Interactions, Specifications, DLR probabilities and the Ruelle Operator in the One-Dimensional Lattice

Abstract

In this paper we consider general continuous potentials and potentials with more regularity on the symbolic space and show how the Dobrushin-Lanford-Ruelle Gibbs measures in the one-dimensional lattice are related to the Gibbs Measure associated to the Ruelle operator. In particular, we show how to obtain the Gibbs Measures for H\"older and Walters potentials considered in Thermodynamic Formalism (for the full shift) via Thermodynamic Limit. An absolutely uniformly summable interaction is associated to every H\"older potential and we show how to construct a DLR specification for any continuous potential. We also show that the Long-Range Ising Model with interactions of the form $1/r^{\alpha}$ for $\alpha>2$ is in the Walter class. We compare Thermodynamic Limit probabilities, DLR probabilities and, motivated by the achieved equivalences, consider the natural concept in Thermodynamic Formalism of dual Gibbs measures. The discussion is conducted from a Ruelle operator perspective, but it should be noted that our results are for the lattice $\mathbb{N}$ and not the usual lattice $\mathbb{Z}$. Employing the newfound connection, we prove some uniform convergence theorems for finite volume Gibbs measures. One of the primary aims of this work is to provide a "dictionary" translating potentials in Thermodynamic Formalism into interactions. Thus one of the contributions is to clarify for both the Dynamical Systems and Mathematical Statisical Mechanics communities how these (seemingly distinct) concepts of Gibbs measures are related.