Graph Homomorphisms and Phase Transitions Bell Laboratories 2c-379 Lucent Technologies 700 Mountain Ave

We model physical systems with \hard constraints" by the space Hom(G;H) of homomorphisms from a locally nite graph G to a xed nite constraint graph H. For any assignment of positive real activities to the nodes of H, there is at least one Gibbs measure on Hom(G;H); when G is in nite, 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 , and characterize those H which, by admitting more than one such construction, exhibit phase transition behavior.