Moderate Deviations for Random Field Curie-Weiss Models

The random field Curie-Weiss model is derived from the classical Curie-Weiss model by replacing the deterministic global magnetic field by random local magnetic fields. This opens up a new and interestingly rich phase structure. In this setting, we derive moderate deviations principles for the random total magnetization Sn, which is the partial sum of (dependent) spins. A typical result is that under appropriate assumptions on the distribution of the local external fields there exist a real number m, a positive real number λ, and a positive integer k such that (Sn−nm)/nα satisfies a moderate deviations principle with speed n1−2k(1−α) and rate function λx2k/(2k)!, where 1−1/(2(2k−1))<α<1.