FK(q)-Weighted Planar Maps

• FK(q)-Weighted Planar Maps are statistical ensembles defined by assigning weights based on the Fortuin–Kasteleyn cluster model, integrating combinatorial map theory with Potts and loop models.
• They feature a continuum description via multi-component Coulomb gas actions and imaginary Liouville field theory, enabling precise computations of geometric correlation functions and scaling exponents.
• Lattice realizations, height representations, and free energy analyses validate these models’ connections with fully-packed loop configurations, proper colorings, and critical Potts systems.

The FK($q$)-weighted planar maps refer to statistical ensembles in which the faces of planar maps (combinatorial embeddings of graphs in the plane) are endowed with local weights derived from the Fortuin-Kasteleyn (FK) cluster model, with parameter $q$, often interpreted through their geometrical representation in terms of fully-packed loop (FPL) models on regular lattices. The %%%%2%%%% parameter governs the weight given to loops or clusters per configuration and is central to the continuum theory describing the scaling limits of critical planar systems. The $FK(q)$-weighted settings have deep connections with $O(n)$ models, the Potts model, and the classification of critical 2D statistical systems in terms of conformal field theory (CFT), notably via their Coulomb gas and imaginary Liouville field-theoretic descriptions.

1. Model Definitions and FK($q$) Loop Correspondence

A FK($q$)-weighted planar map is constructed by assigning to each configuration a weight that depends on $q^{\#\mathrm{clusters}}$, generalizing the planar map enumeration by including Potts/loop degrees of freedom. In the equivalent loop formulation on a regular lattice (such as honeycomb or square), one considers the $O(n)$ loop model with loop-weight $n$ related to the cluster parameter $q$ by $q = n^2$. Specifically, in the FPL($n=2$)—or equivalently, the FK($q=4$)—case, the configuration sum is

$$Z_{\mathrm{FPL}} = \sum_{C} n^{\#\,\mathrm{loops}(C)}.$$

Each fully-packed configuration consists of nonintersecting loops covering all vertices, with the loop weight $n$ associated to each component. In the critical limit, these models map exactly to the $q$-state Potts model on the self-dual critical line, where the FK cluster representation is governed by the same fugacities (Wang et al., 2014).

Such models are also in one-to-one correspondence with proper 4-colorings (for $n=2$; $q=4$) and integer-valued height functions on the dual lattice (Peled et al., 2017, Liu et al., 2010). The universality of their scaling limits, as $q$ is varied, is established via the continuum field theories.

2. Field-Theoretic and Coulomb Gas Descriptions

The continuum limit of FK($q$)-weighted planar maps (and specifically FPL models with $n=2$) is captured by a multi-component Coulomb gas action. For FPL-O(2) on the honeycomb, the field content decomposes into an imaginary Liouville sector and a free boson sector:

$$A[\varphi]= \int d^2 x \, 8\pi^{-1}\sqrt{|g|} \left[2\,(\partial_\mu\phi_1)^2 + 4i q R(x)\phi_1 + e^{2i\phi_1/b} \right] + \frac{2}{3}\int d^2x\,8\pi^{-1}\sqrt{|g|}(\partial_\mu\phi_2)^2,$$

where $\varphi = (\phi_1,\phi_2)$ and the compactification is set by the loop weight via $n = -2\cos(\pi b^2)$, with $b=1$ for $n=2$ (Dupic et al., 2019). The central charge for this theory is

$c = 2 - 6(1/b - b)^2,$

giving $c=2$ for $n=2$, and $c=1$ for the single-component FPL model (Dupic et al., 2019, Liu et al., 2010).

This formalism fully encodes the geometry of clusters and loops and allows the computation of universal observables, such as correlation functions and conformal dimensions:

$\Delta_p = p^2 - \frac{1}{4}(1/b - b)^2,$

where $p$ labels the electric charge of a vertex operator $V_{Q\pm p\rho}=e^{i(Q\pm p\rho)\cdot \varphi}$, corresponding to the insertion of a modified loop weight $n'=2\cos(2\pi b p)$ (Dupic et al., 2019).

