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

Abstract

Let $A=({\xi }_{ij})$ be an $n\times n$ random matrix with independent, not necessarily identically distributed, real entries satisfying \[ \mathbb Eξ_{ij}=0,\qquad \mathbb Eξ_{ij}^{2}=1,\qquad \sup_{z\in\mathbb R}\mathbb P(|ξ_{ij}-z|<a)\le b \] 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 \[ \mathbb P\left( s_{n+1-l}(A)>Ct\frac l{\sqrt n} \right) \le \exp\!\left(-c\min\{tl,n\}\right) \] 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 $l/\sqrt{n}$ with an exponential upper-tail estimate. Combined with the lower bound of the rectangular least singular value bound, this gives $s_{n+1-l}(A)\asymp l/\sqrt{n}$ 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 $N-n+l\le (1-\delta )N$.

35 pages