The smallest singular value for rectangular random matrices with Lévy entries

Abstract

Let $X=(x_{ij})\in {\mathbb{R}}^{N\times n}$ be a rectangular random matrix with i.i.d. entries (we assume $N/n\to \mathbf{a}>1$), and denote by $\ddot{I}{ƒ}_{min}(X)$ its smallest singular value. When entries have mean zero and unit second moment, the celebrated work of Bai-Yin and Tikhomirov show that $n^{-\frac{1}{2}}\ddot{I}{ƒ}_{min}(X)$ converges almost surely to $\sqrt{\mathbf{a}}-1.$ However, little is known when the second moment is infinite. In this work we consider symmetric entry distributions satisfying $\mathbb{P}(|x_{ij}|>t)\sim t^{-\hat{I}\pm }$ for some $\hat{I}\pm \in (0,2)$, and prove that $\ddot{I}{ƒ}_{min}(X)$ can be determined up to a log factor with high probability: for any $D>0$, with probability at least $1-n^{-D}$ we have $$C_1{n}^{\frac{1}{\hat{I}}\pm }(\log n{)}^{\frac{2(\hat{I}\pm -2)}{\hat{I}}}\pm \le \ddot{I}{ƒ}_{min}(X)\le C_2{n}^{\frac{1}{\hat{I}}\pm }(\log n{)}^{\frac{\hat{I}\pm -2}{2\hat{I}\pm }}$$ for some constants $C_1,C_2>0$. The upper bound was derived in a recent work of Bao, Lee and Xu \cite{bao2024phase2} but the lower bound is new and answers a problem posed in that paper in a weaker form. This appears to be the first determination of $\ddot{I}{ƒ}_{min}(X)$ in the $\hat{I}\pm$-stable case with a correct leading order of $n$, as previous anti-concentration arguments only yield lower bound $n^{\frac{1}{2}}$. The same lower bound holds for $\ddot{I}{ƒ}_{min}(X+B)$ for any fixed rectangular matrix $B$ with no assumption on its operator norm. The case of diverging aspect ratio is also computed.

27 pages. Changed the exponent of log(n) and improved presentation