Euclidean Gibbs Measures of Quantum Crystals: Existence, Uniqueness and a Priori Estimates

Summary We give a review of recent results obtained by the authors on the existence, uniqueness and a priori estimates for Euclidean Gibbs measures corresponding to quantum anharmonic crystals. Especially we present a new method to prove existence and a priori estimates for Gibbs mesures on loop lattices, which is based on the alternative characterization of Gibbs measures in terms of their logarithmic derivatives through integration by parts formulas. This method allows us to get improvements of essentially all related existence results known so far in the literature. In particular, it applies to general (non necessary translation invariant) interactions of unbounded order and infinite range given by many-particle potentials of superquadratic growth. We also discuss different techniques for proving uniqueness of Euclidean Gibbs measures, including Dobrushin’s criterion, correlation inequalities, exponential decay of correlations, as well as Poincaré and log-Sobolev inequalities for the corresponding Dirichlet operators on loop lattices. In the special case of ferromagnetic models, we present the strongest result of such a type saying that uniqueness occurs for sufficiently small values of the particle mass.