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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 $S_n$, 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 $(S_n-nm)/n^α$ satisfies a moderate deviations principle with speed $n^{1-2k(1-α)}$ and rate function $λx^{2k}/(2k)!$, where $1-1/(2(2k-1)) < α< 1$.