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Gaussian fields and interfaces for triangular-lattice Lipschitz heights

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Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments \(0,\pm2\) and two-arc boundary values \(\pm1\), and zero-boundary integer Lipschitz heights weighted by fixed \(x\in[1/\sqrt2,1]\). Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal SLE4, establishing Schramm’s real-field/interface predictions.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

Three current papers; three replaced by later editions.

The paper states

Theorem 1.1

In The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data, 6 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Theorem 1.2 in Gaussian free-field limits of weighted integer Lipschitz heights () and Theorem 1.1 in Uniform real Lipschitz surfaces on the triangular lattice ().

Let D be a bounded smooth simply connected planar domain with two distinct marked boundary points. Let D_\delta be simply connected triangular-lattice polygonal approximations with uniformly converging marked boundary parametrizations and edge-midpoint marks, and let h_\delta have the uniform odd-height law (1.1). There is a constant \sigma>0, depending only on the unit triangular-lattice model, such that

h_\delta\ \Longrightarrow\ \bar g+\sigma\Phi_D \qquad\text{in }\mathcal D'(D).

For every f\in C_c^\infty(D), \EE h_\delta(f)\to\int_D\bar g f. Consequently

\frac{h_\delta-\EE h_\delta}{\sigma} \ \Longrightarrow\ \Phi_D \qquad\text{in }\mathcal D'(D).

All finite vectors of test-function integrals converge jointly to the corresponding Gaussian vectors, and all their mixed moments converge. The constant is specified by the absolutely convergent finite-volume series in Proposition 6.1; in the spectral convention below, \sigma=4\pi\sqrt{c_*}.

Reported proof verification
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Papers and explanations
Three current papers, each readable here as a page beside its PDF.

Papers

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Gaussian free-field limits of weighted integer Lipschitz heights

6 October 2026, current edition.

Read it herePDF

This version supplies the annulus proof of the circuit estimate used in the discrete argument; see the previous version.

Earlier editions: 25 September 2026

We prove a Gaussian free field scaling limit for weighted integer Lipschitz heights on the triangular lattice. 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 \(x\in[1/\sqrt2,1]\). The convergence holds as a random distribution on every bounded C2 Jordan domain under inside lattice approximations with uniformly convergent boundary parametrizations. This includes the uniform height model and the predicted critical endpoint.

The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data

6 October 2026, current edition.

Read it herePDF

This version derives the conditional cut and loop estimates from the companion weighted annulus circuit theorem at $x=1$; see the previous version.

Earlier editions: 25 September 2026

We prove that the centered uniform odd integer height function on triangular-lattice approximations of a smooth simply connected domain, with neighboring differences zero or two and boundary values +1 and −1 on two arcs, converges to a universal multiple of the Dirichlet Gaussian free field. This resolves the field part of Schramm’s Problem 2.2. We give an absolutely convergent finite-volume formula for the normalization. The proof combines reflection positivity, a spectral sum rule, and boundary comparison with Gaussian moment identities.

Uniform real Lipschitz surfaces on the triangular lattice

6 October 2026, current edition.

Read it herePDF

This version repairs the boundary-attachment, landing-comparison, and multiscale separation arguments, including the conditional walk and hookup steps, and supplies terminal-side identification and harmonic-mass nonreturn for whole-curve convergence; see the previous version.

Earlier editions: 25 September 2026

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. On approximations of smooth simply connected domains, the centered height field converges as a random distribution to a multiple of the Dirichlet Gaussian free field. At one tuned two-arc boundary amplitude, the zero-height interface converges in uniform curve distance to chordal SLE\(_4\). We identify the relation between the field variance and the boundary height in terms of an implicit stationary tangent-flux coefficient.

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