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$Y=(y_{ij})$是$M\times N$矩阵，其元素独立同分布，均值为$0$，方差为$N^{-1}$。我们证明了其分布存在由$y_{ij}$尾部厚度引发的相变。具体而言，假设$y_{ij}$对称分布，且当$x\to\infty$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$N^{-\frac23}$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$N^{1-\frac{\alpha}{2}}$阶的确定性平移，这在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，该方法最初针对Lévy Wigner矩阵的谱体区域提出。除了从谱体到谱边缘扩展带来的各类技术复杂性，我们的推导还需要两个要素：基于简单而有效的矩阵子式论证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。