Does anyone know if the problem of random matrices over (say) $\mathbf{Z}_p$ have been studied?
Here I mean something quite specific. One could do the following, namely, since $\mathbf{Z}_p$ is compact with a natural measure, look at random elements in $M_N(\mathbf{Z}_p)$ and then ask about the distribution of several obvious quantities as $N$ goes to $\infty$. For example, one can consider the rank of $M \mod p$, which translates into an elementary counting problem over $\mathbf{F}_p$. However, I don’t mean this, that would just be rubbish for my purposes. What I am looking for is something that models a random compact operator, and then I want to understand the behavior of the normalized eigenvectors as the eigenvalue
$\lambda \rightarrow 0$. To be concrete, let $B = \mathbf{Q}_p \langle T \rangle$ be the Tate algebra corresponding to the open unit ball. Then consider a “random” compact operator $U$ acting on $B$. What does random mean? This is a good question, to which I do not know the answer. But let me give several properties that it should satisfy. Because the ball $B$ is a disk, it is “dimension 2 as a real manifold”, and so — imagining that our compact operator is a $p$-adic avatar of $e^{-\nabla}$ for the Laplacian $\nabla$ — the eigenvalues of $U$ should satisfy Weyl’s Law:
$N(T):=\{ \# \lambda \ \| \ -v(\lambda) < T \} \sim \displaystyle{ \frac{\mathrm{Vol}(B)}{4 \pi}} \cdot T.$
Here $v(\lambda)$ denotes the valuation of $\lambda \in \overline{\mathbf{Q}}_p$. Ignoring the volume factor, this just means that the Fredholm determinant $\det(1 - U T)$ has a Newton Polygon with certain quadratic growth. I’m not sure exactly what ensembles one can come up with to define such operators, which is one of my questions. Let us also assume, although this may not be necessary, that $U$ is semi-simple and admits nice convergent spectral expansions. We can’t quite insist that $U$ is a self-adjoint operator, because one doesn’t have p-adic Hilbert spaces. For such an operator, what behavior should one expect of the normalized eigenvalues $\phi_j$ of $U$? For example, suppose one knows that the number of zeros of $\phi_j$ goes to infinity. What limit distribution should the zeros of $\phi_j$ satisfy when $\lambda \rightarrow 0$? (Somewhat troubling here is that the eigenvalues will lie in $\overline{\mathbf{Q}}_p$ in general and $\overline{\mathbf{C}}_p$ has compactness issues…)