The paper “Modeling λ-invariants by p-adic random matrices,” with Akshay Venkatesh and Sonal Jain, just got accepted by Comm. Pure. Appl. Math. But I forgot to blog about it when we finished it! (I was a little busy at the time with the change in my personal circumstances.)
Anyway, here’s the idea. As I’ve already discussed here, one heuristic for the Cohen-Lenstra conjectures about the p-rank of the class group of a random quadratic imaginary field K is to view this p-part as the cokernel of g-1, where g is a random generalized symplectic matrix over Z_p. In the new paper, we apply the same philosophy to the variation of the Iwasawa p-adic λ-invariant.
The p-adic λ-invariant of a number field K is closely related to the p-rank of the class group of K; in fact, Iwasawa theory more or less gets started from the theorem that the p-rank of the class group of 
for some constants 
So in the paper we do two things. On the one hand, we study the dimension of the generalized 1-eigenspace of a random generalized symplectic matrix, and from this we derive the following conjecture: for each p > 2 and r >= 0, the probability that a random quadratic imaginary field K has p-adic λ-invariant r is
Note that this decreases like 
The second part of the paper tests this conjecture numerically, and finds fairly good agreement with the data. A novelty here is that we compute p-adic λ-invariants of K for small p and large disc(K); previous computational work has held K fixed and considered large p. It turns out that you can do these computations reasonably efficiently by interpolation; you can compute special values L(s,chi_K) transcendentally for many s; given a bunch of these values, determined to a certain p-adic precision, you can compute the initial coefficients of the p-adic L-function with some controlled p-adic precision as well, and, in particular, you can provably locate the first coefficient which is nonzero mod p. The location of this coefficient is precisely the λ-invariant. This method shows that, indeed, large λ-invariants do pop up! For instance, the 3-adic λ-invariant of 
Some questions still floating around:
- Should one expect an upper bound
for each odd p? Certainly such a bound is widely expected for the p-rank of the class group.
- In the experiments we did, the convergence to the conjectural asymptotic appears to be from below. For the 3-ranks of class groups of quadratic imaginary fields, this convergence from below was conjectured by Roberts to be explained by a secondary main term with negative coefficient. Roberts’ conjecture was proved this year — twice! Bhargava, Shankar, and Tsimerman gave a proof along the lines of Bhargava’s earlier work (involving thoughful decompositions of fundamental domains into manageable regions, and counting lattice points therein) and Thorne and Taniguchi have a proof along more analytic lines, using the Shintani zeta function. Anyway, one might ask (prematurely, since I have no idea how to prove the main term correct!) whether the apparent convergence from below for the statistics of the λ-invariant is also explained by some kind of negative secondary term.
Quomodocumque 
Hey Jordan, thanks for the shoutout. I forget if I pointed this out earlier, but there is an interesting related paper of Bob Hough. (I don’t see how to embed a link — but it is the unique paper by that name on the arXiv.) He gets a “proof” of Roberts’ conjecture for 3-torsion in imaginary quadratic fields, albeit with a too-big error term. Although his methods don’t apply to cubic fields or to 3-torsion in real quadratic fields, they do apply to k-torsion for odd k > 3! Moreover, he is working on getting the distribution in arithmetic progressions (where screwball behavior appears in the secondary term). He is unlikely to get good enough error terms to prove anything along these lines, but his heuristics along with some PARI/GP data seem likely to offer some convincing conjectures.