Intermediate Singular Values of Random Matrices under Second-Moment and Anti-Concentration Assumptions

Let $A=({\xi }_{ij})$ be an $n\times n$ random matrix with independent, not necessarily identically distributed, real entries satisfying $$\mathbb{E}{\xi }_{ij}=0,\mathbb{E}{\xi }_{ij}^2=1,\underset{z\in \mathbb{R}}{\sup }\mathbb{P}(\mathrm{\mid }{\xi }_{ij}-z\mathrm{\mid }<a)\le b$$ for fixed $a>0$ and $b\in (0,1)$. We prove that, for every $\delta \in (0,1)$, there are constants $c,C>0$, depending only on $a,b,\delta$, such that $$\mathbb{P}(s_{n+1-l}(A)>Ct\frac{l}{\sqrt{n}})\le \exp \text{\hspace{0.17em}⁣}(-c\min \{tl,n\})$$ for every $t\ge 1$ and every $1\le l\le (1-\delta )n$. Thus, with no moment assumption beyond variance, all but a fixed proportion of the largest singular values satisfy the optimal upper bound of order $l\mathrm{/}\sqrt{n}$ with an exponential upper-tail estimate. Combined with the lower bound of the rectangular least singular value bound, this gives $s_{n+1-l}(A)\asymp l\mathrm{/}\sqrt{n}$ with failure probability exponentially small in $l$. The same argument gives the rectangular scale $\sqrt{N+1}-\sqrt{n-l+1}$ for $N\times n$ matrices whenever $N-n+l\le (1-\delta )N$.