Critical Liouville quantum gravity and CLE$_4$

TL;DR

This paper investigates the structure of critical Liouville quantum gravity surfaces coupled with CLE$_4$ loops, revealing their conditional independence and describing the joint distribution of loop lengths via a stable process, extending prior work.

Contribution

It establishes the conditional independence of LQG disks given loop lengths and characterizes the joint law of these lengths in the critical case, using a limiting approach from subcritical cases.

Findings

Critical LQG surfaces are conditionally independent given loop lengths.

The joint distribution of loop lengths is described by a 3/2-stable process.

Results extend previous subcritical analyses to the critical case.

Abstract

Consider a critical ($\gamma=2$) Liouville quantum gravity (LQG) disk together with an independent conformal loop ensemble (CLE) with parameter $\kappa=4$. We show that the critical LQG surfaces parametrized by the regions enclosed by the CLE$_4$ loops are conditionally independent critical LQG disks given the LQG lengths of the loops. We also show that the joint law of the LQG lengths of the loops is described in terms of the jumps of a certain $3/2$-stable process. Our proofs are via a limiting argument based on the analogous statements for $\gamma \in (\sqrt{8/3},2)$ and $\kappa = \gamma^2 \in (8/3,4)$ which were proven by Miller, Sheffield, and Werner (2020). Our results are used in the construction of a coupling of supercritical LQG with CLE$_4$ in another paper by the same authors.