Rank of Sparse Bernoulli Matrices

Abstract: 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 $t \ge 0$, $$ \mathbb{P} \left{ s_{ n - \beta + 1}(A) \le t n^{-2\beta + \mathfrak{n}(1) }(pn)^{-7} \right} = t + ( 1 + o_\mathfrak{n}(1) ) \mathbb{P} \bigg{ \mbox{either $\beta$ rows or $\beta$ columns of $A_n$ equal $\vec{0}$} \bigg}. $$