How does a large random permutation behave? We will try to answer this question for different classical models of random permutations, such as uniform permutations, pattern-avoiding permutations, Mallows permutations and many others. An appropriate framework to describe the asymptotic behaviour of these pemutations is to use a quite recent notion of scaling limits for permutations, called permutons. We will investigate some first interesting examples of these objects, like the Brownian separable permuton.
Permutons lead to some nice connections between probability theory and combinatorics, and during the talk, we will investigate some of them.
In the last part, we will also present a new and complementary recent local limit approach for the study of large random permutations introduced by the speaker. Indeed, permutons are appropriate to describe the ”global shape” of permutations but not the ”finer details”. These are on the contrary encoded by local limits.
ETH Zürich (Switzerland), Zurich Graduate Colloquium.