## Abstract

Recent works have shown that random triangulations decorated by critical (p = 1/2) Bernoulli site percolation converge in the scaling limit to a^{√}8/3-Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by SLE_{6} in two different ways: • The triangulation, viewed as a curve-decorated metric measure space equipped with its graph distance, the counting measure on vertices, and a single percolation interface converges with respect to a version of the Gromov–Hausdorff topology. • There is a bijective encoding of the site-percolated triangulation by means of a two-dimensional random walk, and this walk converges to the correlated two-dimensional Brownian motion which encodes SLE_{6}-decorated^{√}8/3-LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield (2014); this is sometimes called peanosphere convergence. We prove that one in fact has joint convergence in both of these two senses simul-taneously. We also improve the metric convergence result by showing that the map decorated by the full collection of percolation interfaces (rather than just a single interface) converges to^{√}8/3-LQG decorated by CLE_{6} in the metric space sense. This is the first work to prove simultaneous convergence of any random planar map model in the metric and peanosphere senses. Moreover, this work is an important step in an ongoing program to prove that random triangulations embedded into C via the so-called Cardy embedding converge to^{√}8/3-LQG.

Original language | English (US) |
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Article number | 94 |

Journal | Electronic Journal of Probability |

Volume | 26 |

DOIs | |

State | Published - 2021 |

## Keywords

- Brownian map
- Cardy embedding
- Conformal loop ensemble
- Liouville quantum gravity
- Mating of trees
- Peanosphere
- Percolation
- Schramm-Loewner evolution
- Uniform triangulations

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty