FPL₂ Model: Fully Packed Loop‑O(2)

• The FPL₂ model is a 2D lattice system where fully packed, nonintersecting loops cover every vertex, with each loop weighted by 2.
• It establishes exact mappings to proper 4-colorings, dimers, and the zero-temperature 4-state Potts model, unifying critical phenomena in diverse systems.
• Analytical (Coulomb gas techniques) and numerical methods (worm Monte Carlo) reveal its critical exponents, scaling laws, and logarithmic corrections associated with marginal criticality.

The fully packed loop-$O(2)$ (FPL$_2$) model is a two-dimensional lattice statistical mechanics model comprising configurations of nonintersecting loops that cover every vertex exactly once, each assigned a fugacity $2$ per loop. It provides a unifying framework for understanding critical phenomena in coloring models, spin models, and random geometry, notable for its deep connections to the $4$-coloring problem, the zero-temperature $4$-state Potts model, the dimer model, and Coulomb gas universality. The FPL$_2$ model is integrable on several lattices and serves as an archetype for models exhibiting criticality, conformal invariance, and rich phase diagrams, with critical exponents accessible by Coulomb gas techniques. Certain classes of exact solutions and scaling behaviors are established rigorously; many aspects—including conformal loop ensembles and boundary critical phenomena—remain conjectural or only numerically tested.

1. Definition and Partition Function

On a finite domain $H = (V, E)$ of the hexagonal (or honeycomb) lattice, an FPL configuration $\omega \subseteq E$ is a spanning $2$-regular subgraph, i.e., each vertex has degree exactly $2$ in $\omega$. Thus, $\omega$ is a collection of disjoint simple cycles that cover $V$; no vertices are left uncovered and no edge belongs to more than one cycle. The fully packed loop-$O(n)$ model assigns weight $W(\omega) = n^{L(\omega)} x^{o(\omega)}$ to configuration $\omega$, where $L(\omega)$ is the number of loops and $o(\omega)$ is the number of occupied edges. The fully packed limit corresponds to $x \to \infty$, admitting only degree-2 vertex coverings.

Specializing to $n=2$ and $x=\infty$ defines the FPL$_2$ model: $$Z_{\mathrm{FPL}(2)}(H) = \sum_{\omega \in \mathrm{FPL}(H)} 2^{L(\omega)}$$ with the probability measure

$$\mathbb{P}_H(\omega) = \frac{2^{L(\omega)}}{Z_{\mathrm{FPL}(2)}(H)}, \qquad \omega \in \mathrm{FPL}(H)$$

No further edge-weight factors appear, as $x=\infty$ enforces the fully packed constraint (Peled et al., 2017, Liu et al., 2010).

2. Exact Mappings and Bijections

Several exact mappings connect the FPL$_2$ model to coloring and Potts models:

• Proper 4-colorings: There is a bijection between FPL$_2$ configurations on the hexagonal lattice and proper $4$-colorings of the faces. For a coloring $c$: assign to each edge $e$ the status "occupied" if the colors of its incident faces differ by $\pm1$ modulo $4$. Each vertex is adjacent to exactly two such edges, ensuring a degree-2 covering. Conversely, from an FPL configuration, one can reconstruct a unique (up to global shift) 4-coloring modulo 4 (Peled et al., 2017).
• Dimers and Potts equivalences: For $n=2$, fully packed loops on the honeycomb are in bijection with perfect matchings (dimers) and with the zero-temperature $4$-state Potts antiferromagnet on the triangular lattice and the $3$-state model on the kagomé lattice (Liu et al., 2010).
• Mapping to the $q=4$ Potts model: The completely packed nonintersecting $O(2)$ loop model on the square lattice maps to the critical $q=4$ Potts model, with $q = n^2$ and thus $q=4$ for $n=2$ (Wang et al., 2014).

The number of coloring configurations is combinatorially linked to the number and structure of loops in the FPL$_2$ ensemble; the factor $2^{L(\omega)}$ counts the number of color-pair choices corresponding to each loop (Peled et al., 2017).

3. Exact Solutions and Scaling Properties

Bulk, Surface, and Corner Free Energies

On the square lattice, the FPL$_2$ model admits exact, infinite-product expressions for bulk ($f_b$), surface ($f_s$), and corner ($f_c$) free energies at deformation parameter $q$ (with $n = q+q^{-1}$, $n=2 \Leftrightarrow q=1$). For example, the bulk partition function factorizes as

$$\exp\left[f_b(q)\right] = \frac{1}{q^2}\prod_{k=1}^\infty \left(\frac{(1-q^{8k-4})(1-q^{8k-2})}{(1-q^{8k-6})(1-q^{8k})}\right)^4$$

with $\alpha_m$ periodic in $m$ modulo 8 (Vernier et al., 2011).

In the critical $q \to 1^-$ regime, the free energies develop universal logarithmic divergences fully consistent with conformal field theory (CFT) predictions, including the Cardy–Peschel formula for corner entropies. For the FPL$_2$ model, the central charge is $c=3$, and the divergence of the correlation length $\xi$ is of essential (Kosterlitz–Thouless) type: $$\xi(q) \sim \exp\left[\frac{\pi^2}{4(1-q)}\right], \qquad q\to1^-$$ This implies $\nu = \infty$ (no power-law singularity), and corner free energy diverges as $f_c(q)\sim\frac{3\pi^2}{16(1-q)}$ (Vernier et al., 2011).

