Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations

Abstract Uniform integer-valued Lipschitz functions on a domain of size N the triangular lattice are shown to have variations order $$\sqrt{\log N}$$ log N . The level lines such form loop O (2) model edges hexagonal with edge-weight one. An infinite-volume Gibbs measure for is constructed as thermodynamic limit and be unique. It contains only finite loops has properties indicative scale-invariance: macroscopic appearing at every scale. existence carries over height pinned origin; uniqueness does not. proof based representation via pair spin configurations that satisfy FKG inequality. We prove RSW-type estimates certain connectivity notion in aforementioned model.