Asymptotic variance constant and Gaussian scaling limits for the square-ice height function and generalized solid-on-solid models

The uniform square-ice model (six-vertex model with weights $a=b=c=1$) admits a height function equivalent to a uniformly random graph homomorphism $\phi:\mathbb Z^2\to\mathbb Z$, conjectured to converge under diffusive rescaling to a multiple of the two-dimensional Gaussian free field (GFF). Wu proved a central limit theorem for the height at a point, with a normalizing sequence $\sigma_N$ pinned only between two constant multiples of $(\log N)^{1/2}$, and posed three open questions: (1) prove that $\sigma_N/(\log N)^{1/2}$ converges to a finite positive constant $g$, the missing quantitative input for the conjectured GFF scaling limit at the square-ice point $\Delta=1/2$; (2) determine the precise mixing rate of the height gradients, conjectured to be algebraic in the physical scale; (3) prove Gaussianity of the scaling limit for generalized solid-on-solid models with symmetric convex non-quadratic potentials. Later literature: Duminil-Copin, Kozlowski, Lammers and Manolescu proved GFF convergence for the six-vertex height function only in the regime $\Delta\in[-1,-1/2]$ (with explicit amplitude $\sigma^2=2/\arccos(\Delta)$) and state explicitly that the extension toward the square-ice point is blocked by missing rotational-invariance inputs; the Discrete Gaussian (quadratic potential) case of question (3) is resolved, while for non-quadratic integer-valued potentials only delocalization results are available (GFF universality is proved for continuous-valued gradient models with non-convex potentials, a different class). All three questions remain open.

Background

The six-vertex model on the square lattice places an arrow on each edge, subject to the ice rule: at each vertex, exactly two of the four incident arrows point inward and two outward. Configurations are in bijection with height functions $\phi:\mathbb Z^2\to\mathbb Z$, integer-valued maps whose increments between adjacent vertices are $\pm 1$; for the uniform (square-ice) weights $a=b=c=1$ the induced height function is exactly a uniformly random such graph homomorphism. Weighted six-vertex models are parametrized by $\Delta=(a^2+b^2-c^2)/(2ab)$, with $\Delta\in(-1,1)$ the rough (delocalized) regime; square ice sits at $\Delta=1/2$. A height function viewed at lattice spacing $\delta\to 0$ is expected, in this regime, to converge to $\sigma$ times the Gaussian free field (GFF): the centered generalized Gaussian random field on $\mathbb R^2$ with covariance $\mathbb E[\Gamma(x)\Gamma(y)]=-(2\pi)^{-1}\log|x-y|$, the universal log-correlated scaling limit of two-dimensional random surfaces. The GFF is only a generalized field, so convergence is stated through multi-point correlation functions, finite-dimensional distributions against test functions, and negative-regularity Holder-type topologies; the amplitude $\sigma$ is the analogue of the normalization constant $\sigma_N$ below.

Two results lead to the question. First, Duminil-Copin, Harel, Laslier, Raoufi and Ray proved that the variance of the square-ice height at the center of a box of side $N$ is bounded between two absolute constant multiples of $\log N$ (logarithmic variance), establishing roughness but no limit law. Second, Wu proved a central limit theorem: for the uniform square-ice height function $\phi_N$ on an even domain $D_N$ (containing $[-N,N]^2\cap\mathbb Z^2$) with zero boundary condition, $\phi_N(0)/\sigma_N$ converges in distribution to a standard normal, with the normalizing sequence satisfying $C_1(\log N)^{1/2}\le\sigma_N\le C_2(\log N)^{1/2}$; a multivariate version gives independent normals for finitely many bulk points. The proof uses a martingale central limit theorem, a Russo-Seymour-Welsh (RSW) crossing estimate for height level lines (a uniform-probability bound for annular crossings, imported from the earlier logarithmic-variance work), the FKG inequality for the absolute value height function (a positive-correlation property of $|\phi|$ that survives the ice-rule constraints), a monotone coupling of height functions, and ballot-theorem random walk estimates (bounds for the probability that a random-walk path stays positive). The martingale decomposition writes $\phi_N(0)$ as a sum of increments $\Delta_k=\mathbb E[\phi_N(0)|\mathcal F_k]-\mathbb E[\phi_N(0)|\mathcal F_{k-1}]$ indexed by the dyadic scales $r_k=2^{-k}N$, and takes $\sigma_N^2=\sum_{k=1}^{\log N}\mathbb E[\Delta_k^2]$; in this decomposition the geometric scale is the scale index $k$ (a unit step corresponds to a factor of two in length), while the physical scale is the linear size $r_k=2^{-k}N$ that it represents. Wu's Remark 1.2 notes that the theorem gives Gaussianity of the scaling limit but is insufficient to give the asymptotic convergence of the variance $\sigma_N/(\log N)^{1/2}\to g$ for some $g>0$, which he believes requires new ideas.

