Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times

Understand

Sheffield (2011) introduced an inventory accumulation model which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model.

• He showed that a certain two-dimensional random walk associated with an infinite-volume version of the model converges in the scaling limit to a correlated planar Brownian motion.
• We improve on this scaling limit result by showing that the times corresponding to FK loops (or "flexible orders") in the inventory accumulation model converge in the scaling limit to the $\pi/2$-cone times of the correlated Brownian motion.
• This statement implies a scaling limit result for the joint law of the areas and boundary lengths of the bounded complementary connected components of the FK loops on the infinite-volume planar map.