Percolation on uniform quadrangulations and SLE6 ON 8/3 under square root -Liouville Quantum gravity/ Ewain Gwynne and Jason Miller

Includes bibliographical references

We show that the percolation exploration path for critical (p = 3/4) face percolation on a uniform random quadrangulation with simple boundary converges in the scaling limit to a certain curve-decorated metric measure space. Explicitly, the limiting object is SLE6 on a 8/3under square root-Liouville quantum gravity (LQG) disk, or equivalently SLE6 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 8/3under square root-LQG. To carry out this second step, we prove that SLE6 on 8/3under square root-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 SLE6 on 8/3under square root-LQG. We also establish analogous characterization theorems for SLEκ on γ-LQG surfaces for any κ ∈ (4, 8) and γ = 4/ √ κ ∈ ( √ 2, 2), which we expect may be useful for proving scaling limit results for other statistical mechanics models on random planar maps.