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Sheffield (2011) introduced an inventory accumulation model which encodes arandom planar map decorated by a collection of loops sampled from the criticalFortuin-Kasteleyn (FK) model. He showed that a certain two-dimensional randomwalk associated with an infinite-volume version of the model converges in thescaling limit to a correlated planar Brownian motion. We improve on thisscaling limit result by showing that the times corresponding to FK loops (or"flexible orders") in the inventory accumulation model converge in the scalinglimit to the $\pi/2$-cone times of the correlated Brownian motion. Thisstatement implies a scaling limit result for the joint law of the areas andboundary lengths of the bounded complementary connected components of the FKloops on the infinite-volume planar map. In light of the encoding ofDuplantier, Miller, and Sheffield (2014), the limiting object coincides withthe joint law of the areas and boundary lengths of the bounded complementaryconnected components of a collection of CLE$_\kappa$ loops on an independentLiouville quantum gravity cone.

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