The solid-on-solid (SOS) model is the archetypal random-surface model of integer-valued height functions, introduced by van Beijeren (1975) for the interface of the three-dimensional Ising model. Armstrong and Wu (2023) proved that the scaling limit of the continuous (real-valued height) SOS model on Z^d is a multiple of the Gaussian free field, and stated in the same paper that the scaling limit of the original integer-valued SOS model to a Gaussian free field remains an important open question. The rough phase is now quantitatively established (Fröhlich-Spencer 1981; Lammers-Ott 2023; Ott-Schweiger 2025), and the analogous scaling limit has been proved for the discrete Gaussian model with quadratic potential (Bauerschmidt-Park-Rodriguez 2024), for a tilted and perturbed SOS surface at large inverse temperature (Laslier-Lubetzky 2024), and for the six-vertex height function (Duminil-Copin, Kozlowski, Lammers, Manolescu 2026). The Gaussian free field scaling limit of the unperturbed, zero-tilt, integer-valued SOS model with potential ∣η∣|\eta| remains open.

Background

A random surface is a random function h on the lattice Z^d, and the solid-on-solid (SOS) model is the oldest and most studied one: introduced by van Beijeren (1975) as an idealization of the interface separating the two pure phases of the three-dimensional Ising model, it is the height function h: Z^d -> Z distributed over a finite region with zero boundary data according to

P(h)∝exp⁡(−β∑x∼y∣h(x)−h(y)∣),\mathbb{P}(h) \propto \exp\Big(-\beta\sum_{x\sim y}|h(x)-h(y)|\Big),

where the sum runs over nearest-neighbor pairs and β>0\beta>0 is the inverse temperature. If the heights are allowed to be real-valued instead of integer-valued one gets the continuous SOS model, a member of the family of gradient-interface models (∇ϕ\nabla\phi-interface models), i.e. Gibbs measures on gradient fields whose large-scale theory for uniformly convex potentials is classical; the SOS potential V(η)=∣η∣V(\eta)=|\eta| is not differentiable at the origin and not strictly convex, and the integer constraint is a further, different type of nonconvexity.

In two spatial dimensions the model undergoes a roughening transition: at large β\beta the surface is rigid (the variance of h(x)h(x) stays bounded), while at small β\beta it is rough, with Var(h(x))\mathrm{Var}(h(x)) growing logarithmically in ∣x∣|x|, as for the Gaussian free field (GFF) — the centered Gaussian generalized field on Rd\mathbb{R}^d whose covariance is the Green function of the Laplacian. Fröhlich and Spencer (1981), in their celebrated proof of the Kosterlitz-Thouless transition (the transition of the two-dimensional XY model between a quasi-ordered phase with algebraically decaying correlations and an ordered phase) via the Coulomb gas (the dual description of integer-valued height functions in terms of integer charges interacting through the two-dimensional Coulomb potential), first derived the quantitative logarithmic lower bound on the variance in the rough phase; Brydges and Spencer (2012) developed complementary fluctuation estimates for sub-quadratic gradient actions. Lammers and Ott (2023) gave a much simpler delocalization proof, valid on any planar graph and at any slope, together with an FKG-type correlation inequality (positive association) for ∣h∣|h|; Ott and Schweiger (2025) proved quantitative logarithmic delocalization at high temperature, uniformly in the boundary data and the tilt, for the whole family of pp-SOS models with potential ∣∇h∣p|\nabla h|^p, 0<p≤20<p\le 2. All of these establish the rough phase but say nothing about the limit law.

Armstrong and Wu (2023) settled the scaling limit of the continuous SOS model on Zd\mathbb{Z}^d: there exists a constant a>0a>0 such that, for every smooth compactly supported vector field f\mathbf{f}, the gradient observable FR(∇ϕ)=R−d/2∑x∑ifi(x/R)(ϕ(x+ei)−ϕ(x))F_R(\nabla\phi)=R^{-d/2}\sum_{x}\sum_i f_i(x/R)(\phi(x+e_i)-\phi(x)) converges in law, as R→∞R\to\infty, to a centered Gaussian with variance ∫∇⋅f(x) Ga(x,y) ∇⋅f(y) dx dy\int\nabla\cdot\mathbf{f}(x)\,G_a(x,y)\,\nabla\cdot\mathbf{f}(y)\,dx\,dy, where GaG_a is the Green function of aΔa\Delta. Their proof uses the Helffer-Sjostrand representation — an identity rewriting covariances of the field in terms of the Green function of an elliptic difference equation with random coefficients — and then homogenizes that highly degenerate random equation (shows it behaves like the deterministic equation aΔu=0a\Delta u=0 at large scales), using only qualitative ergodicity of the coefficient field. In the same introduction they state verbatim that 'the scaling limit of the integer-valued SOS model to a Gaussian free field remains an important open question', noting that constraining the model to a discrete set introduces another type of nonconvexity which makes the analysis much more difficult.

