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Schramm's Problem 2.3: Gaussian free field and SLE4_4 limits of uniform real Lipschitz heights on the triangular lattice

Let DD be a smooth simply connected planar domain with two marked boundary points splitting its boundary into arcs A+A_+ and A−A_-, approximated by triangular-lattice polygons of mesh δ\delta. Sample real heights on the vertices uniformly (Lebesgue measure) subject to ∣h(x)−h(y)∣≤1|h(x)-h(y)|\le1 across every edge, with boundary values +λ+\lambda on A+A_+ and −λ-\lambda on A−A_-. This hard-constraint gradient model lies outside the uniformly convex potentials covered by Schramm-Sheffield and Miller. In his ICM 2006 problem collection Schramm asked (Problem 2.3) for its scaling limits: a Gaussian free field for the height and SLE4_4 for the zero-height interface at a suitable boundary amplitude; Problem 2.2 asks the analogous questions for uniform odd integer heights with increments 0,±20,\pm2. As δ→0\delta\to0, does the height converge to a multiple of the Dirichlet Gaussian free field, and does the zero interface converge to chordal SLE4_4 for a tuned λ\lambda?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Statistical mechanics; random surfaces and SLE
Posed by
Oded Schramm (Conformally invariant scaling limits: an overview and a collection of problems, ICM 2006, Problems 2.2 and 2.3)
Year posed
2006
Years open
20y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
24 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Principal manuscript: on approximations of smooth simply connected domains, the centered uniform real Lipschitz height converges as a random distribution to a multiple of the Dirichlet GFF, and at one tuned amplitude λ\lambda the zero-height interface converges in uniform curve distance to chordal SLE4_4; the variance-boundary relation is given through an implicit coefficient, so constants are not explicit. Companion: the field part of Problem 2.2 (uniform odd integer heights, boundary ±1\pm1) with an absolutely convergent normalization formula; the Problem 2.2 interface is not proved. Companion: GFF limits for zero-boundary integer Lipschitz heights weighted by x∈[1/2,1]x\in[1/\sqrt2,1] (loop O(2)O(2)), part of a conjecture discussed by Glazman and Lammers; larger xx is not covered.

What the AI did

The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has three manuscripts of the same date: the principal one on real heights (Problem 2.3), one on the field part of Problem 2.2 for odd integer heights, and one on weighted integer heights (loop O(2) model).

Verification

No independent mathematician has checked this yet. Checked here: the abstract, Section 1 statement and history of the principal manuscript, and the introductions and main theorems of both companions, were read against Schramm's Problems 2.2 and 2.3 as the manuscripts cite them. No Lean formalization exists for this family. The SLE4_4 statement holds at one tuned two-arc amplitude identified only implicitly (through a stationary tangent-flux coefficient), and the interface part of Problem 2.2 is explicitly left open by the companion. Not refereed here.

Sources

Changelog1 change

Discussion