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Marked Gibbs measures via cluster expansion
, 1998
"... We give a sufficiently detailed account on the construction of marked Gibbs measures in the high temperature and low fugacity regime. This is proved for a wide class of underlying spaces and potentials such that stability and integrability conditions are satisfied. That is, for state space we take a ..."
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We give a sufficiently detailed account on the construction of marked Gibbs measures in the high temperature and low fugacity regime. This is proved for a wide class of underlying spaces and potentials such that stability and integrability conditions are satisfied. That is, for state space we take a locally compact separable metric space X and a separable metric space S for the mark space. This framework allowed us to cover several models of classical and quantum statistical physics. Furthermore, we also show how to extend the construction for more general spaces as e.g., separable standard Borel spaces. The construction of the marked Gibbs measures is based on the method of cluster expansion.
ANALYSIS AND GEOMETRY ON MARKED CONFIGURATION SPACES
, 2006
"... We carry out analysis and geometry on a marked configuration space ΩM X over a Riemannian manifold X with marks from a space M. We suppose that M is a homogeneous space M of a Lie group G. As a transformation group A on ΩM X we take the “lifting ” to ΩMX of the action on X×M of the semidirect produc ..."
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We carry out analysis and geometry on a marked configuration space ΩM X over a Riemannian manifold X with marks from a space M. We suppose that M is a homogeneous space M of a Lie group G. As a transformation group A on ΩM X we take the “lifting ” to ΩMX of the action on X×M of the semidirect product of the group Diff0(X) of diffeomorphisms on X with compact support and the group GX of smooth currents, i.e., all C ∞ mappings of X into G which are equal to the identity element outside of a compact set. The marked Poisson measure πσ on ΩM X with Lévy measure σ on X × M is proven to be quasiinvariant under the action of A. Then, we derive a geometry on by a natural “lifting ” of the corresponding geometry on X × M. In particular, we construct a ΩM X gradient ∇Ω and a divergence div Ω. The associated volume elements, i.e., all probability measures µ on ΩM X with respect to which ∇Ω and div Ω become dual operators on L2 (ΩM X; µ), are identified as the mixed marked Poisson measures with mean measure equal to a multiple of σ. As a direct consequence of our results, we obtain marked Poisson space representations of the group A and its Lie algebra a. We investigate also Dirichlet forms and Dirichlet operators connected with (mixed) marked Poisson measures. 1991 AMS Mathematics Subject Classification. Primary 60G57. Secondary 57S10, 54H15 0