Rank deficiency of random matrices

Abstract: Let $M_n$ be a random $n\times n$ matrix with i.i.d. $\text{Bernoulli}(1/2)$ entries. We show that for fixed $k\ge 1$, [\lim_{n\to \infty}\frac{1}{n}\log_2\mathbb{P}[\text{corank }M_n\ge k] = -k.]