Gibbs States on Random Configurations

TL;DR

This paper proves the existence of Gibbs measures for classical particle spin systems with unbounded interactions on random graphs, establishing support properties and measurable selections for these measures.

Contribution

It introduces conditions under which Gibbs measures exist on random configurations and constructs measurable maps for these measures, advancing understanding of spin systems on random graphs.

Findings

Gibbs measures exist for almost all random configurations.

Support properties of Gibbs measures are characterized.

Measurable selection maps for Gibbs measures are constructed.

Abstract

We study a class of Gibbs measures of classical particle spin systems with spin space $S=\mathbb{R}^{m}$ and unbounded pair interaction, living on a metric graph given by a typical realization $\gamma $ of a random point process in $\mathbb{R}^{n}$. Under certain conditions of growth of pair- and self-interaction potentials, we prove that the set $\mathcal{G}(S^{\gamma})$ of all such Gibbs measures is not empty for almost all $\gamma $, and study support properties of $\nu_{\gamma}\in \mathcal{G}(S^{\gamma})$. Moreover we show the existence of measurable maps (selections) $\gamma \mapsto \nu_{\gamma}$ and derive the corresponding averaged moment estimates.