Regular $g$-measures are not always Gibbsian

TL;DR

This paper demonstrates that regular g-measures, which depend on past information, are not always equivalent to Gibbs measures, which depend on both past and future, by providing a specific counterexample.

Contribution

It provides the first explicit example of a regular g-measure that is continuous and non-null but not Gibbsian, resolving an open problem in the field.

Findings

Existence of g-measures that are not Gibbsian.

Counterexample with a chain with variable-length memory.

Clarification of the relationship between g-measures and Gibbs measures.

Abstract

Regular $g$-measures are discrete-time processes determined by conditional expectations with respect to the past. One-dimensional Gibbs measures, on the other hand, are fields determined by simultaneous conditioning on past and future. For the Markovian and exponentially continuous cases both theories are known to be equivalent. Its equivalence for more general cases was an open problem. We present a simple example settling this issue in a negative way: there exist $g$-measures that are continuous and non-null but are not Gibbsian. Our example belongs, in fact, to a well-studied family of processes with rather nice attributes: It is a chain with variable-length memory, characterized by the absence of phase coexistence and the existence of a visible renewal scheme.