Estimation of local microcanonical averages in two lattice mean-field models using coupling techniques

We consider an application of probabilistic coupling techniques which provides explicit estimates for comparison of local expectation values between label permutation invariant states, for instance, between certain microcanonical, canonical, and grand canonical ensemble expectations. A particular goal is to obtain good bounds for how such errors will decay with increasing system size. As explicit examples, we focus on two well-studied mean-field models: the discrete model of a paramagnet and the mean-field spherical model of a continuum field, both of which are related to the Curie-Weiss model. The proof is based on a construction of suitable probabilistic couplings between the relevant states, using Wasserstein fluctuation distance to control the difference between the expectations in the thermodynamic limit.