Stochastic dynamics of compact spins: ergodicity and irreducibility (Q5937956)

Let \(M\) be a compact Riemannian manifold and \(Z^d\) an integer lattice. The semigroup \(T_t =e^{-tH_\mu}\) in \(L^2 (\mu)\) where \(\mu\) is a Gibbs measure on \(M^Z{^d}\) and \(H_\mu\) is the corresponding Dirichlet operator is studied. The main result of the paper is the equivalence of irreducibility of the Dirichlet form associated with \(\mu\) and extremality of \(\mu\).