Liouville quantum gravity as a mating of trees

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 C ∪ {∞}. In this embedding, the measure is Liouville quantum gravity (LQG) with parameter γ ∈ (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 ∈ (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 ∈ (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.

Regularity of the SLE$_4$ uniformizing map and the SLE$_8$ trace

“…Namely, if one draws the pair of paths (η 1 , η 2 ) on top of an independent LQG surface called a weight-4 quantum cone, then the quantum surfaces parameterized by the two components of C \ (η 1 ∪ η 2 ) are independent quantum wedges of weight 2. This result was proved for κ < 4 previously in [9]; the purpose of Section 3 is to establish the case κ = 4. If we let ψ be a conformal transformation which takes the component of C \ (η 1 ∪ η 2 ) which is to the left of η 2 to the strip S = R × (0, π) sending 0 to −∞, ∞ to +∞, then there is an explicit description for the field which describes the surface parameterized by S .…”

“…It has since been proved to arise as a scaling limit in several cases [47,26,43,48]. SLE has also been the subject of intensive study as it has deep connections to the Gaussian free field (GFF) [44,8,32] and Liouville quantum gravity (LQG) [46,9].…”

“…The strategy taken in the works [40,29,19,20,15,5,41,1] is based on estimating the derivatives of either the solution to (1.1) or its time-reversal. The main results we will establish in this article will be based on a completely different method, in particular making use of the relationship between certain types of SLE κ curves and LQG [46,9].…”

“…We will collect a number of preliminaries in Section 2. In Section 3 we establish a version of one of the welding results from [9] in the critical case that κ = 4 and in Section 4 we prove bounds for exit times for SLE when it is parameterized by quantum length. We will prove the main estimates which will be used to prove our results in Section 5.…”

“…We will now give an overview of the strategy to prove our main theorems. The description which follows assumes the reader has some basic familiarity with the welding results of [46,9]; we describe these results in more detail in Section 2. We begin with the case κ = 4.…”