Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice

Abstract

In this paper, we describe several different meanings for the concept of Gibbs measure on the lattice $\mathbb{N}$ in the context of finite alphabets (or state space). We compare and analyze these ''in principle" distinct notions: DLR-Gibbs measures, Thermodynamic Limit and eigenprobabilities for the dual of the Ruelle operator (also called conformal measures). Among other things we extended the classical notion of a Gibbsian specification on $\mathbb{N}$ in such way that the similarity of many results in Statistical Mechanics and Dynamical System becomes apparent. One of our main result claims that the construction of the conformal Measures in Dynamical Systems for Walters potentials, using the Ruelle operator, can be formulated in terms of Specification. We also describe the Ising model, with $1/r^{2+\varepsilon}$ interaction energy, in the Thermodynamic Formalism setting and prove that its associated potential is in Walters space -we present an explicit expression. We also provide an alternative way for obtaining the uniqueness of the DLR-Gibbs measures.