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 $γ$ 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 $\mathcal{G}(S^γ)$ of all such Gibbs measures is not empty for almost all $γ$, and study support properties of $ν_γ\in \mathcal{G}(S^γ)$. Moreover we show the existence of measurable maps (selections) $γ\mapsto ν_γ$ and derive the corresponding averaged moment estimates.

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