Markov Random Fields, Markov Cocycles and The 3-colored Chessboard

TL;DR

This paper explores Markov random fields that do not satisfy the conditions of the Hammersley-Clifford theorem, using cocycle formalism to analyze fields like the 3-colored chessboard and those lacking finite-range interactions.

Contribution

It introduces Markov cocycles to extend the Hammersley-Clifford theorem to new classes of Markov fields, including the 3-colored chessboard.

Findings

Extended Hammersley-Clifford theorem to certain non-Gibbsian fields

Characterized Markov fields supported on the 3-colored chessboard

Constructed shift-invariant Markov fields without finite-range interactions

Abstract

The well-known Hammersley-Clifford theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbor interaction. In this paper we study Markov random fields for which the proof of the Hammersley-Clifford theorem does not apply. Following Petersen and Schmidt we utilize the formalism of cocycles for the homoclinic equivalence relation and introduce "Markov cocycles", reparametrisations of Markov specifications. The main part of this paper exploits this to deduce the conclusion of the Hammersley-Clifford theorem for a family of Markov fields which are outside the theorem's purview where the underlying graph is $\mathbb{Z}^d$. This family includes all Markov random fields whose support is the d-dimensional "3-colored chessboard". On the other extreme, we construct a family of shift-invariant Markov random fields which are not given by any finite range shift-invariant interaction.