The nearest resolved cases show where the boundary of current methods lies. The discrete Gaussian model — integer heights with the strictly convex quadratic potential βη2/2\beta\eta^2/2 — was settled at high temperature in two dimensions by Bauerschmidt, Park and Rodriguez (2024) using rigorous renormalization-group (multiscale inductive) methods that rely on the analyticity and strict convexity of the quadratic potential and do not extend to ∣η∣|\eta|. Laslier and Lubetzky (2024) proved the first scaling limit for a tilted SOS-type surface: at large fixed β\beta on a two-dimensional torus with a nonzero boundary slope and a perturbing potential on a positive fraction of sites, the infinite-volume gradient measure exists and its scaling limit is the full-plane GFF. Duminil-Copin, Kozlowski, Lammers and Manolescu (2026) proved GFF convergence for the height function of the six-vertex model (a model of arrows on lattice edges with six allowed local configurations) at −1≤Δ≤−1/2-1\le\Delta\le-1/2, a genuinely different integer-valued interaction of vertex type rather than edge-potential type. None of these covers the unperturbed, zero-tilt SOS model with potential ∣η∣|\eta| in its rough phase.

Problem Statement

Prove, or refute, that the integer-valued solid-on-solid model has a Gaussian free field scaling limit in its rough phase, in the same sense in which Armstrong and Wu settled the continuous model.

Precise target (positive resolution): for the SOS model on Zd\mathbb{Z}^d with d≥2d\ge 2, heights in Z\mathbb{Z} and potential V(η)=β∣η∣V(\eta)=\beta|\eta|, sampled from the gradient Gibbs state obtained as the weak limit of the finite-volume Gibbs measures with zero boundary data (Sheffield's gradient-field formalism), at inverse temperature β\beta in a delocalized regime — in d=2d=2, for all sufficiently small β>0\beta>0, the regime in which logarithmic delocalization is proven (Fröhlich-Spencer, Lammers-Ott, Ott-Schweiger) — prove that there exists a constant a>0a>0 such that, for every f=(f1,…,fd)∈Cc∞(Rd;Rd)\mathbf{f}=(f_1,\dots,f_d)\in C_c^\infty(\mathbb{R}^d;\mathbb{R}^d), the random variables

FR(∇h):=R−d/2∑x∈Zd∑i=1dfi(x/R) (h(x+ei)−h(x)),R≥1,F_R(\nabla h):=R^{-d/2}\sum_{x\in\mathbb{Z}^d}\sum_{i=1}^d f_i(x/R)\,(h(x+e_i)-h(x)),\quad R\ge 1,

converge in law, as R→∞R\to\infty, to a centered normal random variable with variance

∫Rd∇⋅f(x) Ga(x,y) ∇⋅f(y) dx dy,\int_{\mathbb{R}^d}\nabla\cdot\mathbf{f}(x)\,G_a(x,y)\,\nabla\cdot\mathbf{f}(y)\,dx\,dy,

where GaG_a is the Green function of the elliptic operator aΔa\Delta in Rd\mathbb{R}^d. In d=2d=2 an alternative sufficient formulation is the convergence of the centered height field, under diffusive rescaling, to aa times the Gaussian free field. The constant aa (the renormalized stiffness, i.e. the effective macroscopic elasticity of the interface) need not be computed explicitly; its existence and positivity suffice, as in the continuous theorem. The minimal case constituting a resolution is the physically central one, d=2d=2 at sufficiently small β\beta; a proof valid in higher dimensions or on a wider identified delocalized interval is accepted a fortiori. A rigorous refutation would be a proof that, in some delocalized regime, the limit exists but is not of this Gaussian form, or that the limit fails to exist.

