Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-97-17

December 1997

Gibbs Measures and Dismantleable Graphs

Graham R. Brightwell and Peter Winkler


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(Tr,H), where Tr is the (r+1)-regular tree.

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