### Journée-séminaire de combinatoire

#### (équipe CALIN du LIPN, université Paris-Nord, Villetaneuse)

Le 13 septembre 2022 à 14h00 en B107 & visioconférence, Zéphyr Salvy nous parlera de : Random planar maps decomposed into blocks: a phase study

Résumé : Maps come with different shapes, such as trees or triangulations with many more edges. Many classes of maps have been enumerated (2-connected maps, trees, quadrangulations...), notably by Tutte, and a phenomenon of universality has been demonstrated: for the majority of them, the number of elements of size $n$ in the class has an asymptotic of the form $\kappa\, \rho^{-n} \, n^{-5/2}$, for a certain $\kappa$ and a certain $\rho$. Nevertheless, there are classes of degenerate'' maps whose behaviour is similar to that of trees, and whose number of elements of size $n$ has an asymptotic of the form $\kappa\, \rho^{-n} \, n^{-3/2}$, as for example outerplanar maps. This dichotomy of behaviour is not only observed for enumeration, but also for metrics. Indeed, in the tree'' case, the distance between two random vertices is in $\sqrt{n}$, against $n^{1/4}$ for uniform planar maps of size $n$. This work focuses on what happens between these two very different regimes. We highlight a model depending on a parameter $u \in \mathbb{R}^*_+$ which exhibits the expected behaviours, and a transition between the two: depending on the position of $u$ with respect to $u_C$, the behaviour is that of one or the other universality class. More precisely, we observe a subcritical'' regime where the scale limit of the maps is the Brownian map, a supercritical'' regime where it is the Brownian tree and finally a critical regime where it is the $3/2$ stable tree.

[Slides.pdf] [vidéo]

 Dernière modification : Wednesday 12 October 2022 Contact pour cette page : Cyril.Banderier at lipn.univ-paris13.fr