Moderate deviations for random field Curie-Weiss models

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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.