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Delocalization of uniform graph homomorphisms from Z^2 to Z

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Graph homomorphisms from the Zd lattice to Z are functions on Zd whose gradients equal 1 in absolute value. These functions are the height functions corresponding to proper 3-colorings of Zd and, in two dimensions, corresponding to the 6-vertex model (square ice). We show that the model delocalizes in two dimensions, having no translation-invariant Gibbs measures for the uniform sampling subject to boundary conditions. We also obtain additional results in higher dimensions including the facts that every ergodic Gibbs measure is extremal and that the ergodic Gibbs measures are stochastically ordered. The proof follows interesting but little known arguments presented by Scott Sheffield in Random surface which are adapted and simplified to the present settings.

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