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