Characterizations of SLE$_κ$ for $κ\in (4,8)$ on Liouville quantum gravity

Ewain Gwynne, Jason Miller

Published 2017-01-18Version 1

We prove that SLE$_\kappa$ for $\kappa \in (4,8)$ on an independent $\gamma=4/\sqrt{\kappa}$-Liouville quantum gravity (LQG) surface is uniquely characterized by the form of its LQG boundary length process and the form of the conditional law of the unexplored quantum surface given the explored curve-decorated quantum surface up to each time $t$. We prove variants of this characterization for both whole-plane space-filling SLE$_\kappa$ on a $\gamma$-quantum cone (which is the setting of the peanosphere construction) and for chordal SLE$_\kappa$ on a single bead of a $\frac{3\gamma}{2}$-quantum wedge. Using the equivalence of Brownian and $\sqrt{8/3}$-LQG surfaces, we deduce that SLE$_6$ on the Brownian disk is uniquely characterized by the form of its boundary length process and that the complementary connected components of the curve up to each time $t$ are themselves conditionally independent Brownian disks given this boundary length process. The results of this paper will be used in another work of the authors to show that the scaling limit of percolation on random quadrangulations is given by SLE$_6$ on $\sqrt{8/3}$-LQG with respect to the Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces.