Large deviations principle for Curie–Weiss models with random fields

Abstract

In this paper, we consider an extension of the classical Curie–Weiss model in which the global and deterministic external magnetic field is replaced by local and random external fields which interact with each spin of the system. We prove a large deviations principle for the so-called magnetization per spin Sn/n with respect to the associated Gibbs measure, where Sn/n is the scaled partial sum of spins. In particular, we obtain an explicit expression for the rate function, which enables an extensive study of the phase diagram in some examples. It is worth mentioning that the model considered in this paper covers, in particularly, both the case of i.i.d. random external fields (also known under the name of random field Curie–Weiss models) and the case of dependent random external fields generated by e.g. Markov chains or dynamical systems.