BEHAVIOR IN LARGE DIMENSIONS OF THE POTTS AND HEISENBERG MODELS
We extend to the Potts and Heisenberg models some of the results proven in [6] for the Ising model. For both these models we prove that if the interaction is properly normalized, then as the space dimensionality goes to ∞, the spontaneous magnetization converges to the value given by the corresponding Curie-Weiss model (except possibly at the Curie-Weiss transition point, in the case of the Potts model.) For the Potts model we prove also that the ordered phases approach product measures. The proofs are based on the convergence of the free energy to the Curie-Weiss value and on infrared bounds. A consequence of our result for the q-state Potts model is an asymptotic upper bound for the transition temperature, which for q > 2 is better than the one obtained by the conventional use of infrared bounds, or comparison inequalities between different Potts models.