[Paper Review] Mating of trees for random planar maps and Liouville quantum gravity: a survey

Ewain Gwynne, Nina Holden+1 more|arXiv (Cornell University)|Oct 10, 2019

Stochastic processes and statistical mechanics160 references45 citations

TL;DR

This survey presents mating-of-trees bijections for decorated random planar maps and the continuum mating-of-trees theorem linking Liouville quantum gravity (LQG) with SLE to a pair of correlated Brownian motions, and surveys their many applications.

ABSTRACT

We survey the theory and applications of mating-of-trees bijections for random planar maps and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield (2014). The latter theorem gives an encoding of a Liouville quantum gravity (LQG) surface decorated by a Schramm-Loewner evolution (SLE) curve in terms of a pair of correlated linear Brownian motions. We assume minimal familiarity with the theory of SLE and LQG. Mating-of-trees theory enables one to reduce problems about SLE and LQG to problems about Brownian motion and leads to deep rigorous connections between random planar maps and LQG. Applications discussed in this article include scaling limit results for various functionals of decorated random planar maps, estimates for graph distances and random walk on (not necessarily uniform) random planar maps, computations of the Hausdorff dimensions of sets associated with SLE, scaling limit results for random planar maps conformally embedded in the plane, and special symmetries for $\sqrt{8/3}$-LQG which allow one to prove its equivalence with the Brownian map.

Motivation & Objective

• Motivate and explain the theory of mating-of-trees and its continuum counterpart.
• Review discrete bijections that encode decorated planar maps as two paired trees and their scaling limits.
• Illustrate how the continuum mating-of-trees theorem encodes a gamma-LQG surface with an SLE curve via correlated Brownian motion.
• Survey applications to convergence, dimensions, embeddings, and special symmetries, and outline open problems.

Proposed method

• Describe discrete Mullin bijection for spanning-tree decorated maps and its contour-walk encoding.
• Describe Bernardi–Holden–Sun bijection for site-percolated loopless triangulations and the associated boundary-length process.
• Define the Peano curve and the corresponding two-coordinate contour walk that encodes decoration.
• State and outline the continuum mating-of-trees theorem of Duplantier–Miller–Sheffield (DMS21) relating gamma-LQG and SLE to a correlated Brownian motion.
• Discuss variants: disk and ordinary SLE cases, and conformal welding of quantum wedges.
• Summarize applications including convergence results, mated-CRT maps, and embeddings.

Experimental results

Research questions

• RQ1 How can decorated random planar maps be encoded by a mating-of-trees framework and what is the continuum analog?
• RQ2 What is the exact continuum encoding of a gamma-LQG surface decorated by SLE in terms of correlated Brownian motions?
• RQ3 What are the implications of the mating-of-trees theorem for scaling limits, graph distances, and Hausdorff dimensions in LQG/SLE settings?
• RQ4 How do discrete bijections correspond to continuum objects like peanospheres and mated-CRT maps, and what are their applications to embeddings and dimensions?
• RQ5 What open problems remain in the interplay between random planar maps, LQG, and SLE?

Key findings

• A continuum encoding exists for a gamma-LQG surface decorated by an SLE curve via a two-dimensional Brownian motion with correlation -cos(pi gamma^2/4).
• Mating-of-trees bijections in the discrete setting converge to the continuum mating-of-trees framework, linking random maps to LQG.
• The theory underpins convergence results for random planar maps toward LQG and justifies conformal embeddings and scaling limits.
• Applications include bounds on graph distances, random walk behavior on decorated maps, and Hausdorff dimension computations for SLE-related sets.
• Special symmetries at gamma = sqrt(8/3) yield equivalence with the Brownian map in certain frameworks.
• The survey outlines extensions to finite-volume and infinite-volume settings and connects to mated-CRT maps and Tutte embeddings.