Uniform real Lipschitz surfaces on the triangular lattice
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…
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