Fully Packed Loop-O(n) Model

Updated 8 December 2025

The fully packed loop-O(n) model is a statistical ensemble of non-intersecting closed loops on lattices, with each loop carrying a fugacity n.
It unifies concepts across statistical mechanics, combinatorics, and quantum field theory by linking Potts, dimer, and random planar map models.
Transfer-matrix and Monte Carlo methods reveal its exact solutions, critical exponents, and universal scaling behavior in dense and ordered phases.

The fully packed loop- $O(n)$ model is a central object in two-dimensional statistical mechanics, combinatorics, and quantum field theory. It describes ensembles of non-intersecting closed loops that densely cover a regular or random lattice, parameterized by a fugacity $n$ per loop. The model unifies connections to the Potts model, vertex models, random planar maps, and integrable systems, with broad implications for universality, critical phenomena, and integrable combinatorics.

1. Formal Definition and Fundamental Properties

The fully packed loop- $O(n)$ (FPL- $O(n)$ ) model is most rigorously defined on trivalent (degree 3) or tetravalent (degree 4) graphs, such as the honeycomb and square lattices, or on the duals of triangulations in the case of random maps. A FPL configuration is a subset of edges such that every vertex has degree exactly 2—i.e., the configuration is a 2-factor. The partition function on a finite graph $G=(V,E)$ is

$Z_{G,n} = \sum_{A \subset E \atop d_v(A) = 2 \forall v} n^{c(A)},$

where $c(A)$ is the number of connected components (loops) in the configuration $A$ and $d_v(A)$ denotes the degree at vertex $v$ in subgraph $A$ (Liu et al., 2010, Wang et al., 2014, Vernier et al., 2011).

On regular lattices (honeycomb, square), or planar triangulations (random maps), the sum is over all edge configurations in which every vertex is visited by exactly two loop segments. Each loop receives a fugacity $n$ . For the honeycomb lattice, the model is sometimes reformulated in terms of its equivalent dimer model on the same lattice, in which perfect matchings correspond bijectively to fully packed loop configurations at $n=1$ (Glazman et al., 16 Dec 2024).

2. Lattices, Mappings, and Generalizations

Regular Lattices (Hexagonal, Square):

On the honeycomb lattice, the FPL- $O(n)$ model is strictly defined and exhibits deep relations to dimer configurations, Potts antiferromagnets, and coloring models. The $n=1$ case corresponds precisely to dimers; $n=2$ accommodates exact mappings to zero-temperature Potts models on the kagome and triangular lattices (Liu et al., 2010).
On the square lattice, additional “crossing” and “cubic” vertices may be included, allowing for a generalized Eulerian graph model. Still, the fully packed restriction always ensures each site has even degree, typically 2 or 4 (Wang et al., 2014).

Random Planar Maps:

For planar triangulations and related random maps, the dual of the map is equipped with fully packed loops: each triangle of the primal map is crossed by a (dual) loop segment, so every site of the dual map has degree 2. This version is equivalent to the self-dual Fortuin-Kasteleyn model at $q = n^2$ (Berestycki et al., 5 Dec 2025, Borot et al., 2012).

Equivalences:

On the square lattice (branch 1 in the exact solution classification), the FPL- $O(n)$ model is equivalent to the self-dual line of the $Q=n^2$ Potts model (Wang et al., 2014).
The FPL- $O(1)$ model on the hexagonal lattice and perfect matchings (dimers) are in exact bijection (Glazman et al., 16 Dec 2024).

3. Integrability, Exact Results, and Critical Behavior

Transfer Matrix and Exact Solutions:

Transfer-matrix techniques yield the dominant eigenvalues governing the free energy and allow computation of critical exponents via finite-size scaling (Wang et al., 2014, Vernier et al., 2011).
Integrability is present in several cases: the honeycomb FPL- $O(n)$ model is solvable by the Bethe Ansatz for certain values of $n$ ; the square-lattice FPL- $O(n)$ model admits several one-parameter exactly solvable branches (Schultz, Perk–Schultz, Rietman, etc.), each corresponding to specific choices of vertex weights (Wang et al., 2014).

Universal Properties and Phase Diagram:

For $|n|<2$ , the model lies in the so-called “dense” critical phase, with conformal anomaly $c=13-6g-6/g,~g=1-(1/\pi)\arccos(n/2)$ , and explicit scaling dimensions for the magnetic and thermal sectors: $X_h=1-(3g/8)-(1/2g),~X_t=4/g-2$ (Wang et al., 2014).
For $n>2$ , the model undergoes a first-order phase transition into an ordered state (checkerboard or cubic–dominated phase, depending on the allowed vertices) (Wang et al., 2014).
On random triangulations, the critical regime matches Nienhuis' exact results for the hexagonal lattice; the string susceptibility exponent $2-\theta$ (with $\theta=(1/\pi)\arccos(n/2)$ ) interpolates continuously from $3/2$ to $2$ as $n$ increases from $0$ to $2$ (Berestycki et al., 5 Dec 2025).

Boundary and Corner Effects:

Bulk, surface, and corner free energies have been computed to high order by Enting’s finite lattice method and transfer-matrix expansion, with explicit infinite-product parameterizations in terms of the quantum-group parameter $n=q+1/q$ (Vernier et al., 2011).
The critical behavior of corner free energies matches Cardy–Peschel’s conformal field theory predictions; for example, the divergence of the corner free energy as $q\to 1^{-}$ encodes the central charge $c$ .

