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