Critical behaviour of the fully packed loop-$O(n)$ model on planar triangulations

TL;DR

The work bridges analytic combinatorics and probabilistic bijections to analyze the fully packed loop-O(n) model on planar triangulations and its self-dual FK(q) counterpart. By building a dictionary between Sheffield's hamburger-cheeseburger framework and gasket-based resolvent methods, the authors derive an exact partition function F_ℓ, establish its critical asymptotics, and obtain precise loop/cluster perimeter tails in the FK(q) maps. They solve the self-dual resolvent equation via Wiener–Hopf factorisation, obtaining explicit endpoint γ_+ and density ρ(y), then translate these results into sharp exponents for typical loops and clusters. Overall this work confirms planar-map analogues of classic lattice predictions (Nienhuis-type) and rigorously justifies an ansatz central to analytic combinatorics of loop models, advancing understanding of criticality and scaling in random planar maps decorated by loop ensembles.

Abstract

We study the fully packed loop-$O(n)$ model on planar triangulations. This model is also bijectively equivalent to the Fortuin--Kasteleyn model of planar maps with parameter $q\in (0,4)$ at its self-dual point. These have been traditionally studied using either techniques from analytic combinatorics (based in particular on the gasket decomposition of Borot, Bouttier and Guitter arXiv:1106.0153) or probabilistic arguments (based on Sheffield's hamburger-cheeseburger bijection arXiv:1108.2241). In this paper we establish a dictionary relating quantities of interest in both approaches. This has several consequences. First, we derive an exact expression for the partition function of the fully packed loop-$O(n)$ model on triangulations, as a function of the outer boundary length. This confirms predictions by Gaudin and Kostov. In particular, this model exhibits critical behaviour, in the sense that the partition function exhibits a power-law decay characteristic of the critical regime at this self-dual point. This can be thought of as the planar map analogue of Nienhuis' predictions for the critical point of the loop-$O(n)$ model on the hexagonal lattice. Finally, we derive precise asymptotics for geometric features of the FK model of planar maps when $0 < q <4$, such as the exact tail behaviour of the perimeters of clusters and loops. This sharpens previous results of arXiv:1502.00450 and arXiv:1502.00546. A key step is to use the above dictionary and the probabilistic results to justify rigorously an ansatz commonly assumed in the analytic combinatorics literature.

Figures (4)

• Figure 1: Construction of the Tutte map. (a) The planar map $\mathfrak{m}$, with its oriented root edge. (b) We take the set of open edges $\omega$ (in blue) to be all the edges but one, and draw its dual $\omega^{\dagger}$ (in red). Then, we draw an edge (dashed) between each face (i.e. dual vertex) and any incident primal vertex. Considering only the dashed edges gives the quadrangulation $Q(\mathfrak{m})$, and keeping all the (blue, red and black) edges gives the Tutte triangulation $T(\mathfrak{m},\omega)$. (c) We colour the triangles blue and red according to the type of their (unique) coloured edge. We did not colour the infinite triangular face of the map (exterior to the drawing) which is also a blue triangle. The root triangle (dark blue) is the triangle to the right of the root edge of $T(\mathfrak{m},\omega)$. We draw the loops separating primal and dual components of the maps in purple.
• Figure 2: The Mullin--Bernardi--Sheffield bijection, applied to the map in \ref{['fig:Tutte']}. We start from the loop crossing the root triangle in \ref{['fig:Tutte']}(c). Then, we enter unvisited components recursively using the following rule: flip the edge of the last traversed triangle whose companion triangle is not visited by the exploration. In the present case, we only flip primal (blue) edges to dual (red) edges; these edges are drawn in dotted line on the picture. Finally, we read the word from the space-filling exploration: every triangle with a solid edge corresponds to either $\mathsf{h},\mathsf{H}$ (blue) or $\mathsf{c},\mathsf{C}$ (red) depending on whether it is the first/second time that the associated quadrangle is visited. If the triangle has an edge in dotted line (i.e. the edge has been flipped in the aforementioned procedure), we replace the order by an $\mathsf{F}$. We stress that there is a red (fictional) triangle outside of the picture that we did not represent in the drawing. The exploration shown in the figure corresponds to the word $w=\mathsf{h}\mathsf{h}\mathsf{h}\mathsf{h}\mathsf{h}\mathsf{c}\mathsf{c}\mathsf{H}\mathsf{C}\mathsf{H}\mathsf{H}\mathsf{c}\mathsf{H}\mathsf{H}\mathsf{F}\mathsf{F}$.
• Figure 3: The ring partition function $A^{(1\to 2)}_{k,k'}$ from colour 1 (blue) to colour 2 (red). It accounts for the weight of all the triangles crossed by the purple loop.
• Figure 4: Loops, clusters and envelopes. The triangle at $0$ is in grey in the bottom picture and is assumed to be an $\mathsf{F}$. (a) The corresponding typical loop $\mathfrak{L}(0)$ is shown in bold purple (other loops are shown in pale purple). The typical filled-in cluster $\mathfrak{K}(0)$ is the whole loop-decorated triangulation in the middle (in bold). The envelope $\mathfrak{e}(0)$ is the whole loop-decorated triangulation inside the red component, with its root face lying outside the drawing. (b) The envelope is the whole loop-decorated triangulation in this bottom figure, with Sheffield's exploration in purple. We have not represented the full exploration but only the one that corresponds to the reduced walk. In particular, there is a smaller red component (to the right of the exploration) that should be visited by the space-filling exploration.

Theorems & Definitions (27)

• Theorem 1.1: Expression and asymptotics for the partition function
• Corollary 1.2
• Theorem 1.3: Exponents for loops and clusters
• Proposition 3.1: Skeleton words are filled-in clusters
• proof
• Corollary 3.2: Reduced walk expression for $|\partial \mathfrak{K}(0)|$
• proof
• Proposition 3.3: Reduced walk expression for $|\mathfrak{L}(0)|$
• proof
• Proposition 3.4: Typical filled-in cluster marginals