Random matrix statistics and zeroes of $L$-functions via probability in $\lambda$-rings

We introduce a theory of probability in $\lambda$-rings designed to efficiently describe random variables valued in multisets of complex numbers, varieties over a field, or other similar enriched settings. A key role is played by the $\sigma$-moment generating function based on the plethystic exponential, which allows us to describe distributions and argue with independence in a way that is as simple as classical probability theory. As a first application, we use this theory to obtain a concise description of the asymptotic $\sigma$-moment generating functions describing distributions of eigenvalues of Haar random matrices in compact classical groups. Beyond the theory of probability in $\lambda$-rings, the proof uses only classical invariant theory. Using our description we reprove the results of Diaconis and Shahshahani on the joint distributions of traces of powers of matrices, and we also treat symmetric groups. Next, we use Poonen's sieve to establish equidistribution results for the zeroes of $L$-functions in some natural families: simple Dirichlet characters for $\mathbb{F}_q(x)$ and the vanishing cohomology of smooth hypersurface sections. We give concise descriptions of the asymptotic $\sigma$-moment generating functions in these families, then compare them to the associated random matrix distributions. These equidistribution results are sideways in that we fix $q$ and take the degree $d$ to infinity, as opposed to Deligne equidistribution for fixed $d$ as $q \rightarrow \infty$, and the large $d$-limits are related to explicit descriptions of stable homology with twisted coefficients.