Markov and stability properties of equilibrium states for nearest- neighbor interactions
Consider models on the lattice \({\mathbb{Z}}^ d\) with finite spin space per lattice point and nearest-neighbor interaction. Under the condition that the transfer matrix is invertible we use a transfer-matrix formalism to show that each Gibbs states is determined by its restriction to any pair of adjacent (hyper)planes. Thus we prove that (also in multiphase regions) translationally invariant states have a global Markov property. The transfer-matrix formalism permits us to view the correlation functions of a classical d-dimensional system as obtained by a linear functional on a noncommutative (quantum) system in (d-1)-dimensions. More precisely, for reflection positive classical states and an invertible transfer matrix the linear functional is a state. For such states there is a decomposition theory available implying statements on the ergodic decompositions of the classical state in d dimensions. In this way we have stability properties of \({\mathbb{Z}}^ d_{ev}\)-ergodic states and the absence of certain types of breaking of translational invariance.
scientific article; zbMATH DE number 4043022