Delocalization of Uniform Graph Homomorphisms from $${\mathbb {Z}}^2$$ to $${\mathbb {Z}}$$

Graph homomorphisms from the $${\mathbb {Z}}^d$$ lattice to {Z}}$$ are functions on whose gradients equal one in absolute value. These height corresponding proper 3-colorings of and, two dimensions, 6-vertex model (square ice). We consider uniform model, obtained by sampling uniformly such a graph homomorphism subject boundary conditions. Our main result is that delocalizes having no translation-invariant Gibbs measures. Additional results higher dimensions and include fact every measure which ergodic under even translations extremal these measures stochastically ordered.