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]

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