Chains with Complete Connections and One-Dimensional Gibbs Measures

Roberto Fernandez (Laboratoire de mathematiques Raphael Salem, Universite de Rouen)
Gregory Maillard (Laboratoire de mathematiques Raphael Salem, Universite de Rouen)

Abstract


We discuss the relationship between one-dimensional Gibbs measures and discrete-time processes (chains). We consider finite-alphabet (finite-spin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions generalize well known equivalence results between ergodic Markov chains and fields, as well as the known Gibbsian character of processes with exponential continuity rate. Our arguments are purely probabilistic; they are based on the study of regular systems of conditional probabilities (specifications). Furthermore, we discuss the equivalence of uniqueness criteria for chains and fields and we establish bounds for the continuity rates of the respective systems of finite-volume conditional probabilities. As an auxiliary result we prove a (re)construction theorem for specifications starting from single-site conditioning, which applies in a more general setting (general spin space, specifications not necessarily Gibbsian).

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Pages: 145--176

Publication Date: February 25, 2004

DOI: 10.1214/EJP.v9-149

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