Gibbs scheme in the theory of random fields

TL;DR

This paper redefines Gibbs random fields using a purely probabilistic approach, clarifying their structure and demonstrating their compatibility with physical concepts without relying on the traditional DLR-definition.

Contribution

It introduces a new probabilistic definition of Gibbs random fields, expanding the theoretical framework and showing their natural connection to statistical physics.

Findings

Provides a probabilistic framework for Gibbs random fields

Shows the compatibility of physical and probabilistic perspectives

Embeds key results of statistical physics into the new theory

Abstract

The purpose of this work is to expand and clarify the concept of the class of Gibbs random fields and give its structure the form accepted in the theory of random processes. It is possible thanks to the proposed purely probabilistic definition of the Gibbs random field without refereing to any physical notion. However, we do not oppose to each other the physical and probabilistic points of view on mathematical statistical physics; on the contrary, we show their natural compatibility within the framework of the suggested Gibbs scheme. The outlines of the corresponding theory are presented. In this theory, the DLR--definition is not used. At the same time, the related existence theorem provides one of adequate ways to construct Gibbs random fields (in the sense of probabilistic definition). The results of mathematical statistical physics are embedded in the theory of Gibbs random fields under development as its important (if not the most important) part. We describe in general terms the range of those problems that are more natural to solve using our definition. For example, the problem of the Gibbsian description of known classes of random fields as well as questions of validity of limit theorems of probability theory.