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

TL;DR

This paper establishes the scaling limits of the critical FK model on finite random planar maps, showing convergence to correlated Brownian motion and linking discrete FK loop functionals to quantum counterparts on Liouville quantum gravity surfaces.

Contribution

It provides the first finite-volume scaling limit results for the FK model on random planar maps, connecting discrete loop functionals to continuum quantum geometries.

Findings

Random walk converges to a conditioned correlated Brownian motion

FK loop boundary lengths and areas converge to quantum functionals

Finite-volume results mirror infinite-volume scaling theorems

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).