Gibbs random fields with unbounded spins on unbounded degree graphs

Gibbs random fields corresponding to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs with a certain summability property are constructed. It is proven that the set of tempered Gibbs random fields is non-void and weakly compact, and that they obey uniform exponential integrability estimates. In the second part of the paper, a class of graphs is described in which the mentioned summability is obtained as a consequence of a property, by virtue of which vertices of large degree are located at large distances from each other. The latter is a stronger version of a metric property, introduced in [Bassalygo, L. A. and Dobrushin, R. L. (1986). \textrm{Uniqueness of a Gibbs field with a random potential--an elementary approach.}\textit{Theory Probab. Appl.} {\bf 31} 572--589]

A Phase Transition in a Quenched Amorphous Ferromagnet

Quenched thermodynamic states of an amorphous ferromagnet are studied. The magnet is a countable collection of point particles chaotically distributed over $\mathbb{R}^d$, $d\geq 2$. 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 particle density (i.e., 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

Gibbs states over the cone of discrete measures

We construct Gibbs perturbations of the Gamma process on \mathbbm{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 \mathbbm{R}^d and then analyze conditions for their existence. Our approach works also for general L\'evy processes instead of Gamma measures. To this end, we need only the assumption that the first two moments of the involved L\'evy 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.Comment: Keywords: Gamma process, Poisson point process, discrete Radon measures, Gibbs states, DLR equation, Mecke identity, FK

Gibbs states of continuum particle systems with unbounded spins : existence and uniqueness

We study an infinite system of particles chaotically distributed over a Euclidean space Rd. Particles are characterized by their positions x∈Rd and an internal parameter (spin) σx∈Rm and interact via position-position and (position dependent) spin-spin pair potentials. Equilibrium states of such system are described by Gibbs measures on a marked configuration space. Due to the presence of unbounded spins, the model does not fit the classical (super-) stability theory of Ruelle. The main result of the paper is the derivation of sufficient conditions of the existence and uniqueness of the corresponding Gibbs measures