Characterizations of SLE$_{\kappa}$ for $\kappa \in (4,8)$ on Liouville quantum gravity
Abstract
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 curvedecorated quantum surface up to each time $t$. We prove variants of this characterization for both wholeplane spacefilling SLE$_\kappa$ on an infinitevolume LQG surface and for chordal SLE$_\kappa$ on a finitevolume LQG surface with boundary. 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 are used in another paper by the same 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 GromovHausdorffProkhorovuniform topology, the natural analog of the GromovHausdorff topology for curvedecorated metric measure spaces.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 DOI:
 10.48550/arXiv.1701.05174
 arXiv:
 arXiv:1701.05174
 Bibcode:
 2017arXiv170105174G
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Mathematics  Complex Variables
 EPrint:
 79 pages, 9 figures