Liouville quantum gravity as a mating of trees

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). We present an explicit and canonical way to embed the sphere in ${\mathbf C} \cup \{ \infty \}$. In this embedding, the measure is Liouville quantum gravity (LQG) with parameter $γ\in (0,2)$, and the curve is space-filling SLE$_{κ'}$ with $κ' = 16/γ^2$. Achieving this requires us to develop an extensive suite of tools for working with LQG surfaces. We explain how to conformally weld so-called "quantum wedges" to obtain new quantum wedges of different weights. We 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 SLE$_κ(ρ)$ process with $κ\in (0,4)$. We also establish a Lévy tree description of the set of quantum disks to the left (or right) of an SLE$_{κ'}$ with $κ' \in (4,8)$. We show that given two such trees, sampled independently, there is a.s. a canonical way to "zip them together" and recover the SLE$_{κ'}$. The law of the CRT pair we 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.

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