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We show the existence of a variational principle for graph homomorphisms from to a d-regular tree. The technique is based on a discrete Kirszbraun theorem and a concentration inequality obtained through the dynamics of the model. As another consequence of the concentration inequality we also obtain the existence of a continuum of translation-invariant ergodic gradient Gibbs measures for graph homomorphisms from to a regular tree. The method is sufficiently robust such that it could be applied to other discrete models with a quite general target graphs.