Gibbs States on Random Configurations

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.