Liouville quantum gravity as a mating of trees

By Bertrand Duplantier, Jason Miller, Scott Sheffield. 2014

239 citations recorded by the source corpus.

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.

View Paper: https://www.semanticscholar.org/paper/51d5acbbfaef66519d725fbdceadf924cd322151

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Citations in the corpus

• Liouville Quantum Gravity on the Riemann Sphere (2014, background): “To sum up, in what follows, we will formally understand the measure (2.5) as the image of the product measure P ⊗ dc on H − 1 ( R 2 , ˆ g ) × R by the mapping ( X g , c ) 7→ X g + c , where dc is the Lebesgue measure on R and X g has the law of a GFF X g with vanishing λ g -mean, no matter the choic”