Gibbs Measures and Dismantlable Graphs

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 . 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 dismantlable . For instance, H is dismantlable 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 dismantlable, 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 dismantlable (and not too trivial), then there is some r 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.