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