Mean-field models with short-range correlations

Given an arbitrary finite dimensional Hamiltonian H₀, we consider the model H=H₀+ΔH, where ΔH is a generic fully connected interaction. By using the strong law of large numbers, we easily prove that, for all such models, a generalized Curie-Weiss mean-field equation holds. Unlike traditional mean-field models, the term H₀ gives rise to short-range correlations and, furthermore, when H₀ has negative couplings, first-order phase transitions and inverse transition phenomena may take place even when only two-body interactions are present. The dependence from a non-uniform external field and finite-size effects are also explicitly calculated. Partially, these results were derived long ago by using min-max principles but remained almost unknown.