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Gibbs measures and dismantlable graphs. (English) Zbl 1030.05101
Summary: We model physical systems with “hard constraints” by the space \(\operatorname{Hom}(G,H)\) 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 \(\operatorname{Hom}(G, H)\). 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 \(\operatorname{Hom}(G, H)\). 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 \(\operatorname{Hom}(T_r,H)\), where \(T_r\) is the \((r+ 1)\)-regular tree.

05C75 Structural characterization of families of graphs
60C05 Combinatorial probability
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B05 Classical equilibrium statistical mechanics (general)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
Full Text: DOI
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