Emmanuel Kowalski pointed me to a very interesting recent paper by Kenneth Maples, a grad student at UCLA working under Terry Tao. One heuristic justification for the Cohen-Lenstra conjectures, due to Friedman and Washington, relies on the remarkable fact that if M is a random nxn matrix in M_n(Z_p), the distribution of coker(M) among finite abelian p-groups approaches a limit as n goes to infinity; so it makes sense to talk about “the cokernel of a large random matrix” without specifying the size. (There’s a fuller discussion of Friedman-Washington in this old post.)

Maples shows that the requirement that M is random — that is, that the entries of M are independently drawn from Z_p with additive Haar measure – is much stronger than necessary. In fact, he shows that when the entries of M are drawn independently from *any* distribution on Z_p satisfying a mild non-degeneracy condition, the distribution of coker(M) converges to the so-called Cohen-Lenstra distribution, as in Friedman-Washington. That’s pretty cool! I don’t know any arithmetic circumstance that would naturally produce exotic distributions of this kind, but the result helps bolster one’s psychological sense that the Cohen-Lenstra distribution provides the only sensible notion of “cokernel of random matrix,” in some robust sense.

Universality of random matrix laws is a very active and fast-moving topic, but Maples’ result is the first universality result for p-adic matrices that I know of. More generally, I think there’s a lot to be gained by understanding how well the richly developed theory of random large real and complex matrices carries over to the p-adic case.

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M_n or GL_n (which is what you say in your older post)? The Haar measures are rather different!

In the case of Maples’ paper, M_n. If I remember correctly, Friedman and Washington treat both the case of a random matrix in M_n and (g-1) where g is random in GL_n, and find that in the limit they have the same cokernel statistics.