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

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.

arXiv comment70 pages, 12 figures. Minor changes. To appear in Panoramas et Synthèses

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• Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes
• The skew Brownian permuton: a new universality class for random constrained permutations
• Local convergence of large random triangulations coupled with an Ising model
• Random Permutations -- A geometric point of view
• Liouville quantum gravity surfaces with boundary as matings of trees
• Conformal welding of quantum disks
• Probabilistic conformal blocks for Liouville CFT on the torus
• Tensor Field Theories: Renormalization and Random Geometry
• Mating of trees for critical Liouville quantum gravity
• Subdivergence-free gluings of trees
• Environment seen from infinite geodesics in Liouville Quantum Gravity
• Hamiltonian cycles on bicolored random planar maps
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