Random integral matrices and the Cohen-Lenstra heuristics

We prove that given any $\epsilon>0$, random integral $n\times n$ matrices with independent entries that lie in any residue class modulo a prime with probability at most $1-\epsilon$ have cokernels asymptotically (as $n\rightarrow\infty$) distributed as in the distribution on finite abelian groups that Cohen and Lenstra conjecture to be the distribution for class groups of imaginary quadratic fields. This shows the Cohen-Lenstra distribution is universal for finite abelian groups given by generators and random relations---that the distribution of quotients does not depend on the way in which we choose (sufficiently nice) relations. This is a refinement of a result on the distribution of ranks of random matrices with independent entries in ${\Bbb Z}/ p{\Bbb Z}$. This is interesting especially in light of the fact that these class groups are naturally cokernels of square matrices. We also prove the analogue for $n\times (n+u)$ matrices.