Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM9717December 1997 
Graham R. Brightwell and Peter Winkler
Abstract
We model physical systems with ``hard constraints'' by the space Hom(G,H) of homomophisms from a locally finite graph G to a fixed finite constraint graph H. Two homomorphisms are deemed to be adjacent if they differ on a single site of G.
We investigate what appears to be a fundamental dichotomy of constraint graphs, by giving various characterizations of a class of graphs that we call dismantleable. For instance, H is dismantleable if and only if, for every G, any two homomorphisms from G to H which differ at only finitely many sites are joined by a path in Hom(G,H). If H is dismantleable, then, for any G of bounded degree, there is some assignment of activities to the nodes of H for which there is a unique Gibbs measure on Hom(G,H). On the other hand, if H is not dismantleable (and not too trivial), then there is some d such that, whatever the assignment of activities on H, there are uncountably many Gibbs measures on Hom(T_{r},H), where T_{r} is the (r+1)regular tree.
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