Square Graph Height Function in Lattice Models

• Square graph height function is a fundamental object defined on Z² with integer assignments governed by arrow orientations and parity constraints, modeling random surfaces.
• It exhibits logarithmic variance and, at criticality, converges to Gaussian Free Field behavior, revealing massless fluctuations over large domains.
• Analytical tools like the FKG inequality, duality, and renormalization underpin its study, enabling precise estimates in domino tilings and forcing numbers.

The square graph height function serves as a foundational object in the study of random surfaces, statistical mechanics, and combinatorial tilings on the square lattice. It arises in multiple contexts—including the square-ice (six-vertex) model, domino tilings, and dimer models—where it encodes local and global structure through integer-valued assignments on vertices or faces subject to specific constraints. Rigorous investigation of the height function reveals deep connections to random graph homomorphisms, Gaussian free field (GFF) fluctuations, and combinatorial properties such as the forcing number in perfect matchings.

1. Combinatorial Definitions and Lattice Models

The square graph height function is most prominently defined on $\mathbb Z^2$, the square lattice, or its toroidal quotient $(\mathbb Z/n\mathbb Z)^2$. In the square-ice model—a specialization of the six-vertex model with $a=b=c=1$—a configuration is an assignment of arrow orientations to each edge of the dual graph, subject to the ice rule: exactly two arrows point into and two out of each dual vertex (Duminil-Copin et al., 2019). Each admissible configuration is weighted equally under the uniform model.

The height function $h: \mathbb{Z}^2 \to \mathbb{Z}$ is constructed by fixing a reference vertex $v_0$ with $h(v_0) = 0$ and incrementing or decrementing $h$ across each edge according to the arrow orientation on the corresponding dual edge: crossing left to right yields $+1$, otherwise $-1$. The resulting function satisfies $|h(u)-h(v)|=1$ for all nearest-neighbor pairs, and parity constraints: even vertices $(i,j)$ with $i+j$ even receive even heights, odd vertices receive odd heights.

In the context of domino tilings and dimer models, the height function is similarly defined via rules governing increments across edges, typically set by local bipartite orientation and the placement of dimers. For domino tilings, $H_T$ satisfies a parity (mod 4) condition, matching-edge increment, and Lipschitz bound—each determined by the combinatorics of the perfect matching (Aliyev et al., 2024, Chhita, 2011, Boutillier et al., 2017).

2. Variance and Fluctuations of Height Functions

A central result in the square-ice model is the logarithmic variance of the height function between distant points. Specifically, for the uniform square-ice measure on the $n \times n$ torus $\mathbb{T}_n$, there exist constants $c, C > 0$ such that for all $u, v \in \mathbb{T}_n$ (Duminil-Copin et al., 2019):

$$c \log \|u-v\|_1 \leq \phi_{\mathbb{T}_n} \left[ (h(u) - h(v))^2 \right] \leq C \log \|u-v\|_1.$$

In infinite volume, $\mathrm{Var}[h(0)-h(x)] = \Theta(\log |x|)$ as $|x|\to \infty$. This establishes a strong form of roughness, distinguishing the height function in two dimensions as a massless random field.

In dimer models on the square lattice, at critical (uniform weight) points, the height fluctuations converge to the GFF, exhibiting similar logarithmic variance. Off-critical perturbations, where edge weights are slightly altered and the scaling window is exploited, can produce non-Gaussian fluctuations characterized by modified Bessel $K$ function kernels (Chhita, 2011).

3. Key Structural Properties and Inequalities

Several analytical tools underscore the structural behavior of the height function:

• FKG Inequality: For any increasing functions $F, G$, the square-ice height field exhibits positive association, i.e., $$\phi_D^{B, \kappa}[F(h)G(h)] \geq \phi_D^{B, \kappa}[F(h)]\phi_D^{B, \kappa}[G(h)]$$ and analogously for $|h|$ (Duminil-Copin et al., 2019).
• Duality and RSW-Type Crossing Estimates: Duality principles relate crossing events above or below certain height thresholds. The Russo–Seymour–Welsh (RSW) estimate adapts percolation methods to the occurrence of crossings of elevated height, supporting renormalization arguments required for variance bounds.
• Renormalization of Loop Events: Event probabilities for loops surrounding subdomains obey quadratic recurrence relations, leading to dichotomies between delocalization and localization phases.
• Tail Estimates: Superpolynomial decay in the tail of $h(0)$ controls the upper bound for the variance.

