On the characterization of a finite random field by conditional distribution and its Gibbs form

TL;DR

This paper demonstrates how methods from mathematical statistical physics can characterize finite random fields through their conditional distributions, establishing conditions for their existence, uniqueness, and Gibbsian nature, including applications to Markov random fields.

Contribution

It introduces a potential-free axiomatic approach to identify Gibbsian finite random fields and provides necessary and sufficient conditions for their existence and uniqueness.

Findings

Finite random fields can be characterized by their one-point conditional distributions.

Any finite random field is Gibbsian under the proposed axiomatic framework.

The approach applies to Markov random fields, extending classical results.

Abstract

In this paper, we show that the methods of mathematical statistical physics can be successfully applied to random fields in finite volumes. As a result, we obtain simple necessary and sufficient conditions for the existence and uniqueness of a finite random field with a given system of one-point conditional distributions. Using the axiomatic (without the notion of potential) definition of Hamiltonian, we show that any finite random field is Gibbsian. We also apply the proposed approach to Markov random fields.