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-c/n and that of a 1 is c/n. As n--> 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 -> 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.