In this paper, we study the smallest non-zero eigenvalue of the sample covariance matrices $\mathcal{S}(Y)=YY^*$, where $Y=(y_{ij})$ is an $M\times N$ matrix with iid mean $0$ variance $N^{-1}$ entries. We prove a phase transition for its distribution, induced by the fatness of the tail of $y_{ij}$'s. More specifically, we assume that $y_{ij}$ is symmetrically distributed with tail probability $\mathbb{P}(|\sqrt{N}y_{ij}|\geq x)\sim x^{-\alpha}$ when $x\to \infty$, for some $\alpha\in (2,4)$. We show the following conclusions: (i). When $\alpha>\frac83$, the smallest eigenvalue follows the Tracy-Widom law on scale $N^{-\frac23}$; (ii). When $2<\alpha<\frac83$, the smallest eigenvalue follows the Gaussian law on scale $N^{-\frac{\alpha}{4}}$; (iii). When $\alpha=\frac83$, the distribution is given by an interpolation between Tracy-Widom and Gaussian; (iv). In case $\alpha\leq \frac{10}{3}$, in addition to the left edge of the MP law, a deterministic shift of order $N^{1-\frac{\alpha}{2}}$ shall be subtracted from the smallest eigenvalue, in both the Tracy-Widom law and the Gaussian law. Overall speaking, our proof strategy is inspired by \cite{ALY} which is originally done for the bulk regime of the L\'{e}vy Wigner matrices. In addition to various technical complications arising from the bulk-to-edge extension, two ingredients are needed for our derivation: an intermediate left edge local law based on a simple but effective matrix minor argument, and a mesoscopic CLT for the linear spectral statistic with asymptotic expansion for its expectation.

翻译：本文研究了样本协方差矩阵 $\mathcal{S}(Y)=YY^*$ 的最小非零特征值，其中 $Y=(y_{ij})$ 是一个 $M\times N$ 矩阵，其元素 $y_{ij}$ 独立同分布，均值为 $0$，方差为 $N^{-1}$。我们证明了其特征值分布存在由 $y_{ij}$ 尾部厚度引发的相变。具体而言，假设 $y_{ij}$ 对称分布，当 $x\to\infty$ 时尾部概率满足 $\mathbb{P}(|\sqrt{N}y_{ij}|\geq x)\sim x^{-\alpha}$，其中 $\alpha\in (2,4)$。我们得到以下结论：(i) 当 $\alpha>\frac83$ 时，最小特征值在尺度 $N^{-\frac23}$ 上服从 Tracy-Widom 分布；(ii) 当 $2<\alpha<\frac83$ 时，最小特征值在尺度 $N^{-\frac{\alpha}{4}}$ 上服从高斯分布；(iii) 当 $\alpha=\frac83$ 时，分布由 Tracy-Widom 与高斯的插值表示；(iv) 当 $\alpha\leq \frac{10}{3}$ 时，除 MP 律左边缘外，还需从最小特征值中减去阶为 $N^{1-\frac{\alpha}{2}}$ 的确定性偏移，这一结论在 Tracy-Widom 律和高斯律中均成立。总体而言，我们的证明策略受 \cite{ALY} 启发，该工作最初针对 Lévy Wigner 矩阵的体区域展开。除了由体区域到边缘区域扩展带来的各种技术复杂性外，我们的推导需要两个关键要素：基于简单有效的矩阵子式论证得到的中间左边缘局部律，以及包含期望渐近展开的线性谱统计量的介观中心极限定理。