Gibbs random fields with unbounded spins on unbounded degree graphs

TL;DR

This paper constructs and analyzes Gibbs random fields with real-valued spins on unbounded degree graphs, proving their existence, compactness, and exponential integrability under certain summability conditions.

Contribution

It introduces a framework for Gibbs fields on unbounded degree graphs and establishes their fundamental properties, extending previous metric conditions to a broader class of graphs.

Findings

Non-empty set of tempered Gibbs random fields established

Proved weak compactness of the Gibbs fields

Derived uniform exponential integrability estimates

Abstract

Gibbs random fields corresponding to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs with a certain summability property are constructed. It is proven that the set of tempered Gibbs random fields is non-void and weakly compact, and that they obey uniform exponential integrability estimates. In the second part of the paper, a class of graphs is described in which the mentioned summability is obtained as a consequence of a property, by virtue of which vertices of large degree are located at large distances from each other. The latter is a stronger version of a metric property, introduced in [Bassalygo, L. A. and Dobrushin, R. L. (1986). \textrm{Uniqueness of a Gibbs field with a random potential--an elementary approach.}\textit{Theory Probab. Appl.} {\bf 31} 572--589].