Markov and stability properties of equilibrium states for nearest- neighbor interactions
type: Item
instance of: scholarly article
title: Markov and stability properties of equilibrium states for nearest- neighbor interactions
zbMATH Open document ID: 0639.60108
DOI: 10.1007/BF01224126
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B05
zbMATH DE Number: 4043022
zbMATH Keywords: nearest-neighbor interaction
zbMATH Keywords: invariant states
zbMATH Keywords: global Markov property
zbMATH Keywords: decomposition theory
zbMATH Keywords: translational invariance
author: Roelof Kuik
MaRDI profile type: Publication
cites work: Remarks on the global Markov property
cites work: On the existence of the real time evolution in Euclidean lattice gauge theories
cites work: A \(C^*\)-algebra of the twodimensional Ising model
cites work: On a \(C^*\)-algebra approach to phase transition in the two-dimensional Ising model
cites work: Axioms for Euclidean Green's functions
cites work: The global Markov property for equilibrium states which are determined by correlations in a strip
cites work: Q3854185
description: scientific article; zbMATH DE number 4043022