Supercritical Liouville quantum gravity and CLE$_4$

TL;DR

This work establishes the first rigorous link between CLE$_4$ and supercritical Liouville quantum gravity (LQG) on the disk for ${\mathbf c}_{\mathrm L}\in(1,25)$, introducing a canonical supercritical LQG disk and showing a coupling with CLE$_4$ such that LQG pieces inside CLE$_4$ loops are conditionally independent given their boundary lengths.A nested CLE$_4$ coupling is developed, yielding a Markov-type structure, and a dual description links ${\mathbf c}_{\mathrm L}$ with $26-{\f c}_{\mathrm L}$ via an orthogonal rotation of underlying fields; the inner vs outer LQG boundary lengths along CLE$_4$ loops are shown to differ by a random multiplicative factor.The paper also describes explicit boundary-length laws via a $3/2$-stable process and proposes a loop-decorated random planar map model (O(2) fully packed loop model) with a gasket-based scaling-limit conjecture toward CLE$_4$-decorated supercritical LQG disks, along with a broad program of open problems and Liouville-action interpretations.

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 $γ$ with $|γ|=2$. 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 $κ=4$ (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.

Figures (2)

• Figure 1: Illustration of the statement of Theorem \ref{['thm-main-intro']}. The precise version of the theorem (Theorem \ref{['thm-cle4-coupling']}) says the following. Consider LQG surfaces obtained by restricting the field $\Phi$ to the regions $U_\ell$ enclosed by the CLE$_4$ loops, then viewing the pairs $(U_\ell,\Phi|_{U_\ell})$ modulo $Q$-LQG coordinate change, where $Q \in (0,2)$ is as in \ref{['eqn-lqg-parameters']} (see Definition \ref{['def-lqg-surface']}). Then these LQG surfaces are conditionally independent supercritical LQG disks given their boundary lengths.
• Figure 2: The iterative procedure for sampling a planar map decorated by the fully packed $O(2)$ loop model. (1) The initial gasket $G$. (2) For each $f\in\mathcal{F}(G)$, we sample a ring $R$ decorated by a loop $\ell$ and identify the outer boundary of the ring with the inner boundary of $f$ (the inner face of each ring is shown in light yellow). Then, we sample another gasket with outer boundary length equal to the inner boundary length of $\ell$. (3) We identify the outer boundary of each second-generation gasket with the inner boundary of its corresponding ring. Note that two edges on the inner boundary of one of the rings get identified to the same multiplicity-two edge on the boundary of one of the gaskets. (4) We iterate the procedure inside each face of the second-generation gaskets. In this case, each of the second-generation rings has zero inner boundary length, so the process terminates and the resulting loop-decorated planar map is a sample from \ref{['eqn-planar-map-O(2)']}.

Theorems & Definitions (38)

• Definition 1.1
• Theorem 1.2: Main result, informal statement
• Remark 1.3
• Remark 1.4: Physics motivation
• Definition 2.1
• Definition 2.2
• Definition 2.3: Critical LQG disk
• Definition 2.4: Standard embedding
• Remark 2.5
• Lemma 2.6