Fractal and Scaling Dimensions

Coulomb gas predictions and precise numerical studies yield the following universal exponents for the hexagonal lattice:

• Loop correlation exponent: $x_{\mathrm{loop}} = \frac{3g-2}{2g}$, with $g=4/3$, so $x_{\mathrm{loop}} = \frac{3}{4}$
• Fractal dimension of large loops: $d_f = 1 + \kappa/8 = 3/2$ (corresponding to SLE$_4$)
• Probability of co-membership: The probability that two distant points reside on the same loop decays as $|x-y|^{-3/4}$
• Height representation: The effective action is that of a massless Gaussian field compactified on a circle of radius $R=1$, $$S_{\mathrm{eff}}[h]\approx \frac{g}{4\pi}\sum_{\langle y,z\rangle} (h(y)-h(z))^2$$, with $|h(y)-h(z)|=1$ for adjacent $y$, $z$ (Peled et al., 2017).

On the honeycomb lattice, Monte Carlo and exact solutions yield:

• Magnetic scaling dimension $x_h = h = 1 - 2/g = 1/2$
• Thermal scaling dimension $x_t = 6/g - 1 = 1/2$
• Fractal dimension of loop hulls $D_f = 2 - X_{\mathrm{loop}} \approx 1.518$ Empirical results agree with Coulomb gas predictions to high precision, with observed logarithmic corrections consistent with marginal perturbations at $n=2$ (Liu et al., 2010).

4. Lattice Variants, Vertex Weights, and Phase Diagram

On the square lattice, the model admits further generalization by including vertices with crossing bonds ($x$) or cubic vertices ($c$), but the "pure" FPL$_2$ regime (branch 1 of Wang–Guo–Blöte) restricts to nonintersecting $z$-vertices (straight or turn), each with weight 1. The partition function then reduces to a sum over nonintersecting loop configurations with weight $2^{N_l}$, $N_l$ the number of loops (Wang et al., 2014).

The phase diagram is governed by:

• At $n=2$, the FPL$_2$ point is marginal between continuous and first-order transitions (the $q=4$ Potts critical point).
• Perturbations via crossing-bond ($x$) or cubic ($c$) vertices are marginal: the scaling dimension $X_c(g) = 1 + \frac{3g}{2} - \frac{1}{2g}$ evaluates to $2$, so corrections are only logarithmic, not changing universality for small $x,c$.
• Larger $|x|$ or $|c|$ move the system into disordered or ordered phases via weak (Kosterlitz–Thouless–type) transitions. No true gapped phase exists arbitrarily close to the pure FPL$_2$ point (Wang et al., 2014).

5. Markov Chain Algorithms and Numerical Studies

The worm Monte Carlo algorithm provides an ergodic, rejection-free method for sampling FPL$_2$ configurations, operating via defect pairs and local updates in the space of Eulerian subgraphs with either two defects or none. This algorithm samples the correct stationary distribution and enables measurement of fractal and magnetic observables, such as the loop hull dimension, face dimension, return time exponents, and staggered coloring dimension. Results from large system sizes (up to $L=240$) confirm Coulomb gas predictions and reveal subtle logarithmic deviations attributed to marginal criticality (Liu et al., 2010).

6. Rigorous Results, Conjectures, and Open Problems

The FPL$_2$ model is exactly solvable in the sense of having closed-form expressions for bulk and boundary free energies (Baxter 1970, Lieb, etc.), and the bijection to 4-colorings is exact. On the torus, the enumeration of colorings was obtained by Baxter. However, substantial questions remain open:

• Uniqueness of the infinite-volume Gibbs measure remains conjectural for $\mathbb{Z}^2$ with periodic boundaries.
• The scaling limit of loops is believed to coincide with CLE$_4$ (Conformal Loop Ensemble with parameter $\kappa=4$), but rigorous proofs are lacking.
• The critical, power-law decay of correlations is established by Coulomb gas and numerics but not rigorously for the hexagonal case.
• Open directions include conformal invariance and CLE convergence, rigorous calculation of exponents, Markov chain mixing rates, finite-temperature perturbations (large but finite $x$), and Berezinskii–Kosterlitz–Thouless transition analysis in related coloring and solid-on-solid models (Peled et al., 2017).

7. Summary Table: Core Lattice Models and Connections

Lattice	Equivalent Model at $n=2$	Notable Property
Hexagonal	4-coloring (Potts $q=4$ AFM)	Bijection to proper face-4-colorings
Honeycomb	Dimers, $3$-state kagomé AFM	Bijective with perfect matchings
Square (branch 1)	Critical $q=4$ Potts model	FPL$_2$ point: marginal, $c=1$

These connections illustrate the universality and exact equivalences among the FPL$_2$, coloring, and Potts antiferromagnetic models across different lattice types (Peled et al., 2017, Liu et al., 2010, Wang et al., 2014).

• "Lectures on the Spin and Loop $O(n)$ Models" (Peled et al., 2017)
• "Corner free energies and boundary effects for Ising, Potts and fully-packed loop models on the square and triangular lattices" (Vernier et al., 2011)
• "Worm Monte Carlo study of the honeycomb-lattice loop model" (Liu et al., 2010)
• "Completely packed O($n$) loop models and their relation with exactly solved coloring models" (Wang et al., 2014)