Graph Homomorphisms and Phase Transitions

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 on hom(G, H); when G is infinite, there may be more than 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.