# Gaussian fields and interfaces for triangular-lattice Lipschitz heights > Random Lipschitz height surfaces on the triangular lattice converge to the Gaussian free field. Plain-English explainer with sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-232/ Last updated: 2026-10-14 **Random Lipschitz height surfaces on the triangular lattice converge to the Gaussian free field.** **Trust:** Claimed, not checked · **Papers:** 3 · **First paper:** October 6, 2026 · **Field:** Probability and statistical mechanics · **Entry:** 232 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. Give each point of a triangular grid a height, with neighbors differing by at most one. The papers claim that, seen from far away, these random surfaces look like the Gaussian free field. One paper also claims a boundary curve converges to the random curve SLE₄. Together they address Schramm's field questions for these models. ## At a glance - **First step:** 2006 (Oded Schramm, Problems 2.2 and 2.3 on Lipschitz heights) - **Significance:** Solid. It addresses Schramm's field questions for integer and real Lipschitz heights, plus the real-height interface. - **Trust:** Claimed, not checked - **Field:** Probability and statistical mechanics (3 papers) ## The explanation Picture a floor of triangles, where every point has six neighbors. Give each point a whole-number height. The one rule is that neighbors differ by at most 1. This is an integer Lipschitz height function, a random surface with a hard limit on its slope. Start tiny. One point sits in the middle of six neighbors, all held at height 0. The middle point can be −1, 0 or 1. Pick one at random, each equally likely. The average squared height, called the variance, is then 2/3. Now add a weight x for every edge where the height changes. Height 0 changes no edges and gets weight 1. Heights 1 and −1 change all six edges and get x^6 each. With x = 1/√2, x^6 = 1/8. So height 0 has chance 0.8, and each of ±1 has chance 0.1. The variance drops to 0.2. On a large region these local rules add up to a random landscape. Physicists expect it to look like a Gaussian free field from far away. A Gaussian free field is a random surface, the two-dimensional cousin of Brownian motion. Schramm listed questions of this kind in 2006, for integer and real heights[1,2]. The obstacle is the hard slope rule. Smooth, gently curved interaction rules were already handled by Miller, but a hard cut-off is outside those methods[3,4]. Glazman and Lammers studied how much these surfaces wander. Their 2025 paper still called the free field limit a conjecture[5]. Knowing the size of the wandering alone does not give the limit[6]. The first paper uses the weight x from the example[7]. The contours where the height changes form loops, a model physicists call the loop O(2) model[8]. The paper claims a free field limit for every x from 1/√2 to 1, with height 0 on the boundary[9]. That range includes the uniform case x = 1, and the predicted critical end point x = 1/√2[10]. The field is divided by a constant that depends only on x. The paper gives no formula for that constant[11]. The surface is averaged over small patches, with no extra log rescaling[12]. The second paper uses odd heights that step by 0 or 2. The boundary is held at +1 on one arc and −1 on the other. It claims this resolves the field part of Schramm's integer problem[1]. Its tools include reflection positivity, a spectral sum rule and boundary comparison[13]. The curve where the height crosses zero is a separate question, not settled here[14]. The third paper lets heights be any real numbers, with neighbors still within 1 of each other[2]. It claims a free field limit for the centered heights. It also claims that, at one tuned boundary height, the zero-level curve converges to SLE₄[15]. SLE₄ is a specific random curve linked to the free field. The paper calls this a resolution of Schramm's real-valued problem. Approximating the hard rule by smooth ones does not, by itself, give this result[16]. The tuning involves a stiffness constant, and the paper gives no closed formula for it[17]. How sure is this? No Lean formalization is listed for these papers. They are long and technical, and they build on several recent preprints. No specialist review of them has been published. **What it doesn't show.** This does not settle the interface question for the integer two-arc model. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Solid if the proofs hold. No Lean formalization is listed and no specialist review is published; the normalizing constants have no closed formula. ## Words used - **Lipschitz height function:** A height on each grid point, with neighbors differing by at most 1. - **Gaussian free field:** A random surface that is the two-dimensional cousin of Brownian motion. - **Variance:** The average squared distance from the mean. - **Loop O(2) model:** A model of random non-crossing loops, here the contours where the height changes. - **SLE₄:** A specific random curve, linked to the zero level lines of the Gaussian free field. - **Reflection positivity:** A symmetry property under mirror images that gives useful inequalities. ## The original papers (3) - [Gaussian free-field limits of weighted integer Lipschitz heights](https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf) - October 6, 2026 - [The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data](https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf) - October 6, 2026 - [Uniform real Lipschitz surfaces on the triangular