Duality for general attractive spin systems with applications in one dimension
Duality has become an important tool in the theory of interacting spin systems in \({\mathbb{Z}}^ d\). In this paper a new version of dual processes is introduced, whose paths take their values in the collection of finite subsets instead of the elements of \({\mathbb{Z}}^ d\). They promise to be applicable to more general classes of processes than the previous ones, which were limited to special cases. As an example, the author generalizes results of \textit{R. Durrett} and \textit{D. Griffeath} [ibid. 11, 1-15 (1983; Zbl 0508.60080)], which they proved for the nearest neighbour one-dimensional contact process to all attractive nearest neighbour one-dimensional spin systems with critical behaviour. These results concern exponential convergence starting from all 1's to an equilibrium measure, which has exponentially decaying correlation functions.
exponentially decaying correlation functions
scientific article; zbMATH DE number 3976064