I always felt there ought to be a good theory of p-adic random matrices paralleling the very-well-worked out real and complex story. I wrote a paper years ago with Jain and Venkatesh that proposed that a very reasonable model for the lambda-invariant of a random quadratic field would be the number of eigenvalues x of a large random matrix A over Z_p such that x-1 is not a unit. (The p-primary part of the class group, the quantity studied in the Cohen-Lenstra conjecture, is modeled by the cokernel of A-1, which is clearly related but not at all the same thing.

Anyway, we worked out a few simple statistics of random matrices over Z_p but got no further. Now Jiahe Shen and Roger van Peski have done much more! They are well on their way to a fully satisfying theory that mimics the archimedean story. For instance, they can exactly compute the pair correlation showing the familiar repulsion of eigenvalues from each other. And the pair correlation function has a somewhat mysterious (to me, and I think to them too) appearance of a theta function…

One reason we like to model arithmetic quantities by random matrices is that, in the function field case, Frobenius acts on etale cohomology via a p-adic matrix we know nothing about, and since we know nothing about it, we like to treat it as random in Haar measure. Is that justified? Maybe not, but here’s the good news: Shen has a followup paper where he shows (again, consistent with what happens in archimedean-land) that the eigenvalue statistics they find don’t depend sensitively on the exact distribution of the matrices, but rather are universal over a rather broad class of distributions. So that, I think, should make one feel more confident that Frobenii, too, have eigenvalues that look this way, and moreover that the analogous distributions in the number field case, where there’s no literal p-adic matrix at all, might follow suit.

The method here is interesting: we learned, over time, that a good way to get a handle on a random group G is to study the expected number of surjections from G to H, as H varies over groups (or groups in various special classes) — these are the natural moments of a distribution on groups. And in practice, they are often the exact statistics one is able to compute by geometric means. (See Will Sawin and Melanie Wood’s work for a much, much, much more elaborated categorical point of view on what counts as a moment.) Shen’s paper takes the view that a good notion of “moment” for a random p-adic polynomial P is the distribution of the p-adic valuation of the resultant of P with some fixed polynomial Z. These “Z-moments” turn out to also be very computable for characteristic polynomials of random matrices, and they also turn out to provide actionable information about eigenvalues. Nice!