Extremality and the global Markov property. I: The Euclidean fields on a lattice
The problem whether a Gibbs measure for an attractive Markov specification \({\mathcal E}\equiv \{E^{\omega}_{\Lambda^ c}\}_{\omega \in \Omega,\Lambda \in {\mathcal F}_ 0}\) is extremal and has the global Markov property is considered. For the case \(\Omega\equiv \times_{i\in \Gamma}Y_ i\), where \(Y_ i\) is a partially ordered Polish space, \(\Gamma\) a countable set with the family \({\mathcal F}_ 0\) of its finite subsets, we give two sufficient conditions assuring \(\mu\) to be extremal and globally Markov. The first condition says that any Gibbs measure \(\mu\) constructed as a limit of a sequence of measures \(E^{\omega_ 0}_{\Lambda^ c_ m}(\Lambda_ m/\Gamma)\) with a \(\mu\)-a.e. dominating at infinity configuration \(\omega_ 0\in \Omega\), possesses both properties. This condition is shown to be applicable to any FKG-maximal Gibbs measure of a compact attractive Markov specification. In particular this includes the FKG-maximal Gibbs measures defining the Euclidean fields on the lattice. The second condition essentially says, that if a Gibbs measure \(\mu\) can be constructed as a limit of measures \(E^{\omega_ 0}_{\Lambda^ c_ m}(\Lambda_ m/\Gamma)\) with a configuration \(\omega_ 0\in \Omega\) such that with \(\mu\)-probability one any configuration of the system is a small fluctuation of \(\omega_ 0\in \Omega\), then \(\mu\) also possesses both the properties. This second criterion is applied to show that the corresponding properties are valid also for non FKG-maximal Gibbs measures for a large class of local specifications of statistical mechanics [see J. Stat. Phys. 43, 687-705 (1986)].
scientific article; zbMATH DE number 4001115