In this chapter, we describe a model of random planar maps weighted by self-dual Fortuin-Kasteleyn (FK) percolation. This can be thought of as a canonical discretisation of Liouville quantum gravity. We start with some generalities about planar maps and then introduce the FK random map model, which depends on a parameter $q\ge 0$, before explaining the conjectured connection to Liouville quantum gravity. A fundamental tool for studying such random planar maps is Sheffield’s (hamburger-cheeseburger) bijection. We first explain it carefully for tree-decorated maps (the special case of the FK model of planar maps with $q=0$), which correspond under this bijection to random walk excursions in the quarter-plane. We then explain its generalisation to $q\ge 0$ in detail. This is first used to show that the maps possess an infinite volume limit in the local topology. Then, a theorem of Sheffield gives a scaling limit result for these maps. One consequence is that a phase transition takes place at $q=4$. Furthermore, it allows one to compute some associated critical exponents when $q<4$ (which are consistent with the KPZ relation of Chapter 3). These arguments are a discrete analogue of the “mating of trees” perspective on Liouville quantum gravity described in Chapter 9.

In this chapter, we take forward the ideas developed in Chapter 8 and show that if one explores a $\gamma$-quantum cone via a certain space-filling SLE with parameter $\kappa =16/{\gamma }^2$ 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 $\gamma$ (equivalently, on the parameter $\kappa$ 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.