Eigenvalues of random p-adic matrices

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,…