We model physical systems with "hard constraints" by the space Hom(G, H) of homomorphisms from a locally finite graph G to a fixed finite constraint graph H. For any assignment #lambda# of positive real activities to the nodes of H, there is at least one Gibbs measure oh Hom(G, H); when G is infinite, there may be more that one. When G is a regular tree, the simple, invariant Gibbs measures on Hom(G, H) correspond to node-weighted branching random walks on H. We show that such walks exist for every H and #lambda#, and characterize those H which, by admitting more than one such construction, exhibit phase transition behavior.
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