The frontier on the antiferroelectric side is relevant: Duminil-Copin, Kozlowski, Krachun, Manolescu and Oulamara proved rotational invariance of the critical random-cluster model with cluster weight $1\le q\le 4$ at large scales, and Duminil-Copin, Kozlowski, Lammers and Manolescu combined such inputs with a spectral analysis to prove that the six-vertex height function with $a=b=1$, $\sqrt 3\le c\le 2$ (i.e. $\Delta\in[-1,-1/2]$) converges to $\sigma\cdot$GFF with explicit $\sigma^2=2/\arccos(\Delta)$. Delocalization with logarithmic variance is known for the whole range $a=b=1$, $1\le c\le 2$ (which includes square ice), but this does not identify a limit law.

The third strand concerns generalized solid-on-solid (SOS) models: random integer height functions $\phi:\mathbb Z^2\to\mathbb Z$ with Gibbs weight $\exp(-\beta\sum_{x\sim y} V(\phi(x)-\phi(y)))$ for a symmetric convex potential $V$ (e.g. $V(x)=x^2/2$, the Discrete Gaussian model, or $V(x)=|x|$). At roughening (small $\beta$), the scaling limit is conjectured to be a GFF; this is proved only for the quadratic potential, via the renormalization group work of Bauerschmidt, Park and Rodriguez and the multi-point central limit theorem of Park, while for non-quadratic potentials only (qualitative or quantitative) delocalization results are available, notably the absolute-value FKG inequality of Lammers and Ott and the quantitative delocalization at arbitrary tilt of Ott and Schweiger. (For continuous-valued gradient interface models, a different class of random surfaces with real-valued heights, GFF scaling limits are now known even for a broad family of even non-convex potentials, by Buchholz, Cotar and Schweiger.) Wu observes that his method plausibly extends to the general-potential Gaussianity question, with RSW-type crossing estimates the missing input.

Problem Statement

Wu's central limit theorem for the uniform square-ice height function leaves three linked open questions, stated in Section 5 of the source paper, directed toward the Gaussian free field universality picture for the uniform square-ice height function and its solid-on-solid generalizations. A complete answer resolves any one of the three questions by rigorous proof or disproof; question (1) is the central one.

(1) Asymptotic variance. Prove that $\sigma_N/(\log N)^{1/2}$ converges to a finite positive constant $g$ (or prove that it does not), where $\sigma_N$ is the normalizing sequence in Wu's central limit theorem for the height at the origin of the square-ice height function on $D_N$ with zero boundary condition, given by $\sigma_N^2=\sum_{k=1}^{\log N}\mathbb E[\Delta_k^2]$ over the dyadic scales $r_k=2^{-k}N$ (the base of the logarithm only rescales the limit, so the claim is base-independent). This is the missing quantitative input for the conjectured GFF scaling limit of the uniform square-ice height function; Wu identifies the sufficient condition $(1/\log N)\sum_{k=1}^{\log N}\mathbb E[\tilde\Delta_k^2]\to g$ on the variances of his (truncated) martingale increments and states that proving it requires new ideas, essentially some approximate scale invariance. A strictly stronger form of answer is a GFF scaling-limit theorem at the square-ice point $\Delta=1/2$ (multi-point convergence, convergence of finite-dimensional distributions against test functions, or convergence in law in a negative-regularity topology) with an identified amplitude.

(2) Mixing rate. Determine the precise mixing rate, that is the rate of decay of correlations, of the gradients of the square-ice height function as a function of the physical scale (the linear distance between the regions considered); the expected answer is algebraic in the physical scale, analogous to the gradients of the discrete Gaussian free field, whose correlations decay polynomially in distance. This would improve on the algebraic-in-geometric-scale bound of Wu's Lemma 3.2, in which the decay is polynomial in the dyadic scale index (each unit of which corresponds to a factor of two in length) rather than in the linear size it represents.

