Uniqueness and global Markov property for Euclidean fields: The case of general polynomial interactions

We give a general method for proving uniqueness and global Markov property for Euclidean quantum fields. The method is based on uniform continuity of local specifications (proved by using potential theoretical tools) and exploitation of a suitable FKG-order structure. We apply this method to give a proof of uniqueness and global Markov property for the Gibbs states and to study extremality of Gibbs states also in the case of non-uniqueness. In particular we prove extremality for ₂⁴ (also in the case of non-uniqueness), and global Markov property for weak coupling ₂⁴ (which solves a long-standing problem). Uniqueness and extremality holds also at any point of differentiability of the pressure with respect to the external magnetic field.