Lipschitz random surfaces: Gaussian fields and a tuned SLE₄ interface

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Family 232 · Probability and statistical mechanics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

Choose a random surface subject only to a local slope restriction: neighboring heights may differ by at most one. Although each slope is bounded, a large surface can wander over many height levels. What is the continuum law of its fluctuations, and what does a contour separating positive from negative heights become?

These questions are basic tests of universality for random surfaces. The expected field is the Gaussian free field, or GFF. In a suitable two-boundary-arc setup, the expected interface is Schramm–Loewner evolution with parameter four.

Family 232 contains three manuscripts about triangular-lattice models. One treats weighted integer heights with zero boundary data. Another treats uniform odd-integer heights with boundary values plus and minus one. The third treats real-valued heights uniformly sampled from a convex polytope, with a specially tuned boundary amplitude for its interface theorem.

The models share a broad physical picture, but their height units, probability measures, boundary conditions, and normalizing constants differ. In particular, the integer field claims do not include an integer-interface theorem.

2. Background and essential definitions

Three versions of a hard slope constraint

The triangular lattice has basis vectors $(1,0)$ and $(1/2,\sqrt3/2)$. Its nearest neighbors lie in three unoriented directions. At mesh $\delta$, multiply all spatial coordinates by $\delta$.

For integer heights, the constraint is

$$h(u)\in\mathbb Z,\qquad |h(u)-h(v)|\leq1\quad(u\sim v).$$

M1 gives an admissible configuration weight $x^{N(h)}$, where $N(h)$ counts unordered adjacent pairs with unequal heights. Thus $x=1$ is uniform, while smaller $x$ penalizes steps. Its boundary layer is fixed at zero.

M2 uses odd integers and neighboring differences zero or two. Two marked boundary points divide the boundary into arcs with heights $+1$ and $-1$. The transformed height $H=(h-1)/2$ has integer values, unit allowed steps, and boundary values zero and minus one. This change of units multiplies fluctuation amplitudes by two.

M3 instead permits real heights with $|h(u)-h(v)|\leq1$. With fixed boundary values, the admissible free coordinates form a bounded convex polytope. Uniform means normalized Lebesgue measure on that polytope, not equal weight on a finite set of configurations.

A hard constraint differs from a smooth energetic penalty. Its interaction is flat where slopes are allowed and infinite outside. Theorems requiring a smooth uniformly convex potential therefore do not apply by direct substitution.

Fields are tested by averages

Let $D$ be the limiting planar domain. The zero-Dirichlet GFF $\Phi_D$ is a centered Gaussian random distribution with covariance

$$\mathbb E[\Phi_D(f)\Phi_D(g)] =\iint_{D\times D}f(z)G_D(z,w)g(w)\,dz\,dw, \qquad G_D=(-\Delta_D)^{-1}.$$

Here $f,g$ are smooth compactly supported test functions, and $\Delta_D$ is the Euclidean Laplacian with zero boundary condition. The Green function has singular part $(2\pi)^{-1}\log(1/|z-w|)$.

A lattice field is compared with this object through Euclidean area integrals against tests. Such averages can have finite limiting fluctuations even though the height at a single point has a variance growing logarithmically as the mesh shrinks. No division by $\sqrt{\log(1/\delta)}$ is used in these distribution-valued limits.

Nonzero boundary data contribute a deterministic harmonic profile. Harmonic measure $\omega_D(z,A)$ is the probability that Brownian motion started at $z$ first exits through boundary arc $A$; it gives the harmonic extension of that arc’s indicator.

3. Development history and previous partial results

Schramm’s problem collection separately asks about the odd-integer model with two arcs and the real-valued Lipschitz model. Each has a field question, and the contour question demands further information beyond field convergence.

The loop representation links integer Lipschitz heights on the triangular lattice to the loop O(2) model on its hexagonal dual. Duminil-Copin, Glazman, Peled, and Spinka established macroscopic loops at the predicted critical point. Glazman and Manolescu proved logarithmic variations for the uniform integer model and developed a useful two-spin representation.

Glazman and Lammers extended delocalization and logarithmic fluctuation estimates across a broader parameter range. Subsequent work of Glazman, Harel, and Zelesko supplied another delocalization approach. These results establish roughness, but logarithmic covariance growth alone does not identify a Gaussian joint law.

For real surfaces with smooth uniformly convex interactions, homogenization and coupling methods already yield GFF behavior. Miller’s bounded-domain theorem is a key predecessor, and his contour universality work extends the discrete Gaussian contour picture to smooth gradient models. The hard flat/infinite interaction considered here lies outside those assumptions.

