Gaussian fields and interfaces for triangular-lattice Lipschitz heights

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.

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.

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.