Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-97-01

October 1997

Graph Homomorphisms and Phase Transitions

Graham R. Brightwell and Peter Winkler


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.

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