Liouville quantum gravity as a mating of trees/ Bertrand Duplantier, Jason Miller and Scott Sheffield

Includes bibliographical references

Introduction -- Background and motivation -- Preliminaries -- Quantum surfaces -- SLE/GFF couplings -- Structure theorems and quantum natural time -- Welding quantum wedges -- Space-filling SLE and Brownian motion -- Trees determine embedding -- Weldings of forested wedges and duality -- Open questions -- A. Quantum disks and spheres as limits -- B. KPZ interpretation

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 C∪{∞}. In this embedding, the measure is Liouville quantum gravity (LQG) with parameter γ∈(0,2), and the curve is space-filling SLEκ′ with κ′=16/γ2.
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 SLEκ(ρ) process with κ∈(0,4). We also establish a Lévy tree description of the set of quantum disks to the left (or right) of an SLEκ′ with κ′∈(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 SLEκ′.
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.