Moderate Deviations for Random Field Curie-Weiss Models

摘要

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 S _n, 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 (S _n-nm) ~α satisfies a moderate deviations principle with speed n ~(1-2k(1-α)) and rate function λx ~(2k)/(2k)!, where 1-1/(2(2k-1))<α<1.