Quantitative estimates of the singular values of random i.i.d. matrices

Let $M$ be an $n\times n$ random i.i.d. matrix. This paper studies the deviation inequality of $s_{n-k+1}(M)$, the $k$-th smallest singular value of $M$. In particular, when the entries of $M$ are subgaussian, we show that for any $\gamma\in (0, 1/2), \varepsilon>0$ and $\log n\le k\le c\sqrt{n}$ \begin{align} \textsf{P}\{s_{n-k+1}(M)\le \frac{\varepsilon}{\sqrt{n}} \}\le \Big( \frac{C\varepsilon}{k}\Big)^{\gamma k^{2}}+e^{-c_{1}kn}.\nonumber \end{align} This result improves an existing result of Nguyen, which obtained a deviation inequality of $s_{n-k+1}(M)$ with $(C\varepsilon/k)^{\gamma k^{2}}+e^{-cn}$ decay.