A rigorous description of the equilibrium thermodynamic properties of an infinite system of interacting ν-dimensional quantum anharmonic oscillators is given. The oscillators are indexed by the elements of a countable set L ⊂ R d , possibly irregular; the anharmonic potentials vary from site to site. The description is based on the representation of the Gibbs states in terms of path measures -the so called Euclidean Gibbs measures. It is proven that: (a) the set of such measures G t is non-void and compact; (b) every µ ∈ G t obeys an exponential integrability estimate, the same for the whole set G t ; (c) every µ ∈ G t has a Lebowitz-Presutti type support; (d) G t is a singleton at high temperatures. In the case of attractive interaction and ν = 1 we prove that |G t | > 1 at low temperatures. The uniqueness of Gibbs measures due to quantum effects and at a nonzero external field are also proven in this case. Thereby, a qualitative theory of phase transitions and quantum effects, which interprets most important experimental data known for the corresponding physical objects, is developed. The mathematical result of the paper is a complete description of the set G t , which refines and extends the results known for models of this type.

We construct Gibbs perturbations of the Gamma process on R d , which may be used in applications to model systems of densely distributed particles. First we propose a definition of Gibbs measures over the cone of discrete Radon measures on R d and then analyze conditions for their existence. Our approach works also for general Lévy processes instead of Gamma measures. To this end, we need only the assumption that the first two moments of the involved Lévy intensity measures are finite. Also uniform moment estimates for the Gibbs distributions are obtained, which are essential for the construction of related diffusions. Moreover, we prove a Mecke type characterization for the Gamma measures on the cone and an FKG inequality for them.

Gibbs states of a spin system with the single-spin space S = R m and unbounded pair interactions is studied. The spins are attached to the points of a realization γ of a random point process in R n . Under certain conditions on the model parameters we prove that, for almost all γ, the set G(S γ ) 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(S γ ) (random Gibbs measures) and derive the corresponding averaged moment estimates.

Quenched thermodynamic states of an amorphous ferromagnet are studied. The magnet is a countable collection of point particles chaotically distributed over R d , d ≥ 2. Each particle bears a real-valued spin with symmetric a priori distribution; the spin-spin interaction is pair-wise and attractive. Two spins are supposed to interact if they are neighbors in the graph defined by a homogeneous Poisson point process. For this model, we prove that with probability one: (a) quenched thermodynamic states exist; (b) they are multiple if the intensity of the underlying point process and the inverse temperature are big enough; (c) there exist multiple quenched thermodynamic states which depend on the realizations of the underlying point process in a measurable way.

We present a new method to prove existence and uniform a priori estimates for Gibbs measures associated with classical particle systems in a continuum. The method is based on the choice of appropriate Lyapunov functionals and on corresponding exponential bounds for the local Gibbs specification. Extensions to infinite range and multibody interactions are included.