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

Ewain Gwynne, Cheng Mao+1 more|arXiv (Cornell University)|Feb 2, 2015

Stochastic processes and statistical mechanicsMathematics21 citations

TL;DR

This paper establishes a scaling limit for the critical Fortuin-Kasteleyn (FK) model on a random planar map by showing that the times corresponding to FK loops—encoded via an inventory accumulation model—converge to the $π/2$-cone times of a correlated planar Brownian motion. The key result links the joint distribution of loop areas and boundary lengths in the FK model to that of CLE$_\kappa$ loops on a Liouville quantum gravity cone.

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 $π/2$-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$_κ$ loops on an independent Liouville quantum gravity cone.

Motivation & Objective

• To establish a scaling limit for the critical Fortuin-Kasteleyn model on a random planar map using an inventory accumulation model.
• To identify the limiting behavior of times corresponding to FK loops (flexible orders) in the model.
• To connect the joint law of loop areas and boundary lengths in the FK model to conformal loop ensembles on Liouville quantum gravity.
• To extend Sheffield's earlier scaling limit result by refining the convergence to include cone times of the Brownian motion.

Proposed method

• Uses an inventory accumulation model with five symbols (burgers and orders) to encode random planar maps decorated with FK loops.
• Applies a bijection (generalizing Mullin's and Berzunza's work) to relate the model to planar maps with a root edge and edge set $S$.
• Analyzes the random walk $Z^n = (U^n, V^n)$ derived from the word $X^n$, where $U^n$ and $V^n$ track burger and order counts.
• Employs the Dynkin-Lamperti theorem and regular variation to study the asymptotic behavior of times with no burgers or orders.
• Uses Skorokhod coupling and tightness arguments to prove convergence of rescaled times to cone times of correlated Brownian motion.
• Applies probabilistic estimates and path regularity conditioning to control the behavior of the walk under conditioning on no burgers or orders.

Experimental results

Research questions

• RQ1 How do the times corresponding to FK loops in the inventory accumulation model behave in the scaling limit?
• RQ2 What is the limiting distribution of the joint law of loop areas and boundary lengths in the critical FK planar map?
• RQ3 How do the cone times of the correlated Brownian motion arise as scaling limits of FK loop times?
• RQ4 What is the connection between the FK model on random planar maps and the conformal loop ensemble on Liouville quantum gravity?
• RQ5 Can the convergence of loop times be established under conditioning on the absence of burgers or orders in the word?

Key findings

• The rescaled times corresponding to FK loops converge in distribution to the $\pi/2$-cone times of a correlated planar Brownian motion.
• The limiting joint law of loop areas and boundary lengths matches that of a collection of $\operatorname{CLE}_\kappa$ loops on a Liouville quantum gravity cone with $\gamma = 4/\sqrt{\kappa}$.
• The convergence holds under conditioning on the absence of burgers in the word, ensuring path regularity at large times.
• The proof relies on tightness and Skorokhod coupling to establish joint convergence of the rescaled walk and its cone times.
• The result extends Sheffield's earlier scaling limit by identifying the precise limiting times associated with FK loops.
• The convergence is established for all $a \in \mathbb{Q}_{>0}$ and $r \in \mathbb{Q}_{>0}$, with explicit convergence of rescaled times $\overline{\tau}_n^{a,r}$ to $\overline{\tau}^{a,r}$.