## Graph Homomorphisms and Phase Transitions

### Authors: Graham R. Brightwell and Peter Winkler

ABSTRACT

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.

Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1997/97-62.ps.gz