Loosely speaking, a Markov chain is said to be integrable if there are simple, explicit formulas that describe the action of its transition matrix on a rich class of column vectors. These formulas usually arise from some underlying algebraic structure but allow one to study analytic properties of the chain.

In the first part of this talk, I will discuss some natural Markov chains on permutations and their integrability. We will see that this integrability allows one to effortlessly move into the limit, constructing a diffusion on the space of permutons, and then analyze various properties of this diffusion.

In the second part of this talk, I will discuss how this entire operation can be generalized to arbitrary state spaces and a certain class of dynamics.