Gibbs states on random configurations

Abstract

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.

A Phase Transition in a Quenched Amorphous Ferromagnet

“…In a similar way, one can prove the statements mentioned above if the underlying graph is as in the random connection model, see [18,31,33] or a tempered Gibbs random field, see [17,Corollary 3.7]. The only condition is that the graph almost surely has the summability property as in Proposition 4, see [14] for more detail.…”

“…By similar arguments, one can show that G t (β|γ ) is compact in the L-topology. Finally, let us mention that u > 2 in (14) and θ in Proposition 4 should be such that θ(u − 2) > 2. Under this condition the sufficiently fast decay of the tail of χ compensates destabilizing effect of the property (9) of the underlying graph, see [26] for more detail.…”

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

“…In general, the projected measure M does not coincide with the Gibbs measure µ Φ and cannot be described in terms of position-position interactions alone. Thus it is not clear whether the existence result from [10] could be used in order to prove the existence of the annealed Gibbs measure µ. Furthermore, (2.46) indicates that one cannot directly compare (e.g., by means of various correlation inequalities known for measures on S γ , see e.g.…”

“…[3,4,29]. The corresponding Gibbs measures µ γ on the product spaces S γ were constructed in [10]. The relationship between Gibbs measures of these two types can be expressed by the disintegration formula…”

“…This topology has been employed in different frameworks in e.g. [1], [11] and [26]; for a short account of its properties see also [10]. An advantage of the τ -topology is that it makes Γ(X) a Polish space, in contrast to the vague topology inherited from Γ( X) (which is generated by the maps (2.3) with g ∈ C 0 ( X)).…”