Lattice Yang-Mills theories are important models in particle physics. They are defined on the d-dimensional lattice  using a group of matrices of dimension , and Wilson loop expectations are the fundamental observables of these theories. Recently, Cao, Park, and Sheffield showed that Wilson loop expectations can be expressed as sums over certain embedded bipartite maps of any genus. Building on this novel approach, …

Lattice Yang-Mills theory in the large N limit via random surfacesRead More »

What is the behavior of the longest increasing subsequence of a uniformly random permutation? Its length is of order  plus Tracy–Widom fluctuations of order . Its scaling limit is the directed geodesic of the directed landscape.  This talk discusses how this behavior changes dramatically when one looks at universal Brownian-type permutations, i.e., permutations sampled from the Brownian separable permutons. We show that there are explicit constants such that …

Long increasing subsequences in Brownian-type permutationsRead More »

In 1912 Henri Poincaré asked the following simple question: “In how many different ways a simple loop in the plane, called a meander, can cross a line a specified number of times?” Despite many efforts, this question remains very open after more than a century. In this talk, I will present the conjectural scaling limit …

Meanders and Meandric SystemsRead More »

Random geometry and random permutations have been extremely active fields of research for several years. The former is characterized by the study of large planar maps and their continuum limits, i.e. the Brownian map, Liouville quantum gravity surfaces and Schramm–Loewner evolutions. The latter is characterized by the study of large uniform permutations and (more recently) …

Permutations in Random GeometryRead More »

In 1912 Henri Poincaré asked the following question: “In how many different ways a simple loop in the plane, called meander, can cross a line a specified number of times?” Despite many efforts, this question remains open after more than a century. In this talk we construct and study the conjectural scaling limit of uniform …

Permutons, meanders, and SLE-decorated Liouville quantum gravityRead More »

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 introducing the notion of permuton. Permuton convergence has been established for several models of random permutations in various works: we give an overview of some of these …

The skew Brownian permuton: a new universal limit for random constrained permutations and its connections with Liouville quantum gravityRead More »

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 introducing the notion of permutons. Permuton convergence has been established for several models of random permutations in various works: we give an overview of some of these results, …

The skew Brownian permutonRead More »

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 investigated in the last decade. In the first part of the talk, we introduce a new …

Local limits for permutations and generating treesRead More »

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 …

Random permutations: A geometric point of viewRead More »

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

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 …

The feasible region for consecutive patterns of permutations is a cycle polytopeRead More »

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 …

Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processesRead More »

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 …

Phase transition for almost square permutationsRead More »

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 …

An approach through random walks to local limits of permutations encoded by generating treesRead More »

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 …

Square permutations are typically rectangularRead More »

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, …

What is… a permuton?Read More »

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 …

Local convergence for random permutations: the case of uniform pattern-avoiding permutations.Read More »