Moderate deviations for random field Curie-Weiss models

Abstract

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 \(\lambda\), and a positive integer \(k\) such that \((S_n-nm)/n^{\alpha}\) satisfies a moderate deviations principle with speed \(n^{1-2k(1-\alpha)}\) and rate function \(\lambda x^{2k}/(2k)!\), where \(1-1/(2(2k-1)) < \alpha < 1\).