There is a simple way to “glue together” a coupled pair of continuum random trees (CRTs) to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the “interface” between the trees). The authors present an explicit and canonical way to embed the sphere in \(\mathrm{C} \cup\{\infty\}\). In this embedding, the measure is a form of Liouville quantum gravity (LQG) with parameter \(\gamma \in (0,2)\), and the curve is space-filling \(\mathrm{SLE}_{\kappa'}\) with \(\kappa'=16/\gamma^{2}\).

Achieving this requires the authors to develop an extensive suite of tools for working with LQG surfaces. They explain how to conformally weld so-called “quantum wedges” to obtain new quantum wedges of different weights. They construct finite-volume quantum disks and spheres of various types and give a Poissonian description of the set of quantum disks cut off by a boundary-intersecting \(\mathrm{SLE}_{\kappa'}(\rho)\) process with \(\kappa \in (0,4)\).

The authors also establish a Lévy tree description of the set of quantum disks to the left (or right) of an \(\mathrm{SLE}_{\kappa'}\) with \(\kappa' \in(4,8)\). They show that given two such trees, sampled independently, there is a.s. a canonical way to “zip them together” and recover the \(\mathrm{SLE}_{\kappa'}\).

The law of the CRT pair the authors study was shown in an earlier paper to be the scaling limit of the discrete tree/dual-tree pair associated to an FK-decorated random planar map (RPM). Together, these results imply that FK-decorated RPM scales to CLE-decorated LQG in a certain “tree structure” topology.