Astérisque N° 429, 2021 - Grand Format

We show that the percolation exploration path for critical (p = 3/4) face percolation on a uniform random quadrangula-tion with simple boundary converges in the scaling limit to a certain curve-decorated metric measure space. Explicitly, the limiting object is SLEs on a S/3-Liouville quantum gravity (LQG) disk, or equivalently SLEs on the Brownian disk. The topology of convergence is the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces.
We also obtain analogous results for site percolation on a uniform triangulation with simple boundary. We expect that our techniques can be generalized to other variants of percolation on uniform random planar maps. Our proof proceeds by showing tightness of our percolation-decorated random quadrangulation, then showing that every possible subsequential limit must be SLE6 on 8/3-LQG. To carry out this second step, we prove that SLEs on a 8/3-LQG surface is uniquely characterized by a list of simple properties, then check that the subsequential limit must satisfy these properties.
The discrete part of the argument (involving random planar maps) is carried out in the first article of this volume, in which we show tightness and check the hypotheses of the characterization theorem. The continuum part of the argument (involving SLE and LQG) is carried out in the second article, in which we prove the characterization theorem for SLEs on 8/3-LQG. We also establish analogous characterization theorems for SLE, on ry-LQG surfaces for any a E (4,8) and 'y = E (f 2), which we expect may be useful for proving scaling limit results for other statistical mechanics models on random planar maps.