Mating of trees for random planar maps and Liouville quantum gravity: a survey

TL;DR

This survey reviews the mating-of-trees framework connecting random planar maps, Liouville quantum gravity, and SLE curves, highlighting its applications in understanding scaling limits, dimensions, and symmetries of these models.

Contribution

It summarizes the theory and applications of mating-of-trees bijections, emphasizing their role in linking discrete maps with continuum LQG and SLE, and discusses recent advances and results.

Findings

Scaling limits for decorated random planar maps

Hausdorff dimensions of SLE-related sets

Equivalence of 9-LQG with the Brownian map

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.