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 ${\hat{I}}^2$, suppose $p$ satisfies $$\frac{\log (n)}{n}\le p\le c_{\hat{I}}^2$$ where $c_{\hat{I}}^2\in (0,1/2)$ is a ${\hat{I}}^2$-dependentvalue. For $t\ge 0$, $$ \mathbb{P} \left\{ s_{ n - β+ 1}(A) \le t n^{-2β+ \mathfrak{n}(1) }(pn)^{-7} \right\} = t + ( 1 + o_\mathfrak{n}(1) ) \mathbb{P} \bigg\{ \mbox{either $β$ rows or $β$ columns of $A_n$ equal $\vec{0}$} \bigg\}. $$

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