Yuri Kondratiev — Page 6

arXiv1999

Marked Gibbs measures via cluster expansion

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.

math-ph↗

arXiv1998

Classical Limits of Euclidean Gibbs States for Quantum Lattice Models

Models of quantum and classical particles on the d-dimensional cubic lattice with pair interparticle interactions are considered. The classical model is obtained from the corresponding quantum one when the reduced physical mass of the particle tends to infinity. For these models, it is proposed to define the convergence of the Euclidean Gibbs states, when the reduced mass tends to infinity, by the weak convergence of the corresponding Gibbs specifications, determined by conditional Gibbs measures. In fact it is proved that all conditional Gibbs measures of the quantum model weakly converge to the conditional Gibbs measures of the classical model. A similar convergence of the periodic Gibbs measures and, as a result, of the order parameters, for such models with the pair interactions possessing the translation invariance, has also been proven.

math-ph↗