Translation invariant extensions of finite volume measures
| Autor: | Goldstein, S., Kuna, T., Lebowitz, J. L., Speer, E. R. |
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| Rok vydání: | 2015 |
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| Zdroj: | J. Stat. Phys. 166 (2017), 765-782 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10955-016-1595-8 |
| Popis: | We investigate the following questions: Given a measure $\mu_\Lambda$ on configurations on a subset $\Lambda$ of a lattice $\mathbb{L}$, where a configuration is an element of $\Omega^\Lambda$ for some fixed set $\Omega$, does there exist a measure $\mu$ on configurations on all of $\mathbb{L}$, invariant under some specified symmetry group of $\mathbb{L}$, such that $\mu_\Lambda$ is its marginal on configurations on $\Lambda$? When the answer is yes, what are the properties, e.g., the entropies, of such measures? Our primary focus is the case in which $\mathbb{L}=\mathbb{Z}^d$ and the symmetries are the translations. For the case in which $\Lambda$ is an interval in $\mathbb{Z}$ we give a simple necessary and sufficient condition, local translation invariance (LTI), for extendibility. For LTI measures we construct extensions having maximal entropy, which we show are Gibbs measures; this construction extends to the case in which $\mathbb{L}$ is the Bethe lattice. On $\mathbb{Z}$ we also consider extensions supported on periodic configurations, which are analyzed using de~Bruijn graphs and which include the extensions with minimal entropy. When $\Lambda\subset\mathbb{Z}$ is not an interval, or when $\Lambda\subset\mathbb{Z}^d$ with $d>1$, the LTI condition is necessary but not sufficient for extendibility. For $\mathbb{Z}^d$ with $d>1$, extendibility is in some sense undecidable. Comment: 28 pages, LaTex, 3 files; significant amount of new material added |
| Databáze: | arXiv |
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