Examples of extremal lattice fields without the global Markov property (Q1075702)

We give two examples that disprove the conjecture [\textit{S. Goldstein}, Commun. Math. Phys. 74, 223-234 (1980; Zbl 0429.60097)] that every extremal Gibbs state on a lattice system has the global Markov property. Both examples live on a two dimensional lattice, and the Gibbs state is not unique. In the first example, the interaction is extended real valued; in the second one it is real valued but unbounded and not translation invariant. For an example with a translation invariant and bounded interaction, on a three dimensional lattice, see \textit{R. B. Israel}, ibid. 105, 669-673 (1986).