# Talks

## Random permutations: A geometric point of view

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 limiting shape of random permutations: an introduction to permuton convergence.

In this series of two lectures we overview some recent progress in the study of the liming shape of large random (non uniform) permutations.  We start by properly introducing the notion of permuton convergence and by exploring its connection with the convergence of proportion of pattern densities, this being a striking feature of the permuton topology.  In …

## The feasible region for consecutive patterns of permutations is a cycle polytope

We study proportions of consecutive occurrences of permutation patterns of a given size . Specifically, the feasible limits of such proportions on large permutations form a region, called feasible region. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called overlap graph. The latter is a …

## Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes

Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant are well-known to be related to each other through several bijections. In order to study their scaling and local limits, we introduce a further new family of discrete objects, called coalescent-walk processes and we relate them with the other previously …

## Phase transition for almost square permutations

A record in a permutation is an entry which is either bigger or smaller than the entries either before or after it (there are four types of records). Entries which are not records are called internal points. We explore scaling limits (called permuton limits) of uniform permutations in the classes Sq(n,k) of almost square permutations of size n+k with exactly k internal points.We …

## An approach through random walks to local limits of permutations encoded by generating trees

We develop a general technique to study local limits of random permutations in families enumerated through generating trees. We propose an algorithm to sample uniform permutations in such families using random walks and we recover the local behavior of permutations studying local properties of the corresponding random walks. The method applies to families of permutations …

## Square permutations are typically rectangular

We describe the limit (for two topologies) of large uniform random square permutations, i.e., permutations where every point is a record.  First we describe the global behavior by showing these permutations have a permuton limit which can be described by a random rectangle.  We also explore fluctuations about this random rectangle, which we can describe through coupled Brownian motions.  Second, we …

## What is… a permuton?

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, …

## Local convergence for random permutations: the case of uniform pattern-avoiding permutations.

For large combinatorial structures, two main notions of convergence can be defined: scaling limits and local limits. In particular for graphs, both notions are well-studied and well-understood. For permutations only a notion of scaling limits, called permutons, has been recently introduced. The convergence for permutons has also been characterized by frequencies of pattern occurrences. We …