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：本文研究样本协方差矩阵 $\mathcal{S}(Y)=YY^*$ 的最小非零特征值，其中 $Y=(y_{ij})$ 为 $M\times N$ 矩阵，其元素独立同分布且均值为 $0$、方差为 $N^{-1}$。我们证明该特征值的分布存在由 $y_{ij}$ 尾部厚度诱导的相变现象。具体地，假设 $y_{ij}$ 具有对称分布，其尾部概率满足 $\mathbb{P}(|\sqrt{N}y_{ij}|\geq x)\sim x^{-\alpha}$（当 $x\to \infty$），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 $N^{-\frac23}$ 尺度上服从 Tracy-Widom 律；(ii) 当 $2<\alpha<\frac83$ 时，最小特征值在 $N^{-\frac{\alpha}{4}}$ 尺度上服从高斯律；(iii) 当 $\alpha=\frac83$ 时，分布由 Tracy-Widom 律与高斯律的插值给出；(iv) 当 $\alpha\leq \frac{10}{3}$ 时，除 MP 律的左边缘外，需从最小特征值中减去一个 $N^{1-\frac{\alpha}{2}}$ 阶的确定性偏移，该偏移同时适用于 Tracy-Widom 律和高斯律。总体而言，我们的证明策略受启发于 \cite{ALY} 中针对 Lévy Wigner 矩阵谱体区域的研究。除从谱体到谱边扩展带来的技术复杂性外，两个关键要素支撑我们的推导：一是基于简单有效的矩阵子式论证建立的中间左边缘局部律，二是用于线性谱统计量的介观中心极限定理及其期望的渐近展开。