Random Matrix Theory originated in quantum mechanics but the behavior it describes has been observed throughout mathematics, in fields as diverse as number theory, spectral theory and representation theory. I would like to argue that it should be possible to understand these statistics without direct reference to random matrices, by analyzing certain Markov chains.

I will touch upon both the archimedean and non-archimedean side (the structure theorem for abelian p-groups in one case should be viewed as the analogue of the spectral theorem in the other). I will moreover discuss the role played by the Central Limit Theorem.