Scaling limits for the critical Fortuin-Kastelyn model on a random planar map III: finite volume case

Abstract

We prove scaling limit results for the finite-volume version of the inventory accumulation model of Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. In particular, we prove that the random walk associated with the finite-volume version of this model converges in the scaling limit to a correlated Brownian motion $\dot{Z}$ conditioned to stay in the first quadrant for two units of time and satisfy $\dot{Z}(2)=0$. We also show that the times which describe complementary connected components of FK loops in the discrete model converge to the $\pi /2$-cone times of $\dot{Z}$. Combined with recent results of Duplantier, Miller, and Sheffield, our results imply that many interesting functionals of the FK loops on a finite-volume FK planar map (e.g. their boundary lengths and areas) converge in the scaling limit to the corresponding "quantum" functionals of the CLE${}_{\kappa }$ loops on a $4/\sqrt{\kappa }$-Liouville quantum gravity sphere for $\kappa \in (4,8)$. Our results are finite-volume analogues of the scaling limit theorems for the infinite-volume version of the inventory accumulation model proven by Sheffield (2011) and Gwynne, Mao, and Sun (2015).

67 pages, 7 figures

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