Résumé : This talk focuses on classes of planar maps with a weight $u>0$ on certain components called \emph{blocks}. In collaboration with Fleurat, we study the decomposition of generic planar maps into $2$-connected components, revealing a phase transition between the universality classes of maps (converging to the Brownian sphere) and plane trees (converging to the Brownian tree), depending on the value of $u$. We identify a new class with the stable tree of parameter $3/2$ as the scaling limit in the critical case, and obtain precise results on block sizes in each phase. In a subsequent work, I show that it is possible to study many decomposition schemes along similar lines to shed light on a phase transition. I explain how to obtain enumerative results, block sizes and scaling limits for each phase. Finally, with Albenque and Fusy, we studied tree-rooted random planar maps decomposed into tree-rooted $2$-connected blocks, where a spanning tree is drawn simultaneously with the map. This model, which is of interest in theoretical physics, shows new behaviours. We determine the asymptotic behaviour of $2$-connected tree-rooted maps, reveal a phase transition, and study the properties of each phase.

 [Slides.pdf] [vidéo] [arXiv]

Dernière modification : Thursday 27 March 2025

Contact pour cette page : Cyril.Banderier at lipn.univ-paris13.fr