  Let $ A_n $ be an $n \times n$ random matrix with i.i.d Bernoulli($p$)
entries. For a fixed positive integer $\beta$, suppose $p$ satisfies $$ \frac{
\log(n) }{ n } \le p \le c_\beta $$ where $c_\beta \in ( 0, 1/2 )$ is a
$\beta$-dependentvalue. For $t \ge 0$, $$
  \mathbb{P} \left\{
  s_{ n - \beta + 1}(A)
  \le
  t n^{-2\beta + o_\mathfrak{n}(1) }(pn)^{-7}
  \right\}
  =
  t +
  ( 1 + o_\mathfrak{n}(1) ) \mathbb{P} \bigg\{
  \mbox{either $\beta$ rows or $\beta$ columns of $A_n$ equal $\vec{0}$}
\bigg\}. $$