3. Geometric Correlation Functions and Imaginary Liouville Theory

FK($q$)-weighted planar maps enable the direct computation of geometric correlation functions. Two- and three-point functions of insertions that locally modify loop fugacities are given by

$$G_{n'}(x_1,x_2) \propto |x_1 - x_2|^{-4\Delta_p}, \qquad G_{n_1,n_2,n_3}(x_1,x_2,x_3) = C_n(n_1, n_2, n_3) \prod_{i<j} |x_{ij}|^{-\beta_{ij}}.$$

Here, $C_n(n_1,n_2,n_3)$ is identified with the three-point amplitude of imaginary Liouville (IL) CFT, expressed in terms of Upsilon functions $\Upsilon_b(z)$ and the normalization $\mathcal{A}_b$ (Dupic et al., 2019): $$C_b(p_1,p_2,p_3) = \mathcal{A}_b\,\frac{ \prod_{\epsilon_i = \pm 1} \Upsilon_b(\mu_b + \epsilon_1 p_1 + \epsilon_2 p_2 + \epsilon_3 p_3)} { \sqrt{\prod_{j=1}^3 \Upsilon_b(b+2p_j)\Upsilon_b(b-2p_j)} },$$ where $\mu_b = (b+b^{-1})/2$.

Transfer-matrix diagonalizations confirm the agreement between the lattice model amplitudes and IL predictions, demonstrating universality across different lattice loop models for critical FK($q$) weights (Dupic et al., 2019).

4. Lattice Realizations, Height Representation, and Scaling Exponents

In the lattice realization, FK($q$)-weighted (FPL–O(2)) configurations correspond bijectively to integer-valued 1-Lipschitz height functions on the dual triangular lattice, and also to proper 4-colorings (Peled et al., 2017, Liu et al., 2010). The mapping is as follows:

Model	Loop weight $n$	FK parameter $q$	Dual lattice height/coloring
FPL–O(2) (honeycomb)	2	4	Proper 4-colorings
FPL (square)	2	4	Two-species loop/height

Scaling exponents for geometric observables are dictated by the Coulomb gas parameter $g$ (through $n = -2\cos(\pi g/4)$). For $n=2$ ($g=4$), key observables are:

• Hull (loop) dimension $X_\mathrm{loop} = 1 - 2/g = 1/2$
• “Face” (cluster-magnetic) $X_\mathrm{face} = 1/8$
• Fractal dimension of loops $D_\mathrm{loop} = 2 - X_\mathrm{loop} = 3/2$ (Liu et al., 2010)

Monte Carlo and transfer-matrix studies exhibit close agreement with these exact values (Liu et al., 2010, Wang et al., 2014).

5. Exact Solutions, Free Energies, and Boundary Contributions

For the FK($q$)-weighted fully-packed O(2) model on the square lattice, exact solutions for bulk, surface, and corner free energies are available in terms of infinite products in the variable $q$, where $n = q + q^{-1}$ and $q\to 1^{-}$ corresponds to $n=2$, the critical case (Vernier et al., 2011). The expansions display periodic integer exponents reflecting the combinatorics of loop coverings. For bulk, surface, and corner contributions: $$e^{f_b} = \prod_{k=1}^\infty (1-q^k)^{\alpha_k}, \quad e^{f_s},\,e^{f_c}\;\text{similar,}$$ with period-8 and period-16 patterns in the exponents. In the critical limit $q \to 1^{-}$ (i.e., $n=2$), the corner free energy exhibits an essential singularity,

$$e^{f_c} \sim \frac{1}{4} \exp\left[\frac{3\pi^2}{16}\frac{1}{1-q}\right],$$

consistent with Cardy–Peschel predictions for universal corner contributions and central charge $c=3$ (Vernier et al., 2011).

6. Universality, Field-Theoretic Conjectures, and Related Models

Numerical evidence supports the conjecture that geometric correlators and three-point amplitudes for non-intersecting critical loop models (FK($q$)-weighted planar maps for appropriate $q$) are universally described by the imaginary Liouville theory, independent of lattice, detailed weights, or total central charge (Dupic et al., 2019). This universality extends to the fully-packed O(2) loop models on both honeycomb and square lattices (critical $q=4$) as well as to other Temperley–Lieb and dilute O($n$) loop ensembles. A plausible implication is the independence of the imaginary Liouville description from specifics of the bulk CFT provided the model is at a suitable critical point (Dupic et al., 2019).

7. Open Problems and Research Directions

Key unresolved problems include the rigorous establishment of conformal invariance and Schramm-Loewner Evolution (SLE) limits for FK($q$)-weighted fully packed models on regular lattices; the precise classification of translation-invariant Gibbs measures; extension to more general weights and boundary conditions; and understanding of mixing times for Glauber–type dynamics (Peled et al., 2017). Exact amplitude and scaling exponent confirmation, particularly for two- and three-point functions in finite geometries, remain active areas of numerical and analytical investigation.

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References:

(Dupic et al., 2019, Peled et al., 2017, Wang et al., 2014, Dabholkar et al., 2023, Liu et al., 2010, Vernier et al., 2011)