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

TL;DR

This paper refines the understanding of the scaling limits of FK model loops on random planar maps, showing convergence of loop times to cone times of a correlated Brownian motion, linking to CLE and Liouville quantum gravity.

Contribution

It demonstrates that FK loop times in the inventory model converge to cone times of a correlated Brownian motion, enhancing previous scaling limit results.

Findings

FK loop times converge to $rac{C0}{2}$-cone times of Brownian motion

Joint law of areas and boundary lengths of FK loops matches CLE$_BA$ on quantum cones

Results connect FK model scaling limits to CLE and Liouville quantum gravity

Abstract

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. In light of the encoding of Duplantier, Miller, and Sheffield (2014), the limiting object coincides with the joint law of the areas and boundary lengths of the bounded complementary connected components of a collection of CLE$_\kappa$ loops on an independent Liouville quantum gravity cone.