Dobrushinʹs compactness criterion for euclidean gibbs measures

We prove existence and uniform á priori estimates for Euclidean Gibbs measures corresponding to certain quantum systems with unbounded spins, pair potentials of superquadratic growth, and infinite radius of interaction. The quantum particles are indexed by the elements of a countable, possibly irregular, set L ⊂ ∝d. We use Dobrushinʹs criterion and give a direct construction of appropriate compact functions on (infinite dimensional) loop spaces. For the quantum systems on L := ∝d, with the superquadratic interactions of finite range, a new uniqueness result is established by means of the Dobrushin-Pechersky criterion.