Consider a large random permutation satisfying some constraints or biased according to some statistics. What does it look like? In this seminar we make sense of this question by presenting the notion of permuton convergence. Then we answer the question for different choices of random permutation models.

We mainly focus on two examples of permuton convergence, introducing the “Brownian separable permuton” and the “Baxter permuton”. We also discuss several connections between various limiting permutons and other probabilistic objects, such as the Continuum Random Tree and the coalescent flows of some perturbed versions of the Tanaka SDE.

At the end of the talk we present some conjectures on a new universal limit for random permutations called the "skew Brownian permuton".