AI 中文总结
研究证明随机矩阵理论的通用局部点过程由循环方程层次结构刻画，如正弦β点过程是体循环方程唯一解，艾里β点过程是边缘循环方程唯一解，此为通用性提供直接途径，许多模型中方程可由局部定律和分部积分得出。
AI 中文摘要
我们证明了随机矩阵理论的通用局部点过程由其循环方程层次结构所刻画。具体而言，对于每个有理β>0，正弦β点过程是体循环方程层次结构的唯一解，而艾里β点过程是边缘循环方程层次结构的唯一解。这些唯一性结果为通用性提供了直接途径：只需验证系综的相应近似循环方程即可。在许多模型中，这些方程可由局部定律和分部积分得出。
英文摘要
We prove that the universal local point processes of random matrix theory are characterized by their loop equation hierarchies. More precisely, for every rational $β>0$, the $\mathrm{Sine}_β$ point process is the unique solution of the bulk loop equation hierarchy, and the $\mathrm{Airy}_β$ point process is the unique solution of the edge loop equation hierarchy. These uniqueness results provide a direct route to universality: it suffices to verify the corresponding approximate loop equations for the ensemble. In many models, these equations follow from local laws and integration by parts.
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