1 · The essentials
Imagine assigning a random height to every point of a triangular mesh, while keeping neighboring heights close together. These manuscripts study what the resulting landscape looks like as the mesh becomes finer inside a smooth region. They show in the manuscripts that several versions approach the same kind of Gaussian random field, after adjusting the overall height scale. The main manuscript treats whole-number heights fixed to zero at the boundary, with a weight that favors or discourages height changes within a specified range. A companion treats odd whole-number heights with boundary heights of +1 and −1 on two arcs. Another treats real-valued heights and describes the limiting curve separating them at height zero, when opposite boundary heights are specially tuned. The field limits concern spatial averages rather than a height at each individual point. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, picture a quilt of small tiles whose elevations may differ only slightly from those of their neighbors. The manuscripts describe a common large-scale pattern of fluctuations as the tiles shrink, but the limiting object is not an ordinary quilt with a well-defined elevation everywhere.
Why it matters
The results shown in the manuscripts connect several constrained height models to Gaussian free fields under explicit boundary and domain assumptions. The odd-integer manuscript describes its result as resolving the field part of Schramm’s Problem 2.2, while the real-height manuscript describes its field and interface results as resolving Problem 2.3. Together, they address both fluctuations across a region and, in the tuned real-valued case, the shape of a zero-height interface.
2 · Going further
The main problem concerns integer Lipschitz heights on the triangular lattice, with zero boundary values and a factor for each edge on which the height changes. For every fixed , the main manuscript shows convergence, after division by a positive constant depending only on , to the zero-Dirichlet Gaussian free field. The convergence is as a random distribution on every bounded Jordan domain, using inside lattice approximations with uniformly convergent boundary parametrizations. The abstract explicitly includes the uniform height model and what it calls the predicted critical endpoint.
The odd-integer companion considers uniform heights with neighboring differences in and boundary values and on two arcs of a smooth simply connected domain. Its centered field is shown in the manuscript to converge to a universal multiple of the Dirichlet Gaussian free field. It also gives an absolutely convergent finite-volume formula for the normalization. The argument described in its abstract combines reflection positivity, a spectral sum rule, boundary comparison, and Gaussian moment identities; the supplied abstract of the main manuscript does not specify its argument.
The real-valued companion studies uniformly sampled nearest-neighbor Lipschitz heights on triangular-lattice approximations of smooth simply connected domains. It shows in the manuscript that the centered field converges as a random distribution to a multiple of the Dirichlet Gaussian free field. At one tuned opposite two-arc boundary amplitude, it also shows convergence of the zero-height interface, in uniform curve distance, to chordal . It identifies a relation between field variance and boundary height through an implicit stationary tangent-flux coefficient. Thus the companions supply different boundary and height settings, with the real-valued manuscript additionally treating an interface limit.
Useful notions
- Random height functions on a lattice
- Nearest-neighbor height constraints
- Boundary values and centering
- Scaling limits and random distributions
- Gaussian fields
What was known before / what is new
Before
The source places the odd-integer field result in the context of Schramm’s Problem 2.2 and the real-valued field and interface results in the context of Problem 2.3. It also identifies the endpoint included in the main manuscript as a predicted critical endpoint. The supplied text does not give further details about earlier partial results or their history.
New
The main manuscript shows a Gaussian free field limit throughout the fixed parameter range for zero-boundary weighted integer Lipschitz heights. The odd-integer companion supplies a two-arc field limit and an absolutely convergent finite-volume normalization formula. The real-valued companion supplies both a field limit and, at one tuned boundary amplitude, an interface limit, together with an implicit relation between field variance and boundary height.
Limits and assumptions
These statements concern triangular-lattice models under the stated height constraints and boundary conditions. The main result requires fixed , a bounded Jordan domain, and inside approximations with uniformly convergent boundary parametrizations; the companions assume smooth simply connected domains. Field convergence is distributional, not pointwise, and the real-valued interface statement applies at one tuned two-arc boundary amplitude rather than arbitrary boundary heights. The source supplies no conclusions outside these settings and no additional list of remaining open problems. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Lipschitz heights —
- Heights constrained so that neighboring lattice sites cannot differ by more than an allowed amount.
- Centered field —
- A random height field with its expected height subtracted.
- Random distribution —
- A random object described through its averages against smooth test functions rather than necessarily through values at individual points.
- Dirichlet Gaussian free field —
- The Gaussian random distribution appearing as the limiting field with prescribed boundary behavior, with zero-Dirichlet specifying zero boundary data.
- Scaling limit —
- A limiting random object obtained as the lattice mesh becomes finer, with the stated normalization.
- Chordal SLE₄ —
- The random-curve law identified in the real-height manuscript as the limit of the zero-height interface at the tuned boundary amplitude.
Keywords : Lipschitz heights · Gaussian free field · SLE₄
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments and two-arc boundary values , and zero-boundary integer Lipschitz heights weighted by fixed . 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.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Gaussian free-field limits of weighted integer Lipschitz heights
06/10/2026We 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 . 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
06/10/2026We 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
06/10/2026We prove the Gaussian free field and SLE 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. We identify the relation between the field variance and the boundary height in terms of an implicit stationary tangent-flux coefficient.