Jérémie Bettinelli

French National Centre for Scientific Research | CNRS · Institut national des sciences mathématiques et de leurs interactions (INSMI)

Fix an arbitrary compact orientable surface with a boundary and consider a uniform bipartite random quadrangulation of this surface with $n$ faces and boundary component lengths of order $\sqrt n$ or of lower order. Endow this quadrangulation with the usual graph metric renormalized by $n^{-1/4}$, mark it on each boundary component, and endow it wi...

The main purpose of this work is to provide a framework for proving that, given a family of random maps known to converge in the Gromov--Hausdorff sense, then some (suitable) conditional families of random maps converge to the same limit. As a proof of concept, we show that quadrangulations with a simple boundary converge to the Brownian disk. More...

We unify and extend previous bijections on plane quadrangulations to bipartite and quasibipartite plane maps. Starting from a bipartite plane map with a distinguished edge and two distinguished corners (in the same face or in two different faces), we build a new plane map with a distinguished vertex and two distinguished half-edges directed toward...

We unify and extend previous bijections on plane quadrangulations to bipartite and quasibipartite plane maps. Starting from a bipartite plane map with a distinguished edge and two distinguished corners (in the same face or in two different faces), we build a new plane map with a distinguished vertex and two distinguished half-edges directed toward...

We give a short proof that a uniform noncrossing partition of the regular $n$-gon weakly converges toward Aldous's Brownian triangulation of the disk, in the sense of the Hausdorff topology. This result was first obtained by Curien & Kortchemski, using a more complicated encoding. Thanks to a result of Marchal on strong convergence of Dyck paths to...

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family (BD\_L, 0\textless{}L\textless{} infinity) of random metric spaces homeomorphic to the closed unit disk of R^2, the space BD\_L being called the Brownian disk of perimeter L and unit area. These resul...

The Catalan numbers count many classes of combinatorial objects. The most emblematic such objects are probably the Dyck walks and the binary trees, and, whenever another class of combinatorial objects is counted by the Catalan numbers, it is natural to search for an explicit bijection between the latter objects and one of the former objects. In mos...

A sub-problem of the open problem of finding an explicit bijection between alternating sign matrices and totally symmetric self-complementary plane partitions consists in finding an explicit bijection between so-called $(n,k)$ Gog trapezoids and $(n,k)$ Magog trapezoids. A quite involved bijection was found by Biane and Cheballah in the case $k=2$....

We give a different presentation of a recent bijection due to Chapuy and Do\l\k{e}ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier--Di Francesco--Guitter-like generalization of the Cori--Vauquelin--Schaeffer bijection in the context of general nonorientable su...

This is a quick survey on some recent works done in the field of random maps, which, very roughly speaking, are graphs embedded without edge crossings in a surface. We present the main results and tools in this area then summarize the original contributions presented during the conference Journées MAS 2014.

This is a quick survey on some recent works done in the field of random maps.

In this work, we expose four bijections each allowing to increase (or decrease) one parameter in either uniform random forests with a fixed number of edges and trees, or quadrangulations with a boundary having a fixed number of faces and a fixed boundary length. In particular, this gives a way to sample a uniform quadrangulation with n + 1 faces fr...

We define a class a metric spaces we call Brownian surfaces, arising as the scaling limits of random maps on surfaces with a boundary and we study the geodesics from a uniformly chosen random point. These spaces generalize the well-known Brownian map and our results generalize the properties shown by Le Gall on geodesics in the latter space. We use...

We discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence $(\sigma_n)$ of integers such that $\sigma_n/\sqrt{2n}$ tends to some $\sigma\in[0,\infty]$. For every $n \ge 1$, we call $q_n$ a random map uniformly distributed over the set of all r...

We prove that a uniform rooted plane map with n edges converges in distribution after a suitable normalization to the Brownian map for the Gromov-Hausdorff topology. A recent bijection due to Ambj{\o}rn and Budd allows to derive this result by a direct coupling with a uniform random quadrangulation with n faces.

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given g, we consider, for every n ≥ 1, a random quadrangulation qn uniformly distributed over the set of all rooted bipartite quadrangulations of genus g with n faces. We view it as a metric space by endowing its set of vertices with the graph distance. We show t...

In this work, we discuss the scaling limits of two particular classes of maps. In a first time, we address bipartite quadrangulations of fixed positive genus g and, in a second time, planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We view these objects as metric spaces by endowing their sets...

Le premier problème consiste à étudier le placement quasi-optimal de parcs de panneaux solaires dans une région géographique définie. Si on suppose que la puissance d'un parc est directement proportionnelle à sa surface, on voudra maximiser la somme des surfaces des parcs. Néanmoins, la construction de tels parcs solaires est souvent soumise à des...

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every $n \ge 1$, a random quadrangulation $\q_n$ uniformly distributed over the set of all rooted bipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metric space by endowing its set of vertices with the graph dista...

Résumé Au cours de ce rapport réalisé à partir de l'article de Maxim Krikun [3] dans la cadre du cours de Grégory Miermont, nous montrons comment on peut coder l'enveloppe d'une quadrangulation par un certain processus de branchement, et nous donnons quelques appli-cations de cette méthode. Nous nous intéressons particulièrement à la frontière de l...