Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense

Ewain Gwynne, Nina Holden, Xin Sun

Research output: Contribution to journalArticlepeer-review


Recent works have shown that random triangulations decorated by critical (p = 1/2) Bernoulli site percolation converge in the scaling limit to a8/3-Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by SLE6 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 SLE6-decorated8/3-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 to8/3-LQG decorated by CLE6 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 to8/3-LQG.

Original languageEnglish (US)
Article number94
JournalElectronic Journal of Probability
StatePublished - 2021


  • 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


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