Percolation on triangulations: a bijective path to Liouville quantum gravity

TL;DR

This paper establishes a bijective encoding of percolated planar triangulations that links discrete models to Liouville quantum gravity and SLE, demonstrating convergence of key percolation functionals to continuous random objects.

Contribution

It introduces a novel bijection between percolated triangulations and lattice paths, connecting discrete models to LQG and SLE frameworks, and proves convergence results for percolation functionals.

Findings

Percolation exploration tree converges to branching SLE_6

Percolation cycles converge to CLE_6

Counting measure on pivotal points converges to a continuous limit

Abstract

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$_6$. 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.