Propagation of chaos in the random field Curie–Weiss model

We prove quenched propagation of chaos in the Random field mean-field Ising model, also known ad the Random field Curie–Weiss model. We show that in the paramagnetic phase, i.e. in the regime where temperature and distribution of the external field admit a unique minimizer of the expected Helmholtz free energy, quenched propagation of chaos holds. By the latter we mean that the finite-dimensional marginals of the Gibbs measure converge towards a product measure with the correct expectation as the system size goes to infinity. This holds independently of whether the system is in a high-temperature phase or at a phase transition point and alsmost surely with respect to the random external field. If the Helmholtz free energy possesses several minima, there are several possible equilibrium measures. In this case, we show that the system picks one of them at random (depending on the realization of the random external field) and propagation of chaos with respect to a product measure with the same marginals as the one randomly picked holds true. We illustrate our findings in a simple example.