Gibbs states on random configurations

Gibbs states of a spin system with the single-spin space $S=\mathbb {R}^{m}$S=Rm and unbounded pair interactions are studied. The spins are attached to the points of a realization γ of a random point process in $\mathbb {R} ^{n}$Rn. Under certain conditions on the model parameters we prove that, for almost all γ, the set $\mathcal {G}(S^{\gamma })$G(Sγ) of all Gibbs states is nonempty and its elements have support properties, explicitly described in the paper. We also show the existence of measurable selections $\gamma \mapsto \nu _{\gamma }\in \mathcal {G}(S^{\gamma })$γ↦νγ∈G(Sγ) (random Gibbs measures) and derive the corresponding averaged moment estimates.