Convergence of percolation on uniform quadrangulations with boundary to SLE$_{6}$ on $\sqrt{8/3}$-Liouville quantum gravity

TL;DR

This paper proves that the percolation exploration path on random quadrangulations converges to SLE$_6$ on Liouville quantum gravity, establishing a key link between discrete models and their continuous scaling limits.

Contribution

It demonstrates the convergence of percolation interfaces on random quadrangulations to SLE$_6$ on Liouville quantum gravity, extending the understanding of scaling limits in random planar maps.

Findings

Percolation exploration paths converge to SLE$_6$ on quantum disks.

Results apply to face and site percolation on various random maps.

Method extends to models explored via peeling techniques.

Abstract

Let $Q$ be a free Boltzmann quadrangulation with simple boundary decorated by a critical ($p=3/4$) face percolation configuration. We prove that the chordal percolation exploration path on $Q$ between two marked boundary edges converges in the scaling limit to chordal SLE$_6$ on an independent $\sqrt{8/3}$-Liouville quantum gravity disk (equivalently, a Brownian disk). The topology of convergence is the Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. We also obtain analogous scaling limit results for face percolation on the uniform infinite half-plane quadrangulation with simple boundary, and for site percolation on a uniform triangulation with simple boundary. Our method of proof is robust and, up to certain technical steps, extends to any percolation model on a random planar map which can be explored via peeling.