On the Wasserstein distance and the Dobrushin uniqueness theorem

Abstract

In this paper, we revisit the Dobrushin uniqueness theorem for Gibbs measures of lattice systems of interacting particles at thermal equilibrium. In a nutshell, Dobrushin's uniqueness theorem provides a practical way to derive sufficient conditions on the inverse temperature and/or model parameters assuring uniqueness of Gibbs measures by reducing the uniqueness problem to a suitable estimate of the Wasserstein distance between pairs of 1-point Gibbs measures with different boundary conditions. After proving a general result of completeness for the Wasserstein distance, we reformulate the Dobrushin uniqueness theorem in a convenient form for lattice systems of interacting particles described by Hamiltonians that are not necessarily translation-invariant with possibly infinite-range pair-potentials and with general complete metric spaces as single-spin spaces. Our reformulation includes existence, and covers both classical lattice systems and the Euclidean version of quantum lattice systems. Further, a generalization to the Dobrushin-Shlosman theorem for translation-invariant lattice systems is given. Subsequently, we give a series of applications of these uniqueness criteria to some high-temperature classical lattice systems including the Heisenberg, Potts and Ising models. An application to classical lattice systems for which the local Gibbs measures are convex perturbations of Gaussian measures is also given.