  In this paper, we revisit Dobrushin's 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 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 Dobrushin's
uniqueness theorem in a convenient form for lattice systems of interacting
particles described by Hamiltonians that are not necessarily
translation-invariant and with a general complete metric space as single-spin
space. Subsequently, we give a series of applications. After proving a
uniqueness result at high temperature that we extend to the Ising and Potts
models, we focus on classical lattice systems for which the local Gibbs
measures are convex perturbations of Gaussian measures. We show that uniqueness
holds at all temperatures by constructing suitable couplings with the 1-point
Gibbs measures as marginals. The decay of correlation functions is also
discussed.