lattice](https://github.com/openai/math/blob/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf) - October 6, 2026 ## Quoted lines from the paper (17) [1] From [The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data](https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf): > This resolves the field part of Schramm's Problem 2.2. [2] From [Uniform real Lipschitz surfaces on the triangular lattice](https://github.com/openai/math/blob/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf): > We prove the Gaussian free field and SLE$_4$ scaling limits for uniformly sampled real nearest-neighbor Lipschitz heights on the triangular lattice, resolving Schramm's Problem 2.3. [3] From [Uniform real Lipschitz surfaces on the triangular lattice](https://github.com/openai/math/blob/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf): > It does not satisfy those convexity hypotheses. [4] From [The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data](https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf): > The integer-valued law studied here, with its hard constraint on every edge increment, lies outside those hypotheses. [5] From [The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data](https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf): > Their 2025 article discusses the Gaussian free field limit as a conjecture. [6] From [The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data](https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf): > Logarithmic fluctuations alone do not determine a Gaussian scaling limit. [7] From [Gaussian free-field limits of weighted integer Lipschitz heights](https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf): > We weight each unit height change by a parameter $x$. [8] From [Gaussian free-field limits of weighted integer Lipschitz heights](https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf): > The associated contours form the loop $O(2)$ model on the hexagonal lattice. [9] From [Gaussian free-field limits of weighted integer Lipschitz heights](https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf): > With zero boundary values and a factor $x$ for each edge on which the height changes, the field converges after division by a positive constant depending only on $x$ to the zero-Dirichlet Gaussian free field, for every fixed $x\in[1/\sqrt2,1]$. [10] From [Gaussian free-field limits of weighted integer Lipschitz heights](https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf): > This includes the uniform height model and the predicted critical endpoint. [11] From [Gaussian free-field limits of weighted integer Lipschitz heights](https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf): > The theorem is for each fixed $x$; it neither specifies a formula for $\sigma(x)$ nor asserts that this constant is independent of $x$. [12] From [Gaussian free-field limits of weighted integer Lipschitz heights](https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf): > There is no logarithmic rescaling of the amplitude. [13] From [The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data](https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf): > The proof combines reflection positivity, a spectral sum rule, and boundary comparison with Gaussian moment identities. [14] From [The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data](https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf): > The interface assertion is a separate question. [15] From [Uniform real Lipschitz surfaces on the triangular lattice](https://github.com/openai/math/blob/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf): > At one tuned two-arc boundary amplitude, the zero-height interface converges in uniform curve distance to chordal SLE$_4$. [16] From [Uniform real Lipschitz surfaces on the triangular lattice](https://github.com/openai/math/blob/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf): > An approximation by smooth potentials is therefore not, by itself, a solution. [17] From [Uniform real Lipschitz surfaces on the triangular lattice](https://github.com/openai/math/blob/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf): > The stiffness is $vA$; no elementary closed form for it is proposed. ## Other sources - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [Exact Hausdorff gauges for SLE](https://closedform.nishachay.tech/e/openai-math-2026-230/) - Probability and statistical mechanics - [Optimal logarithmic mixing of the Thorp shuffle](https://closedform.nishachay.tech/e/openai-math-2026-238/) - Probability and statistical mechanics - [Sharp singularity rates for symmetric random sign matrices](https://closedform.nishachay.tech/e/openai-math-2026-239/) - Probability and statistical mechanics - [Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling](https://closedform.nishachay.tech/e/openai-math-2026-223/) - Probability and statistical mechanics - [The Mézard–Parisi formula for diluted spin glasses](https://closedform.nishachay.tech/e/openai-math-2026-221/) - Probability and statistical mechanics - [Probability and statistical mechanics](https://closedform.nishachay.tech/science/mathematics/probability-and-statistical-mechanics/) - Field · 29 results - [Open since the 2000s](https://closedform.nishachay.tech/open-since/2000s/) - Era · 48 results - [Claimed, not yet checked](https://closedform.nishachay.tech/trust/claimed/) - Trust · 130 results - [Solid results](https://closedform.nishachay.tech/significance/solid/) - Significance · 50 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20232%5D%20)