Gibbsian Representations of Continuous Specifications: The Theorems of Kozlov and Sullivan Revisited

The theorems of Kozlov and Sullivan characterize Gibbs measures as with positive continuous specifications. More precisely, showed that every specification on symbolic configurations the lattice is generated by a norm-summable interaction. shift-invariant interaction satisfying weaker condition variation-summability. These results were proven in 1970s. An open question since time whether Kozlov’s theorem holds setting, equivalently Sullivan’s conclusion can be improved from variation-summability to norm-summability. We show answer no: there exist specifications are not any On other hand, we give complete proof an extension, suggested Kozlov, characterization configuration spaces arbitrary hard constraints. also present extended version theorem. Aside simplifying some arguments original proof, our new applies various settings covered proof. In particular, it when support hard-core shift or two-dimensional q-coloring for $$q\ge 6$$ .