Miłoś and Peled developed delocalization methods for hard-core surfaces, including uniform Lipschitz constraints, on square tori. Further controlled-gradient results provide inputs for the real model. They do not themselves identify the bounded-domain field and contour limits claimed here.

For the integer arguments, reflection positivity, transfer spectra, and harmonic moment analysis also have substantial precedents in the 2026 six-vertex work of Duminil-Copin, Kozlowski, Lammers, and Manolescu. The triangular models still require their own conditional comparisons, boundary analysis, and normalization; family 225 discusses the distinct square-lattice plane theorem.

4. What the new manuscripts claim

Weighted integer heights with zero boundary values

M1 fixes

$$\frac1{\sqrt2}\leq x\leq1.$$

Its domain is bounded with a $C^2$ Jordan boundary, approximated from inside by simple hexagonal-lattice cycles. The enclosed faces have triangular-lattice centers. Every compact subset is eventually included, and boundary parametrizations converge uniformly.

For the weighted integer law with zero boundary-layer heights, M1 claims a constant $\sigma(x)>0$, independent of the domain and its admissible approximation, such that

$$\frac{h_\delta}{\sigma(x)}\Longrightarrow\Phi_D \quad\text{in the space of distributions on }D.$$

The height is extended constantly across each hexagonal face, and test weights are the actual face integrals. The statement includes tightness, joint convergence of all finite families of smooth-test integrals, and convergence of their mixed moments.

The constant is implicit. No formula for $\sigma(x)$, nor independence from $x$, is asserted. The left endpoint is described as the predicted critical value for this loop model; the field theorem does not give a complete phase diagram outside the displayed interval.

The loop correspondence is exact: unequal neighboring heights create mutually vertex-disjoint hexagonal loops, and choosing each loop’s orientation gives its signed height increment. Forgetting orientations produces weight

$$2^{\#\text{loops}}x^{\#\text{occupied edges}}.$$

M1’s result concerns the field. It does not claim convergence of this entire loop collection.

Uniform odd heights with two boundary arcs

M2 uses bounded smooth simply connected domains and triangular-lattice polygonal approximations with uniformly converging marked boundaries. The discrete marks lie at edge midpoints, so no boundary vertex receives conflicting values.

For uniform odd heights with boundary values $+1$ and $-1$, M2 claims

$$h_\delta\Longrightarrow\bar g+\sigma_{\rm odd}\Phi_D, \qquad \frac{h_\delta-\mathbb Eh_\delta}{\sigma_{\rm odd}} \Longrightarrow\Phi_D.$$

The height is interpolated affinely over triangles. The function $\bar g$ is the bounded harmonic extension of the two boundary values, and expectations of smooth-test integrals converge to the corresponding integrals of $\bar g$. Joint mixed moments converge as well.

The positive constant $\sigma_{\rm odd}$ depends only on this lattice model. The manuscript specifies it through an absolutely convergent series of finite-volume counts and Fourier coefficients. This is an implicit normalization, not an elementary numerical formula.

The source uses an annulus estimate from M1 at $x=1$, but not M1’s weighted field-convergence theorem. The common method does not license identifying the two constants or extending M1 to weighted two-arc data.

Real heights and the tuned zero contour

M3 considers real heights with boundary values $+\lambda_*$ and $-\lambda_*$. It claims the existence of a domain-independent amplitude $\lambda_*\in(0,1/2]$ for which the zero contour between the marked boundary points converges to chordal $\mathrm{SLE}_4$.

The upper bound on $\lambda_*$ makes the adjacent boundary prescriptions compatible with unit slope constraints. After affine interpolation, each triangular face has at most one zero segment almost surely; these segments form loops and the distinguished boundary-to-boundary contour.

The claimed convergence retains the entire oriented curve in uniform distance modulo increasing changes of speed. It also gives the centered field limit

$$\frac{h_\delta-\mathbb Eh_\delta}{\sigma_{\rm real}} \Longrightarrow\Phi_D,$$

with harmonic limiting mean

$$m_D(z)=\lambda_*\bigl(\omega_D(z,A_+)-\omega_D(z,A_-)\bigr).$$

A positive stationary tangent-flux coefficient $A$ determines the constants. With lattice vertex density $v=2/\sqrt3$, the source’s normalization is

$$\sigma_{\rm real}^2=\frac1{vA}, \qquad \lambda_*^2=\frac\pi{8vA} =\frac\pi8\sigma_{\rm real}^2.$$

The coefficient measures the effective macroscopic response of the constrained surface to a small tilt. It is defined through the stationary cell construction, without a proposed elementary closed form. The theorem singles out this tuned amplitude; it does not say that every two-arc amplitude produces ordinary SLE₄.