4. Combinatorics, Connection to Integrable Systems, and Algebraic Structures

Temperley-Lieb and Combinatorics:

On the square lattice, fully packed loops are intimately connected to the Temperley-Lieb algebra and the six-vertex model at special points. The action of Temperley–Lieb operators generates local loop reconnections, and the ground state of the related XXZ spin chain encodes refined combinatorics of FPL configurations (Cantini et al., 2010).
The Razumov–Stroganov conjecture, now proven by combinatorial methods (gyration, involutions), connects the ground state components of the O(1) FPL model with link-pattern enumeration in the square grid (Cantini et al., 2010).

Coloring Models and Eulerian Graphs:

The FPL- $O(n)$ model is equivalent to exactly solvable edge-coloring models, in which each loop is colored, and lattice edges inherit color constraints. In the Perk–Schultz coloring model, the O(n) loop weight emerges from the summation over all colorings compatible with loop connectivity (Wang et al., 2014).
The mapping to coloring models enables exact enumeration for finite sizes and underlies the integrable structures in the model.

5. Random Planar Maps, Critical Exponents, and Universality

Analytic Combinatorics and Probabilistic Methods:

On random planar triangulations, the partition function and scaling exponents for the FPL- $O(n)$ $O (n)$ model have been obtained exactly through two methodological frameworks:
- Gasket decomposition and resolvent analysis—giving Stieltjes representations and explicit spectral densities for boundary-generating functions (Berestycki et al., 5 Dec 2025, Borot et al., 2012);
- Probabilistic bijections (e.g., Sheffield’s hamburger–cheeseburger)—connecting cluster boundaries and loop lengths to hitting-times of random walks (Berestycki et al., 5 Dec 2025).

Critical Exponent Dictionary:

The tail exponents for loop and cluster perimeter distributions have the form $P(|\partial \mathcal K| = \ell) \sim \tilde C \ell^{-(3-2\theta)}$ , with full agreement between combinatorial and probabilistic approaches (Berestycki et al., 5 Dec 2025).
The model interpolates between dense and dilute universality classes by tuning the bending energy (local curvature weight), and supports both explicit and spontaneous domain-symmetry breaking transitions in the presence or absence of coloring asymmetry (Borot et al., 2012).

Phase Diagram and Self-Dual Points:

On random maps, the FPL- $O(n)$ model with $n = \sqrt{q}$ at specific $x_c$ coincides with the self-dual FK( $q$ ) model on triangulations (Berestycki et al., 5 Dec 2025).
For the random–map $Q$ -state Potts model, the critical point is not generically self-dual except for $Q=1$ , and parameter-matching via the nested-loop framework provides the exact transition locus (Borot et al., 2012).

6. Conformal Field Theory and Scaling Limits

Coulomb Gas and Imaginary Liouville:

The continuum limit of the FPL- $O(n)$ model is governed by a two-component Coulomb-Gas field theory, with compactification on the A $_2$ lattice (sl $_3$ root lattice) (Dupic et al., 2019).
The field decomposition yields a sector described by imaginary Liouville theory with central charge $c_{IL}=1-6(1/b-b)^2$ and an independent free boson ( $c=1$ ), so that $c(n)=2-6(1/b-b)^2$ .
Conformal structure constants, three-point functions, and loop-operator scaling dimensions are accessible via this Coulomb-Gas framework. Central to this analysis is the relation $n=-2\cos(\pi b^2)$ between the loop weight and the field-theoretic compactification parameter (Dupic et al., 2019).

Universal Observables:

Loop-weight-changing and multi-point correlation function amplitudes (structure constants) can be extracted both analytically and numerically, with finite-size transfer-matrix computations verifying the predicted scaling and universal amplitudes (Dupic et al., 2019).
Generalization to other non-intersecting loop models (e.g., dilute O(n), Temperley–Lieb) is supported by numerical evidence and conjectures regarding the universality of the imaginary-Liouville description.

7. Dynamical and Algorithmic Results

Monte Carlo Algorithms:

The worm-type Markov chain algorithm provides efficient and provably ergodic sampling for the FPL- $O(n)$ model on bipartite cubic graphs for any $n > 0$ , including the fully packed $(x \to \infty)$ limit (Liu et al., 2010). The algorithm accommodates configurations with two defects, allowing for locality and detailed balance even at large weights.
Dynamical observables (e.g., worm-return times) demonstrate exponents governed by magnetic dimensions, providing concrete dynamical manifestations of the underlying scaling properties (Liu et al., 2010).

Critical Dynamics and Exponents:

The scaling of worm dynamics is governed by loop and magnetic dimensions that coincide (up to statistical error) with Coulomb-gas predictions. This establishes a correspondence between stochastic sampling algorithms and field-theoretical scaling data (Liu et al., 2010).

References:

(Glazman et al., 16 Dec 2024): "On loops in the complement to dimers" (Cantini et al., 2010): "Proof of the Razumov-Stroganov conjecture" (Liu et al., 2010): "Worm Monte Carlo study of the honeycomb-lattice loop model" (Wang et al., 2014): "Completely packed O( $n$ ) loop models and their relation with exactly solved coloring models" (Vernier et al., 2011): "Corner free energies and boundary effects for Ising, Potts and fully-packed loop models on the square and triangular lattices" (Berestycki et al., 5 Dec 2025): "Critical behaviour of the fully packed loop- $O(n)$ model on planar triangulations" (Dupic et al., 2019): "Three-point functions in the fully packed loop model on the honeycomb lattice" (Borot et al., 2012): "Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model"