We consider the assignment (or bipartite matching) problem between $n$ source points and $n$ target points on the real line, where the assignment cost is a concave power of the distance, i.e. |x − y|p, for 0 < p < 1. It is known that, differently from the convex case (p > 1) where the solution is rigid, i.e. it does not depend on p, in the concave case it may varies with p and exhibit interesting long-range connections, making it more appropriate to model realistic situations, e.g. in economics and biology. In the random version of the problem, the points are samples of i.i.d. random variables, and one is interested in typical properties as the sample size n grows. Barthe and Bordenave in 2013 proved asymptotic upper and lower bounds in the range 0 < p < 1/2, which they conjectured to be sharp. Bobkov and Ledoux, in 2020, using optimal transport and Fourier-analytic tools, determined explicit upper bounds for the average assignment cost in the full range 0 < p < 1, naturally yielding to the conjecture that a “phase transition” occurs at p = 1/2. We settle affirmatively both conjectures. The novel mathematical tool that we develop, and may be of independent interest, is a formulation of Kantorovich problem based on Young integration theory, where the difference between two measures is replaced by the weak derivative of a function with finite q-variation.
Joint work with M. Goldman (arXiv:2305.09234).[-]

In this talk I will review the recent developments on weighted distances in scale free random graphs as well as highlight key techniques used in the proofs. We consider graph models where the degree distribution follows a power-law such that the empirical variance of the degrees is infinite, such as the configuration model, geometric inhomogeneous random graphs, or scale free percolation. Once the graph is created according to the model definition, we assign i.i.d. positive edge weights to existing edges, and we are interested in the proper scaling and asymptotic distribution of weighted distances.
In the infinite variance degree regime, a dichotomy can be observed in all these graph models: the edge weight distributions form two classes, explosive vs conservative weight distributions. When a distribution falls into the explosive class, typical distances converge in distribution to proper random variables. While, when a distribution falls into the conservative class, distances tend to infinity with the model size, according to a formula that captures the doubly-logarithmic graph distances as well as the precise behaviour of the distribution of edge-weights around the origin. An integrability condition decides into which class a given distribution falls.
This is joint work with Adriaans, Baroni, van der Hofstad, and Lodewijks.[-]

Planar maps are planar graphs embedded in the sphere viewed modulo continuous deformations. There are two families of bijections between planar maps and lattice paths that are applied to prove scaling limit results of planar maps to so-called Liouville quantum gravity surfaces: metric bijections and mating-of-trees bijections. We will present scaling limit results obtained in this way, including works with Bernardi and Sun and with Albenque and Sun.[-]

In this talk, I will present recent results, obtained in collaboration with Laurent Ménard, about the geometry of spin clusters in Ising-decorated triangulations, and build on previously work obtained in collaboration with Laurent Ménard and Gilles Schaeffer.
In this model, triangulations are sampled together with a spin configuration on their vertices, with a probability biased by their number of monochromatic edges, via a parameter nu. The fact that there exists a combinatorial critical value for this model has been initially established in the physics literature by Kazakov and was rederived by combinatorial methods by Bousquet-Mélou and Schaeffer, and Bouttier, Di Francesco and Guitter.
Here, we give geometric evidence of that this model undergoes a phase transition by studying the volume and perimeter of its monochromatic clusters. In particular, we establish that, when nu is critical or subcritical, the cluster of the root is finite almost surely, and is infinite with positive probability for nu supercritical.[-]

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 results, mainly focusing on the case of pattern-avoiding permutations.
The main goal of the talk is to present a new family of universal limiting permutons, called skew Brownian permuton. This family includes (as particular cases) some already studied limiting permutons, such as the biased Brownian separable permuton and the Baxter permuton. We also show that some natural families of random constrained permutations converge to some new instances of the skew Brownian permuton.
The construction of these new limiting objects will lead us to investigate an intriguing connection with some perturbed versions of the Tanaka SDE and the SDEs encoding skew Brownian motions. We finally explain how it is possible to construct these new limiting permutons directly from a Liouville quantum gravity decorated with two SLE curves. Building on the latter connection, we compute the density of the intensity measure of the Baxter permuton.[-]

Liouville CFT is a conformal field theory developped in the early 80s in physics, it describes random surfaces and more precisely random Riemannian metrics on surfaces. We will explain, using the Gaussian multiplicative chaos, how to associate to each surface $\Sigma$ with boundary an amplitude, which is an $L^2$ function on the space of fields on the boundary of $\Sigma$ (i.e. the Sobolev space $H^{-s}(\mathbb{S}^1)^b$ equipped with a Gaussian measure, if the boundary of $\Sigma$ has $b$ connected components), and then how these amplitudes compose under gluing of surfaces along their boundary (the so-called Segal axioms).
This allows us to give formulas for all partition and correlation functions of the Liouville CFT in terms of $3$ point correlation functions on the Riemann sphere (DOZZ formula) and the conformal blocks, which are holomorphic functions of the moduli of the space of Riemann surfaces with marked points. This gives a link between the probabilistic approach and the representation theory approach for CFTs, and a mathematical construction and resolution of an important non-rational conformal field theory.
This is joint work with A. Kupiainen, R. Rhodes and V. Vargas. [-]

Motivated by the discovery of hard-to-find social networks in epidemiology, we consider the question of exploring the topology of random structures (such as a random graph G) by random walks. The usual random walk jumps from a vertex of G to a neighboring vertex, with providing information on the connected components of the graph G. The number of these connected components is the Betti number $beta_{0}$. To gather further information on the higher Betti numbers that describe the topology of the graph, we can consider the simplicial complex C associated to the graph G: a k-simplex (edge for k = 1, triangle for k = 2, tetrahedron for k = 3 etc.) belongs to C if all the lower (k-1)-simplices that constitute it also belong to C. For example, a triangle belongs to C if its three edges are in the graph G. Several random walks have already been proposed recently to explore these structures. We introduce a new random walk, whose generator is related to a Laplacian of higher order of the graph and to the Betti number betak. A rescaling of the walk for k = 2 (cycle-valued random walk), and on regular triangulation of the torus, is also detailed. We embed the space of chains into spaces of currents to establish the limiting theorem.
Joint work with T. Bonis, L. Decreusefond and Z. Zhang.
https://perso.math.u-pem.fr/tran.viet-chi/[-]

La géométrie stochastique est l'étude d'objets issus de la géométrie euclidienne dont le comportement relève du hasard. Si les premiers problèmes de probabilités géométriques ont été posés sous la forme de casse-têtes mathématiques, le domaine s'est considérablement développé depuis une cinquantaine d'années de part ses multiples applications, notamment en sciences expérimentales, et aussi ses liens avec l'analyse d'algorithmes géométriques. L'exposé sera centré sur la description des polytopes aléatoires qui sont construits comme enveloppes convexes d'un ensemble aléatoire de points. On s'intéressera plus particulièrement aux cas d'un nuage de points uniformes dans un corps convexe fixé ou d'un nuage de points gaussiens et on se focalisera sur l'étude asymptotique de grandeurs aléatoires associées, en particulier via des calculs de variances limites. Seront également évoqués d'autres modèles classiques de la géométrie aléatoire tels que la mosaïque de Poisson-Voronoi.[-]