Gibbsian representations of continuous specifications: the theorems of Kozlov and Sullivan revisited
| Autor: | Sebastián Barbieri, Ricardo Gómez, Brian Marcus, Tom Meyerovitch, Siamak Taati |
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| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
010102 general mathematics Probability (math.PR) 82B03 82B20 37B10 37D35 60G60 Complex system FOS: Physical sciences Statistical and Nonlinear Physics Extension (predicate logic) Mathematical Physics (math-ph) Dynamical Systems (math.DS) Characterization (mathematics) Lattice (discrete subgroup) 01 natural sciences Mathematics::Algebraic Topology 0103 physical sciences FOS: Mathematics 010307 mathematical physics 0101 mathematics Mathematics - Dynamical Systems Mathematical Physics Mathematics - Probability Mathematics |
| DOI: | 10.48550/arxiv.2001.03880 |
| Popis: | The theorems of Kozlov and Sullivan characterize Gibbs measures as measures with positive continuous specifications. More precisely, Kozlov showed that every positive continuous specification on symbolic configurations of the lattice is generated by a norm-summable interaction. Sullivan showed that every shift-invariant positive continuous specification is generated by a shift-invariant interaction satisfying the weaker condition of variation-summability. These results were proven in the 1970s. An open question since that time is whether Kozlov's theorem holds in the shift-invariant setting, equivalently whether Sullivan's conclusion can be improved from variation-summability to norm-summability. We show that the answer is no: there exist shift-invariant positive continuous specifications that are not generated by any shift-invariant norm-summable interaction. On the other hand, we give a complete proof of an extension, suggested by Kozlov, of Kozlov's theorem to a characterization of positive continuous specifications on configuration spaces with arbitrary hard constraints. We also present an extended version of Sullivan's theorem. Aside from simplifying some of the arguments in the original proof, our new version of Sullivan's theorem applies in various settings not covered by the original proof. In particular, it applies when the support of the specification is the hard-core shift or the two-dimensional $q$-coloring shift for $q\geq 6$. Comment: 43 pages and 2 beautiful figures |
| Databáze: | OpenAIRE |
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