Astérisque, n° 427. 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 spacefilling curve (which describes the « interface » between the trees). We present an explicit and canonical way to embed the sphere in C U {∞}. In this embedding, the measure is a form of Liouville quantum gravity (LQG) with parameter γϵ (0, 2), and the curve is space-filling SLE<sub>K'</sub> with K' = 16 / γ<sup>2</sup>.
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<sub>K</sub>(ρ) process with K ϵ (0,4). We also establish a Levy tree description of the set of quantum disks to the left (or right) of an SLE<sub>K'</sub> with K' ϵ (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<sub>K'</sub>.
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.
