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 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.The Curie-Weiss model is a mean-field model of a ferromagnet. Due to its mean-field structure the spatial location of the spins is unimportant. The Hamiltonian of the Curie-Weiss model with external magnetic field h ∈ R can therefore be described

Path-space moderate deviation principles for the random field Curie-Weiss model

“…To conclude, it is worth to mention that our results are consistent with the moderate deviation principles obtained in [LM12] for the random field Curie-Weiss model at equilibrium. Indeed, as prescribed by Thm.…”

“…It is worth to mention that our statements are in agreement with the static results found in [LM12]. The paper is organized as follows.…”

Fluctuations of the Magnetization in the Block Potts Model

“…As usual in large deviations theory, the large deviations rate function derived in [19, §3] is not obtained when setting θ = 0 in Theorem 6, while for θ = 1/2 one obtains as rate function the exponent of the CLT in Theorems 1 and 2. It is folklore knowledge in large and moderate deviation theory that such a smooth transition from the MDP regime to the regime of the CLT is usually true, even though there are counterexamples (see e.g., [22] for a counterexample).…”

Moderate deviations for a Curie–Weiss model with dynamical external field

“…In the case of multiple minima a similar, general transfer principle for conditional measures is still an open question. In the recent preprint [13] such a transfer principle was proved under additional conditions on the inverse temperature for the random field Curie-Weiss model (Lem. 5.2 (ii)).…”