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.