Ergodicity ofL2-Semigroups and Extremality of Gibbs States

We extend the classical results of Holley–Stroock on the characterization of extreme Gibbs states for the Ising model in terms of the irreducibility (resp. ergodicity) of the corresponding Glauber dynamics to the case of lattice systems with unbounded (linear) spin spaces. We first develop a general framework to discuss questions of this type using classical Dirichlet forms on infinite dimensional state spaces and their associated diffusions. We then describe concrete applications to lattice models with polynomial interactions (i.e., the discreteP(ϕ)d-models of Euclidean quantum field theory). In addition, we prove the equivalence of extremality and shift-ergodicity for tempered Gibbs states of these models and also discuss this question in the general framework.