Extremality and the global Markov property. I: The Euclidean fields on a lattice
type: Item
instance of: scholarly article
title: Extremality and the global Markov property. I: The Euclidean fields on a lattice
zbMATH Open document ID: 0617.60048
DOI: 10.1016/0047-259X(87)90104-7
published in: Journal of Multivariate Analysis
publication date: 1987-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 82B20
zbMATH DE Number: 4001115
zbMATH Keywords: attractive specification
zbMATH Keywords: extremality
zbMATH Keywords: global Markov property
zbMATH Keywords: Euclidean quantum fields
zbMATH Keywords: Gibbs measure
zbMATH Keywords: FKG-maximal Gibbs measure
zbMATH Keywords: statistical mechanics
author: Boguslaw Zegarliński
MaRDI profile type: Publication
cites work: Uniqueness and the global Markov property for Euclidean fields. The case of trigonometric interactions
cites work: Compactness and the maximal Gibbs state for random Gibbs fields on a lattice
cites work: Q4144005
cites work: Q3906882
cites work: Remarks on the global Markov property
cites work: Random fields
cites work: Uniqueness and the global Markov property for Euclidean fields: The case of general exponential interaction
description: scientific article; zbMATH DE number 4001115