Gasket Decomposition (Borot, Bouttier, Guitter)

• The paper introduces a combinatorial framework that decomposes loop-decorated triangulations into a rooted planar map (gasket), encoding the geometry and nesting of loops.
• It develops recursive and integral equations for partition functions, revealing critical behavior and loop-perimeter scaling exponents within the fully packed loop-O(n) and FK models.
• The method unifies combinatorial, probabilistic, and spectral techniques to provide actionable insights into universality and critical phenomena in planar map models.

The gasket decomposition introduced by Borot, Bouttier, and Guitter (BBG) provides a combinatorial framework for analyzing the fully packed loop-$O(n)$ model on planar triangulations. This method translates a loop-decorated triangulation into a rooted planar map ("gasket") with additional combinatorial data encoding the embedded loops. The decomposition enables enumeration and asymptotic analysis of models ranging from the fully packed loop-$O(n)$ model to the Fortuin–Kasteleyn (FK) model at its self-dual point. The approach systematically encodes the geometry and nesting of loops within planar maps, yielding recursive and integral equations for partition functions and observables, with implications for map criticality and universality classes (Berestycki et al., 5 Dec 2025).

1. Definition of the Gasket and Decomposition Bijection

Given a rooted planar triangulation $t$ with boundary (root face) of degree $\ell$, decorated by a fully-packed configuration $L$ of non-intersecting simple loops (each visiting every internal face exactly once), the total weight is defined as

$$Z(t,L; x, n) = x^{\#(\text{internal triangles})} n^{\#(\text{loops})},$$

with associated partition function

$$F_\ell = \sum_{(t,L)\ \text{on boundary}\ \ell} Z(t,L; x, n).$$

The gasket $G$ of $(t,L)$ is the submap consisting of edges reachable from the boundary without crossing any loop. Due to the fully packed nature of $L$, $G$ is a rooted planar map. Its faces comprise one external face of degree $\ell$, and, for each loop of $L$ whose external perimeter is $k$, a corresponding internal face of degree $k$. The decomposition is realized via a combinatorial bijection:

$$\{\text{fully-packed}\ (t,L)\ \text{of boundary}\ \ell \} \leftrightarrow \{\text{gasket}\ G\ \text{of boundary}\ \ell, \text{ plus, for each internal face of } G \text{ (degree } k): \text{ring-triangle configuration and an attached } (t',L') \text{ of boundary } k'\}.$$

2. Algorithmic Decomposition: Stepwise Construction of the Gasket

The decomposition proceeds as follows:

1. Identify all edges reachable from the boundary without crossing a loop—these form the gasket $G$, with external boundary of length $\ell$.
2. Each loop in $L$ forms an annular region; removing gasket edges yields, for each loop of perimeter $k$, an internal face of $G$ with degree $k$.
3. Reinsert triangles intersected by the loop, creating a triangular ring of boundary lengths $k$ (outer) and $k'$ (inner), together with a loop-decorated triangulation of boundary $k'$ filling the interior.

In schematic notation:

$$(t,L) \longrightarrow \text{Gasket } G \ + \ \big(\text{ring of } k \leftrightarrow k' \text{ triangles} \times \text{loop–O(n) triangulation of boundary } k'\big)$$

for each face of $G$ of degree $k$.

3. Functional Equations and Fixed-Point Relations

The enumeration yields functional relations for the partition functions. For $x>0$ and loop-weight $n>0$:

• Let $A_{k\to k'}$ denote the number of planar rings of triangles with outer boundary $k$ and inner boundary $k'$.
• The contribution $g_k$ for a face of degree $k$ in the gasket satisfies:

$$g_k = n \sum_{k'\geq 0} A_{k \to k'} x^{k + k'} F_{k'}.$$

Assigning weight $g_{\deg(\text{face})}$ to each face of a rooted map $G$ with boundary $\ell$, the total Boltzmann weight of all gaskets of boundary $\ell$ is $F_\ell$. Defining the resolvent

$W(z) := \sum_{\ell\geq 0} F_\ell z^{-\ell - 1},$

one obtains, by loop-equation/Motzkin-path methods, that $W(z)$ is analytic off a single interval $[\gamma_-, \gamma_+]$ and satisfies

$$W(z + i0) + W(z - i0) + n W(1/x - z) = z, \qquad z \in (\gamma_-, \gamma_+),$$

with asymptotics $W(z) \rightarrow 1/z$ as $z \to \infty$.

