Recent works have shown that random triangulations decorated by critical () Bernoulli site percolation converge in the scaling limit to a -Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by 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 -decorated -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 simultaneously. 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 -LQG decorated by 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 via the so-called Cardy embedding converge to -LQG.
E.G. was partially supported by a Herchel Smith fellowship and a Trinity College junior research fellowship. N.H. was partly supported by a doctoral research fellowship from the Norwegian Research Council and partly supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation. X.S. was supported by Simons Foundation as a Junior Fellow at Simons Society of Fellows and by NSF grants DMS-1811092 and by Minerva fund at Department of Mathematics at Columbia University.
We thank an anonymous referee for helpful comments on the draft.
"Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense." Electron. J. Probab. 26 1 - 58, 2021. https://doi.org/10.1214/21-EJP659