Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{ -1}$, \begin{ align*} { \mathbb P}\big\{ \mbox{ $M_n$ is singular}\big\}&=(1+o_n(1)){ \mathbb P}\big\{ \mbox{ $M_n$ contains a zero row or column}\big\}\\ &=(2+o_n(1))n\,(1-p)^n, \end{ align*} where $o_n(1)$ denotes a quantity which converges to zero as $n\to\infty$. We provide the corresponding upper and lower bounds on the smallest singular value of $M_n$ as well.
Categories: math.PR