Liouville Quantum Gravity as a Mating of Trees
Nathanaël Berestycki, Ellen Powell
Abstract
In this chapter, we take forward the ideas developed in Chapter 8 and show that if one explores a γ -quantum cone via a certain space-filling SLE with parameter κ = 16/γ2 this results in a (stationary) decomposition of the cone into two independent quantum wedges, which are glued along the boundary. Furthermore, as we discover the curve, the relative changes in the boundary lengths evolve like a pair of correlated Brownian motions, where the correlation coefficient depends explicitly on the coupling constant γ (equivalently, on the parameter κ of the SLE). This gives a representation of the quantum cone as a glueing (“mating”) of two correlated continuous random trees, which is a direct continuum analogue of the results on random planar maps obtained in Chapter 4. This connection provides a rigorous justification that decorated random planar map models converge to Liouville quantum gravity in a certain precise sense. In order to explain the main results, we give an extensive description and treatment of whole-plane space-filling SLE, although we do not prove the essential but complex fact that it can be defined as a continuous curve.