5. Why the advance matters

Importance: 5/5. These are basic hard-constraint random surfaces, with no smooth quadratic energy that directly forces Gaussian behavior. The claimed results would strengthen roughness estimates into full field laws and, for the real model, identify the geometry of a distinguished contour.

The weighted interval connects the uniform integer surface to the predicted critical endpoint. The two-arc theorems retain the actual imposed boundary profile, rather than only a translation-invariant bulk covariance. The real-interface claim supplies information that distributional field convergence alone cannot deliver.

The constants also matter. Gaussian universality permits a model-dependent stiffness, and the contour height must be tuned to that same stiffness. The displayed relation explains how field normalization and interface boundary conditions fit together in the real model.

The rating concerns the combined mathematical claims as stated, while preserving the different models and the earlier delocalization and smooth-potential results on which the developments build.

6. The main idea

For integer heights, the manuscripts encode height residues using two spin variables. Appropriate circuits reveal enough information to cut a region off from its exterior while leaving the correct conditional law inside. In the weighted setting, the recorded exploration history must include the relevant spins as well as auxiliary cell statuses; a less informative conditioning need not preserve the required order inequalities.

Reflection through lattice rows creates a positive transfer operator. Its spectral representation constrains possible large-scale covariances. Combining all three triangular-lattice reflection directions forces the claimed inverse-square frequency structure, the Fourier counterpart of a logarithmic Green kernel. A further identity determines a unique positive coefficient.

That still leaves the true boundary condition and Gaussianity. The papers compare fields with separated pinned regions, control rare conditioning events multiplicatively, and move the pins through reflected configurations. Boundary attachment estimates then recover the harmonic profile. Away from collisions, limiting moment functions are harmonic; their logarithmic collision singularities and boundary values lead to the Gaussian pairing rule for higher moments.

The real model uses the geometry of its convex height polytope. Local slack and volume losses control large averages. Reflected Brownian couplings compare boundary-conditioned surfaces, and a stationary response problem identifies the effective stiffness. Small tilts then determine Gaussian fluctuations.

For the contour, conditioning on an explored segment continually changes the polytope. The source supplies conditional covariance and barrier estimates for those changing domains, then determines the effective height gap across the interface. A stopped harmonic observable identifies the SLE₄ driver, while additional separation and nonreturn estimates retain the whole ordered curve. This extra geometric work is why the field and interface conclusions are separate achievements.

7. Scope, limitations, and open questions

The integer models use discrete counting measures; the real model uses Lebesgue volume. Their normalizations are not interchangeable. Nor should the six-vertex multiplier from family 225 be inserted into these triangular-lattice statements.

M1 assumes zero boundary data and an inside approximation of a $C^2$ Jordan domain. M2 and M3 have their own smooth-domain, two-arc approximation conventions. Arbitrary rough boundaries, nonzero macroscopic slopes, or weighted two-arc laws require further results.

The real interface is tuned to one amplitude. The corresponding contour question for uniform odd-integer heights remains separate from M2’s field theorem. The sources do not infer a full loop-ensemble limit merely from the GFF limits.

Pointwise height tightness is not claimed. Finite fluctuations of smooth averages coexist with divergent microscopic variance. Quantitative rates, explicit evaluations of the stiffness constants, and wider boundary or parameter classes remain questions beyond the stated conclusions.

8. Sources and further reading

Schramm’s Problems 2.2 and 2.3 distinguish the integer and real questions. Glazman–Manolescu and Glazman–Lammers give the integer roughness background; Miłoś–Peled and Miller explain important hard-constraint and smooth-potential predecessors.

M1 below is the weighted zero-boundary field theorem, M2 the uniform odd-height two-arc theorem, and M3 the real-valued field and tuned-interface theorem. Their current source statements and normalization conventions are used separately throughout this digest.

Source manuscripts

• M1. OpenAI, Gaussian free-field limits of weighted integer Lipschitz heights (October 6, 2026).
• M2. OpenAI, The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data (October 6, 2026).
• M3. OpenAI, Uniform real Lipschitz surfaces on the triangular lattice (October 6, 2026).

Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.