Scaling limits of critical FK-decorated random planar maps with $q=4$

TL;DR

This work delivers the first rigorous scaling limit for FK($q$)-decorated planar maps at the critical value $q=4$, resolving a long-standing open problem. By exploiting a novel exact correspondence with the fully packed loop-$O(2)$ model on triangulations, the authors derive an explicit partition-function formula and sharp asymptotics that feed into geometric observables such as typical loops, clusters, and envelopes. The core achievement is showing that the burger-count process $\mathcal{S}_n$ scales diffusively while the discrepancy $\mathcal{D}_n$ acquires a logarithmic correction, converging to an independent Brownian motion after an appropriate rescaling: $\left(\frac{\mathcal{S}_{\lfloor nt \rfloor}}{\sqrt{n}}, \frac{\log n}{2\pi\sqrt{n}}\mathcal{D}_{\lfloor nt \rfloor}\right)_{t\in\mathbb{R}} \Rightarrow (B^1_t, B^2_t)_{t\in\mathbb{R}}$. These results, together with the loop/CLE$_4$ and critical LQG identifications in the peanosphere sense, provide a rigorous bridge between FK-decorated maps at criticality and the continuum theories of CLE$_4$ and $\gamma=2$ LQG, marking a significant advance in the understanding of universality classes for random planar maps. The work also establishes exact (logarithmic) geometric exponents for typical loops and clusters at criticality, underscoring the integrable structure of the model. Overall, the paper advances both probabilistic techniques and conformal-field-theory-inspired predictions for random geometry at a pivotal critical point.

Abstract

We establish the first scaling limit for FK($q$)-weighted planar maps in the critical case $q=4$, resolving a problem that has remained open since Sheffield's seminal work arXiv:1108.2241. In that work, Sheffield proved a scaling limit for $q<4$ via the celebrated hamburger-cheeseburger bijection, which initiated the peanosphere (mating-of-trees) approach to Liouville quantum gravity. We prove that, at criticality, the associated burger count $\mathcal{S}$ and discrepancy $\mathcal{D}$ satisfy \[ \left(\frac{\mathcal{S}_{\lfloor nt \rfloor}}{\sqrt{n}}, \frac{\log(n)}{{2π}\sqrt{n}} \mathcal{D}_{\lfloor nt \rfloor}\right)_{t\in\mathbb{R}} \stackrel{\text{d}}{\longrightarrow} (B^1_t, B^2_{t})_{t\in\mathbb{R}}, \] where $B^1$ and $B^2$ are independent two-sided Brownian motions. To the best of our knowledge, no conjecture for the correct discrepancy scaling factor had previously been formulated. Matching the limiting process with the critical mating of trees arXiv:2109.00275, we establish the first rigorous planar map convergence towards CLE$_4$ and critical ($γ=2$) Liouville quantum gravity, in the peanosphere sense. Our proof is based on a novel approach that reveals the exactly solvable nature of the model through a correspondence with the (bicoloured) fully packed loop-$O(2)$ model on triangulations, and yields critical geometric exponents matching the predictions of conformal field theory.

Figures (10)

• Figure 1: Numerical simulations of the hamburger-cheeseburger trajectory for different values of $q$. We took $N=10^6$ in the inventory model and represented the burger count and discrepancy, both rescaled by $\sqrt{N}$. For $q>4$ (right), the discrepancy collapses to $0$, marking a phase transition. This paper is concerned with the critical case $q=4$ (middle), where we show that a logarithmic correction emerges.
• Figure 2: Illustration of the Tutte map. (a) A planar map $\mathfrak{m}$ together with an FK percolation configuration $\mathfrak{p}$ (blue), with an oriented root edge. Closed edges (i.e. edges in $\mathfrak{m}\setminus \mathfrak{p}$) are in pale blue. We draw the dual vertices in red, each corresponding to a face of $\mathfrak{m}$, and connect each dual vertex to its adjacent primal vertices (dashed). The root triangle is the triangle to the right of the root edge (grey). (b) We draw the dual edges of $\mathfrak{p}^\dagger$ (red) between dual vertices. The Tutte map $T(\mathfrak{m},\mathfrak{p})$ is the triangulation that arises from considering all the edges (blue, red and dashed). The interface between primal and dual components corresponds to loops (purple).
• Figure 3: The Mullin--Bernardi--Sheffield bijection, applied to the map in \ref{['fig:Tutte']}. We construct the space-filling exploration (purple) by starting from the loop crossing the root triangle and opening up the other connected components -- the corresponding "fictional" edges that were flipped along the procedure are dotted. The precise word that corresponds to the pair $(\mathfrak{m},\mathfrak{p})$ is $W = \mathsf{h} \mathsf{c} \mathsf{H} \mathsf{h} \mathsf{h} \mathsf{c} \mathsf{H} \mathsf{c} \mathsf{H} \mathsf{C} \mathsf{F} \mathsf{h} \mathsf{h} \mathsf{h} \mathsf{H} \mathsf{C} \mathsf{H} \mathsf{F}$.
• Figure 4: A portion of the infinite FK planar map. Primal (resp. dual) elements are represented in blue (resp. red). In grey is the triangle corresponding to time $0$ in the hamburger-cheeseburger encoding, which is here assumed to correspond to an $\mathsf{F}$ (we drew the corresponding fictional edge in dotted blue). This $\mathsf{F}$ symbol determines a typical loop$\mathfrak{L}(0)$ (purple). We denote by $\mathfrak{c}(0)$ the connected component (in bold) inside the loop and refer to it as the typical cluster, in this case a primal (blue) component. The outside of $\mathfrak{c}(0)$ in this drawing is identified as a single root face for $\mathfrak{c}(0)$. It is readily checked that the perimeter of $\mathfrak{c}(0)$, i.e. the degree of this root face, is $|\partial \mathfrak{c}(0)| = 9$. The envelope$\mathfrak{e}(0)$ is the planar map in shaded yellow (together with the portion of the grey triangle that it includes), with its root face lying outside the yellow region. Note that it has simple boundary.
• Figure 5: A glimpse of the hamburger-cheeseburger exploration (purple) of the planar map in Figure \ref{['fig:outer-bdry']}. The exploration enters through a fictional triangle corresponding to an $\mathsf{F}$-excursion of type $\mathsf{h}$ (the fictional edge is drawn in dotted blue). When drawing the contour of the blue component, the purple exploration gets diverted and fills in some pockets (shaded purple regions) -- we did not represent these pocket diversions.

Theorems & Definitions (85)

• Theorem 1.1: Main result: scaling limit of critical hamburger-cheeseburger walk
• Theorem 1.2: Variance estimate
• Theorem 1.3: Loop and cluster exponents
• Theorem 1.4: Exact expression and asymptotics of $F_\ell$
• Theorem 2.1: Sheffield, SheffieldScott2016QGAI
• Proposition 2.2: Reduced walk expressions of $|\mathfrak{L}(0)|$ and $|\partial\mathfrak{c}(0)|$
• proof
• Lemma 2.3: Skeleton words are clusters
• proof
• Proposition 3.1: Explicit expression of $\hat{\rho}$ given $(A,B)$