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

Understand

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.