Percolation on triangulations: a bijective path to Liouville quantum gravity

Understand

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 \emph{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$_6$.