Liouville quantum gravity as a mating of trees

TL;DR

This paper establishes a canonical way to construct Liouville quantum gravity surfaces with space-filling SLE curves by gluing continuum random trees, developing tools for conformal welding, and connecting discrete models to continuum limits.

Contribution

It introduces a new method to embed LQG surfaces with SLE curves via mating of trees, and develops tools for conformal welding and quantum surface construction.

Findings

Constructed quantum disks and spheres of various types.

Established a Poissonian description of quantum disks cut by SLE.

Connected discrete random planar maps to continuum LQG surfaces.

Abstract

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 $\gamma \in (0,2)$, and the curve is space-filling SLE$_{\kappa'}$ with $\kappa' = 16/\gamma^2$. 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$_{\kappa}(\rho)$ process with $\kappa \in (0,4)$. We also establish a L\'evy tree description of the set of quantum disks to the left (or right) of an SLE$_{\kappa'}$ with $\kappa' \in (4,8)$. We show that given two such trees, sampled independently, there is a.s. a canonical way to "zip them together" and recover the SLE$_{\kappa'}$. 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.