Given a countable graph $\mathcal{G}$ and a finite graph $\mathrm{H}$, weconsider $\mathrm{Hom}(\mathcal{G},\mathrm{H})$ the set of graph homomorphismsfrom $\mathcal{G}$ to $\mathrm{H}$ and we study Gibbs measures supported on$\mathrm{Hom}(\mathcal{G},\mathrm{H})$ . We develop some sufficient and othernecessary conditions on $\mathrm{Hom}(\mathcal{G},\mathrm{H})$ for theexistence of Gibbs specifications satisfying strong spatial mixing (withexponential decay rate). We relate this with previous work of Brightwell andWinkler, who showed that a graph $\mathrm{H}$ has a combinatorial propertycalled dismantlability if and only if for every $\mathcal{G}$ of boundeddegree, there exists a Gibbs specification with unique Gibbs measure. Westrengthen their result by showing that this unique Gibbs measure can be chosento have weak spatial mixing, but we also show that there exist dismantlablegraphs for which no Gibbs measure has strong spatial mixing.
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