Supercritical Liouville quantum gravity and CLE$_4$

Abstract

We establish the first relationship between Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which corresponds to central charge values $\mathbf c_{\mathrm L} \in (1,25)$ or equivalently to complex values of $\gamma$ with $|\gamma|=2$. More precisely, we introduce a canonical supercritical LQG surface with the topology of the disk. We then show that for each $\mathbf c_{\mathrm L} \in (1,25)$ there is a coupling of this LQG surface with a conformal loop ensemble with parameter $\kappa=4$ (CLE$_4$) wherein the LQG surfaces parametrized by the regions enclosed by the CLE$_4$ loops are conditionally independent supercritical LQG disks given their boundary lengths. In this coupling, the CLE$_4$ is neither determined by nor independent from the LQG. Guided by our coupling result, we exhibit a combinatorially natural family of loop-decorated random planar maps whose scaling limit we conjecture to be the supercritical LQG disk coupled to CLE$_4$. We include a substantial list of open problems.