These inequalities provide a foundation for scale iteration techniques—forcing variance to grow logarithmically per dyadic annulus—and for tail probability estimates used in upper bounds (Duminil-Copin et al., 2019).

4. Height Functions in Domino Tilings and Forcing Numbers

In domino tilings of square regions, the height function method facilitates a sharp lower bound on the forcing number—the size of the smallest subset of dominos uniquely specifying the tiling. For a $2n \times 2n$ board, define $g(x) = h_{\max}(x) - h_{\min}(x)$, the maximal difference for functions satisfying the tile constraints and fixed boundary heights. Let $G_{\max} = \max_{x \in R} g(x)$. Then (Aliyev et al., 2024):

$f(R) \geq \lceil G_{\max}/4 \rceil$

On a $2n \times 2n$ square, this bound matches the known optimum $f(R) = n$. The construction employs graph coloring, shortest-path algorithms (multi-source Dijkstra or Bellman-Ford), and the extension of boundary heights via geodesic paths preserving parity and local constraints. The method is polynomial-time computable and exact for squares and various other regions.

5. Limit Shape, Frozen Boundaries, and Gaussian Free Field Scaling

In square-hexagon and square-lattice dimer models, the random height function is quantitatively described by partition functions represented as Schur functions. Asymptotic analysis yields a law of large numbers (limit shape) and a description of the frozen boundary, which is algebraic (“cloud curve”) and characterized by the combinatorics of boundary segments and fundamental-domain edge weights (Boutillier et al., 2017):

• Partition function: $Z = s_\omega(1^N)$ for uniform weights.
• Limit shape: $H_N(\chi, \kappa)/N \to h_*(\chi, \kappa)$, with $h_*$ maximizing a surface tension functional.
• Frozen boundary: Algebraically described by roots and derivatives of rational functions, with degree determined by the structure of weights and boundary.

Fluctuations around the limit shape in the “liquid” region converge under scaling to the massless GFF with Dirichlet boundary conditions. Boundary facets and tangency counts correspond to transitions in the macroscopic slope of the height profile.

6. Off-Critical Scaling and Non-Gaussian Fluctuations

Perturbing the edge weights of the square-lattice dimer model and examining the scaling window produces height fluctuations that are strictly non-Gaussian, except in the critical case where the Gaussian structure is recovered. The field exhibits rotational invariance for certain symmetric choices of mass parameters. The two-point and four-point correlations of height increments are precisely captured by integrals of Bessel $K$ functions:

• Two-point function: $$S_2(\gamma_i, \gamma_j) = \frac{18}{\pi^2}|\lambda|^2 \int_{\gamma_i}\int_{\gamma_j} [B_K(0, |\lambda||v_i-v_j|)^2 + B_K(1, |\lambda||v_i-v_j|)^2] dv_i dv_j$$
• Four-point function: Strictly non-zero in the scaling window—failure of Wick’s theorem signals non-Gaussianity, reverting to GFF statistics as perturbations vanish (Chhita, 2011).

A plausible implication is that such off-critical models interpolate between classical GFF roughness and genuinely non-Gaussian random surfaces, providing a rich ground for further study of universality classes in two-dimensional statistical mechanics.

7. Infinite-Volume Limits and Universality

Both finite-volume and infinite-volume treatments are central. All variance and fluctuation estimates for the square-ice and dimer height functions pass to the infinite lattice via monotonicity and boundary condition methods. The square-ice height exhibits logarithmic variance uniformly as domains expand, and limiting measures on $\mathbb{Z}^2$ are well-defined as the volume $N \to \infty$ (Duminil-Copin et al., 2019).

Conjectures and partial results indicate that, under scaling limits, the square-ice and uniform dimer height fields converge in distribution to the Gaussian Free Field, reinforcing their status as two-dimensional massless random fields. This suggests a powerful universality across models governed by Lipschitz constraints, matchings, and local parity rules.

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Principal References:

• Logarithmic variance for square-ice: (Duminil-Copin et al., 2019)
• Forcing numbers and height functions in domino tilings: (Aliyev et al., 2024)
• Limit shape, height fluctuations, square-hexagon lattices: (Boutillier et al., 2017)
• Off-critical dimer height fluctuations: (Chhita, 2011)