4. Recursive Characterization and the One-Cut Ansatz

The resolvent $W(z)$ and support $[\gamma_-, \gamma_+]$ are uniquely determined by the so-called one-cut lemma: for any nonnegative weights $(g_k)$, there is at most one solution analytic off $[\gamma_-, \gamma_+]$, regular at infinity, and real on the cut. The fixed-point and functional relations are:

• $g_k = n \sum_{k'} A_{k \to k'} x^{k+k'} F_{k'}$
• $F_\ell = \text{Res}_{z=\infty} z^\ell W(z)$

Self-dual (critical) 2-coloring occurs at $x = x_c$, uniquely forcing $\gamma_+ = \frac{1}{2x_c}$; this is confirmed via matching to the hamburger–cheeseburger bijection for the self-dual FK($q$) model ($n = \sqrt{q}$). The singular integral for the spectral density

$$\rho(y) = -\frac{1}{2\pi i} \left( W(y + i0) - W(y - i0) \right), \qquad y \in [\gamma_-, \gamma_+]$$

is reduced via a convolution in the uniformizing variable $$u = \frac{1}{2} \log \frac{\gamma_+ - \gamma_-}{\gamma_+ - y}$$, with Wiener–Hopf factorization yielding explicit forms for $\rho(y)$ and $W(z)$.

5. Exact Solution, Critical Behaviour, and Implications

At the self-dual point $x=x_c$ and $n=2\cos(\pi\theta)$ with $0<\theta<1/2$, the parameters are

$$\gamma_+ = 2^{3/2}\cos(\pi\theta/2), \qquad \gamma_+ - \gamma_- = 2^{3/2}\theta\,\sin(\pi\theta/2).$$

The spectral density is

$$\rho(y) = C (\gamma_+ - y)^{1-\theta} \Big[ \big(\sqrt{2\gamma_+ - \gamma_- - y} + \sqrt{y - \gamma_-}\big)^{2\theta} - \big(\sqrt{2\gamma_+ - \gamma_- - y} - \sqrt{y - \gamma_-}\big)^{2\theta} \Big],$$

with $C$ an explicit normalization constant. The partition function follows as

$$F_\ell = \int_{\gamma_-}^{\gamma_+} \rho(y) y^\ell dy,\quad \ell \geq 0,$$

with asymptotics for large $\ell$:

$$F_\ell \sim \text{const} \cdot \gamma_+^\ell\,\ell^{2-\theta}.$$

This matches predictions of Gaudin and Kostov and characterizes the non-generic critical phase, exhibiting a loop-perimeter exponent $3-2\theta$ (Berestycki et al., 5 Dec 2025). The rigorous confirmation of the one-cut ansatz through probabilistic bijection methods and analytic combinatorics sharpens previous results on cluster and loop perimeter asymptotics.

6. Connections to Related Models and Further Results

The gasket decomposition is bijectively equivalent to methodologies employed in the analysis of the FK model on planar maps (parameter $q \in (0,4)$), particularly at the self-dual point. These techniques provide a "dictionary" relating combinatorial, probabilistic, and spectral quantities of interest. The approach justifies functional ansätze commonly utilized in analytic combinatorics, and the results generalize aspects of the critical behaviour observed in the loop–$O(n)$ model on the hexagonal lattice (Nienhuis universality). The method establishes exact and asymptotic expressions for partition functions, cluster perimeter distributions, and phase exponents. Recent work (Berestycki et al., 5 Dec 2025) confirms, refines, and provides new rigorous asymptotics for loop perimeter and map features, enhancing and supplanting previous results from (Berestycki et al., 2015) and (Gwynne et al., 2015).