[Paper Review] Chains with complete connections and one-dimensional Gibbs measures

Roberto Fernández, Grégory Maillard|arXiv (Cornell University)|May 1, 2003

Mathematical Dynamics and Fractals13 references4 citations

TL;DR

This paper establishes a rigorous probabilistic correspondence between one-dimensional stochastic processes (chains with complete connections) and one-dimensional Gibbs measures by introducing left-interval specifications (LIS), which mirror the specification formalism of statistical mechanics. The key contribution is a constructive, invertible mapping between LIS and specifications under hereditary uniqueness and good future conditions, generalizing known Markov and exponentially decaying continuity rate equivalences.

ABSTRACT

We discuss the relationship between discrete-time processes (chains) and one-dimensional Gibbs measures. 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).

Motivation & Objective

• To clarify the relationship between discrete-time stochastic processes and one-dimensional Gibbs measures in finite-alphabet systems with exclusion rules.
• To generalize existing equivalence results between Markov chains and Gibbs fields beyond the Markovian case.
• To provide a probabilistic framework—using left-interval specifications (LIS)—that unifies the treatment of stochastic processes and Gibbsian specifications.
• To establish conditions under which a process is Gibbsian and vice versa, particularly for non-Markovian processes with summable variation.
• To prove that the Gibbsian character of processes with exponentially decreasing continuity rates is equivalent to their consistency as stochastic processes, resolving an open question.

Proposed method

• Introduces left-interval specifications (LIS) as a probabilistic analog to statistical mechanical specifications, enabling direct comparison with Gibbs measures.
• Uses regular systems of conditional probabilities (specifications) and establishes consistency-preserving maps between LIS and specifications.
• Applies the hereditary uniqueness condition (HUC) and good future (GF) property to ensure unique and well-behaved limits in the construction.
• Employs recursive decomposition of conditional probabilities using maximal elements in finite sets, leveraging order-consistency and normalization properties.
• Proves a reconstruction theorem (Theorem A.4) for extending single-site kernels to full specifications under boundedness and order-consistency, valid for general spin spaces.
• Derives continuity rate estimates linking LIS and associated specifications via recursive identities and inductive arguments.

Experimental results

Research questions

• RQ1 Under what conditions does a stochastic process with complete connections define a Gibbs measure?
• RQ2 When can a one-dimensional Gibbs measure be represented as a consistent stochastic process with transition probabilities?
• RQ3 How do continuity rates of conditional probabilities in the process (LIS) relate to those in the corresponding specification?
• RQ4 What conditions ensure that the mapping between LIS and specifications is invertible?
• RQ5 To what extent do criteria like Dobrushin’s and boundary-uniformity for specifications transfer to the associated stochastic processes?

Key findings

• There exists a well-defined, invertible map between left-interval specifications (LIS) and specifications of Gibbs measures under the hereditary uniqueness condition (HUC) and good future (GF) property.
• For processes with exponentially decreasing continuity rates, the correspondence between LIS and specifications is bijective, generalizing known results for Markov chains.
• The continuity rate of the specification is bounded in terms of the continuity rate of the associated LIS, with explicit estimates provided in Theorem 4.18.
• If a specification satisfies Dobrushin or boundary-uniformity criteria, then the corresponding stochastic process also satisfies analogous criteria, as shown in Theorem 4.17.
• The reconstruction theorem (Theorem A.4) establishes a unique extension of single-site normalized kernels to full specifications under order-consistency and boundedness, valid even for non-Gibbsian kernels.
• The proof of the main theorems relies on inductive decomposition of conditional probabilities using maximal elements in finite index sets, with identities derived from recursive normalization and factorization properties.