发表机构
Northeast Normal University（东北师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究非中心样本协方差矩阵，在不要求可交换性下证明最优尺度局部定律、特征值刚性、离域性及边缘普适性，通过矩阵Dyson方程的稳定性分析实现。
AI 中文摘要
我们考虑实非中心样本协方差矩阵 $\mathcal{W}=YY^\top$，其中 $Y=A+\Sigma^{1/2}X$。这里 $A\in\mathbb{R}^{M\times N}$ 是确定性的，$\Sigma\in \mathbb{R}^{M\times M}$ 是确定性的正定总体协方差矩阵，$X\in\mathbb{R}^{M\times N}$ 具有独立的中心化条目，方差为 $N^{-1}$。我们在不要求 $AA^\top$ 与 $\Sigma$ 可交换的情况下，证明了在最优谱尺度下正则右边缘附近的局部定律。作为推论，我们获得了最右正则边缘处的最优特征值刚性以及相应左右奇异向量的离域性。我们还证明，以高概率，在相邻谱间隙中，超出最优 $N^{-2/3}$ 边缘尺度（至多允许任意小的 $N^\varepsilon$ 损失）没有特征值。最后，我们建立了最右正则边缘的边缘普适性：在中心化和缩放之后，最大特征值收敛到 Tracy--Widom 分布。主要技术成分是对与 $Y$ 的线性化相关的矩阵 Dyson 方程（MDE）的稳定性分析，其自能算子不满足一般 MDE 理论的平坦性条件。利用特殊的块结构，我们将稳定性分析精确地约化为一个二维算子。这种约化产生了谱密度的正则性和正则右边缘处的平方根行为，以及这些边缘附近的尖锐稳定性界。
英文摘要
We consider the real noncentral sample covariance matrices $\mathcal{W}=YY^\top$ with $Y=A+Σ^{1/2}X$. Here $A\in\mathbb{R}^{M\times N}$ is deterministic, $Σ\in \mathbb{R}^{M\times M}$ is a deterministic positive definite population covariance matrix and $X\in\mathbb{R}^{M\times N}$ has independent centered entries with variance $N^{-1}$. We prove local laws near regular right edges down to optimal spectral scales without requiring the commutativity of $AA^\top$ and $Σ$. As a consequence, we obtain optimal eigenvalue rigidity at the rightmost regular edge and delocalization of the corresponding left and right singular vectors. We also show that, with high probability, there are no eigenvalues in the adjacent spectral gap beyond the optimal $N^{-2/3}$ edge scale, up to an arbitrarily small $N^\varepsilon$ loss. Finally, we establish edge universality at the rightmost regular edge: after centering and scaling, the largest eigenvalue converges to the Tracy--Widom distribution. The main technical ingredient is a stability analysis of the matrix Dyson equation (MDE) associated with the linearization of $Y$, whose self-energy operator does not satisfy the flatness condition of the general MDE theory. Exploiting the special block structure, we reduce the stability analysis exactly to a two-dimensional operator. This reduction yields regularity of the spectral density and square-root behavior at regular right edges, together with sharp stability bounds near such edges.
