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? What can you say about the probability distribution of its rank?

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 a small number of entries with a different distribution would strongly effect your answer.