Gaussian fields and interfaces for triangular-lattice Lipschitz heights

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 heightsSource 1,Source 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 methodsSource 3,Source 4. Glazman and Lammers studied how much these surfaces wander. Their 2025 paper still called the free field limit a conjectureSource 5. Knowing the size of the wandering alone does not give the limitSource 6.

The first paper uses the weight x from the exampleSource 7. The contours where the height changes form loops, a model physicists call the loop O(2) modelSource 8. The paper claims a free field limit for every x from 1/√2 to 1, with height 0 on the boundarySource 9. That range includes the uniform case x = 1, and the predicted critical end point x = 1/√2Source 10. The field is divided by a constant that depends only on x. The paper gives no formula for that constantSource 11. The surface is averaged over small patches, with no extra log rescalingSource 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 problemSource 1. Its tools include reflection positivity, a spectral sum rule and boundary comparisonSource 13. The curve where the height crosses zero is a separate question, not settled hereSource 14.

The third paper lets heights be any real numbers, with neighbors still within 1 of each otherSource 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 SLE₄Source 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 resultSource 16. The tuning involves a stiffness constant, and the paper gives no closed formula for itSource 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.