Résumé : We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the mating-of-trees framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to $\sqrt{8/3}$-LQG and SLE${}_6$. In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of $\sqrt{8/3}$-LQG and SLE${}_6$. For instance, we show that the exploration tree of the percolation converges to a branching SLE${}_6$, and that the collection of percolation cycles converges to the conformal loop ensemble CLE${}_6$. We also prove convergence of counting measure on the pivotal points of the percolation. Our results play an essential role in several other works, including a program for showing convergence of the conformal structure of uniform triangulations and works which study the behavior of random walk on the uniform infinite planar triangulation. (Joint work with Olivier Bernardi and Xin Sun.)

 [arXiv]

Dernière modification : Thursday 21 November 2024

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