(3) Gaussianity for generalized solid-on-solid models. Prove (or disprove) that the scaling limit of the generalized solid-on-solid model with Gibbs weight $\exp(-\beta\sum_{x\sim y}V(\phi(x)-\phi(y)))$ is Gaussian for symmetric convex potentials $V$ beyond the quadratic (Discrete Gaussian) case, in the roughening (delocalized) regime of $\beta$; Wu notes that the absolute value FKG inequality is available and that a Russo-Seymour-Welsh estimate is the missing input for extending his method.

References

1. Wei Wu, "A central limit theorem for square ice", Acta Mathematica Sinica, English Series 42(6):1609-1620 (2026), DOI 10.1007/s10114-026-4385-5, arXiv:2206.12058, https://arxiv.org/abs/2206.12058
2. Hugo Duminil-Copin, Karol Kajetan Kozlowski, Piet Lammers, Ioan Manolescu, "Gaussian free field convergence of the six-vertex model with $-1\leq\Delta\leq-\frac12$", arXiv:2603.06268 (2026), https://arxiv.org/abs/2603.06268
3. Hugo Duminil-Copin, Matan Harel, Benoit Laslier, Aran Raoufi, Gourab Ray, "Logarithmic variance for the height function of square-ice", Communications in Mathematical Physics 396:867-902 (2022), DOI 10.1007/s00220-022-04483-x, arXiv:1911.00092, https://arxiv.org/abs/1911.00092
4. Roland Bauerschmidt, Jiwoon Park, Pierre-Francois Rodriguez, "The Discrete Gaussian model, I. Renormalisation group flow at high temperature", The Annals of Probability 52 (2024), DOI 10.1214/23-AOP1658, arXiv:2202.02286, https://arxiv.org/abs/2202.02286
5. Roland Bauerschmidt, Jiwoon Park, Pierre-Francois Rodriguez, "The Discrete Gaussian model, II. Infinite-volume scaling limit at high temperature", The Annals of Probability 52 (2024), DOI 10.1214/23-AOP1659, arXiv:2202.02287, https://arxiv.org/abs/2202.02287
6. Jiwoon Park, "Central limit theorem for multi-point functions of the 2D discrete Gaussian model at high temperature", arXiv:2211.14367 (2022), https://arxiv.org/abs/2211.14367
7. Piet Lammers, Sebastien Ott, "Delocalisation and absolute-value-FKG in the solid-on-solid model", Probability Theory and Related Fields 188:63-87 (2023), DOI 10.1007/s00440-023-01202-y, arXiv:2101.05139, https://arxiv.org/abs/2101.05139
8. Sebastien Ott, Florian Schweiger, "Quantitative delocalization for solid-on-solid models at high temperature and arbitrary tilt", arXiv:2505.16804 (2025), https://arxiv.org/abs/2505.16804
9. Hugo Duminil-Copin, Alex Karrila, Ioan Manolescu, Mendes Oulamara, "Delocalization of the height function of the six-vertex model", arXiv:2012.13750 (2020), https://arxiv.org/abs/2012.13750
10. Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun, Ioan Manolescu, Mendes Oulamara, "Rotational invariance in critical planar lattice models", arXiv:2012.11672 (2020), https://arxiv.org/abs/2012.11672
11. Hong-Bin Chen, Hugo Duminil-Copin, Tiancheng He, Francois Jacopin, Dmitry Krachun, Ioan Manolescu, Jiaming Xia, "Critical exponents for planar random-cluster model with cluster-weight $q=4$", arXiv:2605.30030 (2026), https://arxiv.org/abs/2605.30030
12. Hong-Bin Chen, Jiaming Xia, "Multi-point functions of a full-plane two-state fuzzy $4$-Potts model", arXiv:2605.31374 (2026), https://arxiv.org/abs/2605.31374
13. Simon Buchholz, Codina Cotar, Florian Schweiger, "Gradient Gibbs measures with non-convex potentials and the universality class of the Gaussian free field", arXiv:2608.14526 (2026), https://arxiv.org/abs/2608.14526