In recent years, surprising connections have emerged between the theory of random permutations and random geometric objects arising in probability, statistical mechanics, and quantum physics. On one side are permutons, which describe scaling limits of large permutations; on the other are random planar maps, Schramm–Loewner evolution curves, and Liouville quantum gravity surfaces. In this talk, I will discuss some of these …
We study a natural one-parameter family of random bipolar-oriented planar maps which lies in the -Liouville quantum gravity (LQG) universality class for . For these maps, we identify exact scaling exponents for directed graph distances. Writing for the size of the map, the longest directed paths have lengths comparable to (up to constants), while the …
Last and first passage percolation in two dimensions are classical discrete models of random directed planar Euclidean metrics in the KPZ universality class. Their scaling limit is described by the directed landscape of Dauvergne-Ortmann-Virág. Random planar maps are classical discrete models of random undirected planar fractal metrics in the LQG universality class. Their scaling limit is described by the (undirected) LQG metric of Ding-Dubédat-Dunlap-Falconet …
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 …
The longest increasing subsequence of Brownian separable permutonsRead More »
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 …
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) …
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 …