阵的谱特征，提出新诊断工具
Persistence diagrams of random matrices via Morse theory: universality and a new spectral diagnostic
- 通过莫尔斯理论将矩阵谱间距映射为拓扑条带长度
- 推导出高斯正交系综的持久熵公式，数值验证收敛速度为n^{-0.6}
- 持久熵比传统间距比更优，可检测传统方法忽略的全局扰动
我们证明了在单位球面S^{n-1}上限制二次型f(x) = x^T M x的子水平集滤链的持久图谱完全由对称矩阵M的特征值决定。根据莫尔斯理论，该图谱恰好包含n-1个有限条带，第k个条带位于同调维数k-1，其长度等于第k个特征值间距s_k = λ_{k+1} - λ_k。这一对应将随机矩阵理论（RMT）的普适性转化为持久图谱的普适性：对于高斯正交系综（GOE）矩阵，我们导出了闭式持久熵公式PE = log(8n/π) - 1，数值验证持久统计量的变异系数以n^{-0.6}速率衰减。不同随机矩阵系综（GOE、GUE、Wishart）产生具有区分性的普适持久图谱，构成RMT普适类的拓扑指纹。作为实际应用，我们证明持久熵在区分GOE与GUE矩阵时优于标准间距比⟨r⟩（n=100时AUC分别为0.978与0.952，非重叠置信区间），且能检测⟨r⟩无法察觉的罗森茨魏格-波特模型中的全局谱扰动。这些结果确立持久熵作为捕捉现有RMT工具互补信息的新谱诊断工具。
原文摘要 · Abstract (English)
We prove that the persistence diagram of the sublevel set filtration of the quadratic form f(x) = x^T M x restricted to the unit sphere S^{n-1} is analytically determined by the eigenvalues of the symmetric matrix M. By Morse theory, the diagram has exactly n-1 finite bars, with the k-th bar living in homological dimension k-1 and having length equal to the k-th eigenvalue spacing s_k = λ_{k+1} - λ_k. This identification transfers random matrix theory (RMT) universality to persistence diagram universality: for matrices drawn from the Gaussian Orthogonal Ensemble (GOE), we derive the closed-form persistence entropy PE = log(8n/π) - 1, and verify numerically that the coefficient of variation of persistence statistics decays as n^{-0.6}. Different random matrix ensembles (GOE, GUE, Wishart) produce distinct universal persistence diagrams, providing topological fingerprints of RMT universality classes. As a practical consequence, we show that persistence entropy outperforms the standard level spacing ratio \langle r \rangle for discriminating GOE from GUE matrices (AUC 0.978 vs. 0.952 at n = 100, non-overlapping bootstrap 95% CIs), and detects global spectral perturbations in the Rosenzweig-Porter model to which \langle r \rangle is blind. These results establish persistence entropy as a new spectral diagnostic that captures complementary information to existing RMT tools.
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