Gibbs states on random configurations

Gibbs states of a spin system with the single-spin space 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 Rn. Under certain conditions on the model parameters we prove that, for almost all γ, the set 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 γ↦νγ∈G(Sγ) (random Gibbs measures) and derive the corresponding averaged moment estimates