发表机构
School of Mathematical Sciences, Peking University（北京大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明有限型转置相关随机矩阵在精确第四矩阈值下的强普适性，给出协方差匹配自由高斯极限，并应用于连续轮廓，指出嵌套情形阈值尖锐。
AI 中文摘要
我们证明了在精确第四矩阈值下，由独立的无序对向量原子组装而成的有限族非厄米随机矩阵的强普适性定理。在一个原子内，矩阵颜色和两个端点方向可以具有任意的联合实协方差，但需满足有序类型对上的反转一致性和同类型块中的端点可交换性；该分布也可能依赖于有限多个端点类型。这些矩阵可以附加到任意确定性元组上，该元组与类型投影联合强收敛。对于每个固定的矩阵放大和固定的非交换$*$-多项式，所得元组强收敛到一个显式的协方差匹配自由高斯族。跨类型块由掩码圆变量描述，而同类型块则分裂为独立的端点对称和端点反对称半圆扇区。仅假设对角线外具有有限的径向第四矩，而对角线上具有有限的第二矩即可。收敛模式取决于耦合方式：一个无限阵列的角在共同事件上几乎必然收敛，而固定律非嵌套三角阵列则对每个固定测试依概率收敛。作为应用，我们获得了有限可分离连续左/右轮廓的精确第四矩强极限，包括分别加权的字面转置项。对于嵌套的几乎必然公式，第四矩阈值在维格纳子族上已经是尖锐的；在此阈值下，任意新鲜行不允许对我们的依概率结论进行耦合不变的几乎必然升级。
英文摘要
We prove a strong-universality theorem at the exact fourth-moment threshold for finite families of non-Hermitian random matrices assembled from independent unordered-pair vector atoms. Within one atom, the matrix colors and the two endpoint orientations may have arbitrary joint real covariance, subject to reversal consistency across ordered type pairs and endpoint exchangeability in same-type blocks; the law may also depend on finitely many endpoint types. The matrices may be adjoined to an arbitrary deterministic tuple that converges jointly strongly with the type projections. For every fixed matrix amplification and fixed noncommutative \(*\)-polynomial, the resulting tuple converges strongly to an explicit covariance-matched free Gaussian family. Cross-type blocks are described by masked circular variables, whereas same-type blocks split into independent endpoint-symmetric and endpoint-antisymmetric semicircular sectors. Only a finite radial fourth moment is assumed off the diagonal, and a finite second moment suffices on the diagonal. The mode of convergence depends on the coupling: corners of one infinite array converge almost surely on a common event, while fixed-law nonnested triangular arrays converge in probability for each fixed test. As an application, we obtain exact-fourth-moment strong limits for finite-separable continuous left/right profiles, including separately weighted literal-transpose terms. For the nested almost-sure formulation, the fourth-moment threshold is sharp already on the Wigner subfamily; at this threshold arbitrary fresh rows admit no coupling-invariant almost-sure upgrade of our in-probability conclusion.
Comments33 pages
