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

Source abstract

Let $A=(ξ_{ij})$ 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| 0$ and $b\in(0,1)$. We prove that, for every $δ\in(0,1)$, there are constants $c,C>0$, depending only on $a,b,δ$, 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 $t\ge1$ and every $1\le l\le(1-δ)n$. 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 $N-n+l\le(1-δ)N$.