Moderate deviations for random field Curie-Weiss models

TL;DR

This paper establishes moderate deviations principles for the total magnetization in the random field Curie-Weiss model, revealing the effects of local randomness on the phase structure and fluctuation behavior.

Contribution

It derives the first moderate deviations principles for the dependent spins in the random field Curie-Weiss model, extending classical results to a more complex, random environment.

Findings

Moderate deviations principles depend on local field distributions.

The rate function is polynomial with degree related to the model parameters.

Results highlight the impact of local randomness on phase transitions.

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 $1-1/(2(2k-1)) < \alpha < 1$.