The following topic came up in conversation with my office-mate Lionel: Let $p$ be a fixed prime, $c$ a fixed positive real parameter and $n$ a large number. Consider a random $(0,1)$ matrix with entries in $Z/p$, where the probability of a $0$ is $1-\frac{c}{n}$ and that of a $1$ is $\frac{c}{n}$. As $n\rightarrow \infty$, what is the probability that this matrix is singular? UPDATE: As moonface points out, this probability is $1$. Furthermore, his argument points out that we should expect the corank to be something like $a*n$. So I'll ask more generally what we can say about the behavior of the rank as $n\rightarrow\infty$.



Actually, in our motivating example, the matrix is symmetric and the distribution on the diagonal is different than in the rest of the matrix. I left these details out, but please mention it if this point would strongly effect your answer.