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

Abstract

In the present paper we prove moderate deviations for a Curie-Weiss model with external magnetic field generated by a dynamical system, as introduced by Dombry and Guillotin-Plantard in [C. Dombry and N. Guillotin-Plantard, Markov Process. Related Fields 15 (2009) 1-30]. The results extend those already obtained for the Curie-Weiss model without external field by Eichelsbacher and Löwe in [P. Eichelsbacher and M. Löwe, Markov Process. Related Fields 10 (2004) 345-366]. The Curie-Weiss model with dynamical external field is related to the so called dynamic Z-random walks (see [N.Guillotin-Plantard and R. Schott, Theory and applications, Elsevier B. V., Amsterdam ( 2006).]). We also prove a moderate deviation result for the dynamic Z-random walk, completing the list of limit theorems for this object.

Moderate Deviations for Random Field Curie-Weiss Models

“…The proofs of our main theorems use ideas that can also be found in [19,22,6]. To put it roughly, the proof is mainly divided into four parts:…”

“…The latter is a well-studied object of Laplace's Method (use e. g. a slight generalization of Lemma 2.6 in [22]) and we get…”

“…However, a MDP also inherits properties from the large deviations such as the exponential decay of probabilities. Although the universality of these properties have never been proved one does find more examples for their existence even if the CLT density is not the normal distribution (see e. g. [8,20,21,22]). Note, however that the picture may be different in the setting of disordered models.…”

On the accuracy of the normal approximation for the free energy in the Random Energy Model

“…Note that moderate deviations for mean field models from statistical mechanics have already been studied (cf. e. g. [10,12]). We will show in this article that (3) holds if and only if t N = o( √ N ), that is, (3) holds only in a small range of scalings close to the CLT scaling.…”

Metastates in Mean-Field Models with Random External Fields Generated by Markov Chains

“…over the sites, we now treat a quenched disorder which is correlated and generated by a Markov chain with transition matrix M . For a study of the Curie-Weiss model in a dynamical external field see [8,20].…”