The model must be the unperturbed, zero-tilt SOS model: no additional pinning or perturbing potentials, no boundary slope, no alteration of the potential away from ∣η∣|\eta|. Results for the discrete Gaussian (quadratic potential), for tilted or site-perturbed variants, or variance/delocalization bounds without a limit law do not answer the question.

References

  1. Scott Armstrong and Wei Wu, 'The scaling limit of the continuous solid-on-solid model', 2023, arXiv:2310.13630, DOI 10.48550/arXiv.2310.13630, https://arxiv.org/abs/2310.13630 (source of the open-question statement; no journal version as of September 2026)
  2. H. van Beijeren, 'Interface sharpness in the Ising system', Communications in Mathematical Physics 40(1):1-6, 1975, DOI 10.1007/BF01614092, https://doi.org/10.1007/BF01614092
  3. Jürg Fröhlich and Thomas Spencer, 'The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas', Communications in Mathematical Physics 81(4):527-602, 1981, DOI 10.1007/BF01208273, https://doi.org/10.1007/BF01208273
  4. David Brydges and Thomas Spencer, 'Fluctuation estimates for sub-quadratic gradient field actions', Journal of Mathematical Physics 53(9):095216, 2012, DOI 10.1063/1.4747194, https://doi.org/10.1063/1.4747194
  5. Scott Sheffield, 'Random surfaces', Astérisque 304, 2005, DOI 10.24033/ast.697, https://doi.org/10.24033/ast.697
  6. Piet Lammers and Sébastien Ott, 'Delocalisation and absolute-value-FKG in the solid-on-solid model', Probability Theory and Related Fields 188(1-2):63-87, 2024 (online 2023), DOI 10.1007/s00440-023-01202-y, arXiv:2101.05139, https://arxiv.org/abs/2101.05139
  7. Roland Bauerschmidt, Jiwoon Park and Pierre-François Rodriguez, 'The discrete Gaussian model, I. Renormalisation group flow at high temperature', Annals of Probability 52(4), 2024, DOI 10.1214/23-aop1658, arXiv:2202.02286, https://arxiv.org/abs/2202.02286
  8. Roland Bauerschmidt, Jiwoon Park and Pierre-François Rodriguez, 'The discrete Gaussian model, II. Infinite-volume scaling limit at high temperature', Annals of Probability 52(4), 2024, DOI 10.1214/23-aop1659, arXiv:2202.02287, https://arxiv.org/abs/2202.02287
  9. Benoît Laslier and Eyal Lubetzky, 'Tilted Solid-On-Solid is liquid: scaling limit of SOS with a potential on a slope', 2024, arXiv:2409.08745, https://arxiv.org/abs/2409.08745
  10. Sébastien Ott and Florian Schweiger, 'Quantitative delocalization for solid-on-solid models at high temperature and arbitrary tilt', 2025, arXiv:2505.16804, https://arxiv.org/abs/2505.16804
  11. Hugo Duminil-Copin, Karol Kajetan Kozlowski, Piet Lammers and Ioan Manolescu, 'Gaussian free field convergence of the six-vertex model with -1 <= Delta <= -1/2', 2026, arXiv:2603.06268, https://arxiv.org/abs/2603.06268
  12. Thibault Durand and Piet Lammers, 'The BKT transition and surface tension differentiability', 2026, arXiv:2605.28473, https://arxiv.org/abs/2605.28473
  13. Loren Coquille, Paul Dario and Arnaud Le Ny, 'Quantitative delocalisation for the Gaussian and q-SOS long-range chains', 2024, arXiv:2412.15782, https://arxiv.org/abs/2412.15782
  14. Mark Sellke, 'Localization of Random Surfaces with Monotone Potentials and an FKG-Gaussian Correlation Inequality', 2024, arXiv:2402.18737, https://arxiv.org/abs/2402.18737
  15. J. Bricmont, J.-R. Fontaine and J. L. Lebowitz, 'Surface tension, percolation, and roughening', Journal of Statistical Physics 29(2):193-203, 1982, DOI 10.1007/BF01020782, https://doi.org/10.1007/BF01020782
  16. Joseph Chen and Eyal Lubetzky, 'The limiting law of the Discrete Gaussian level lines', 2025, arXiv:2509.04333, https://arxiv.org/abs/2509.04333
  17. Milind Hegde, Yujin H. Kim and Christian Serio, 'Scaling limit and tail bounds for a random walk model of SOS level lines', 2025, arXiv:2502.10384, https://arxiv.org/abs/2502.10384