Large deviations principle for Curie-Weiss models with random fields

TL;DR

This paper establishes a Large Deviations Principle for the magnetization in a generalized Curie-Weiss model with local, random external fields, providing explicit rate functions and analyzing phase diagrams.

Contribution

It extends classical Curie-Weiss models by incorporating local, random external fields and derives an explicit LDP rate function for the magnetization.

Findings

Explicit LDP rate function for magnetization per spin.

Applicable to i.i.d. and dependent random external fields.

Enables detailed phase diagram analysis.

Abstract

In this article we consider an extension of the classical Curie-Weiss model in which the global and deterministic external magnetic field is replaced by local and random external fields which interact with each spin of the system. We prove a Large Deviations Principle for the so-called {\it magnetization per spin} $S_n/n$ with respect to the associated Gibbs measure, where $S_n/n$ is the scaled partial sum of spins. In particular, we obtain an explicit expression for the LDP rate function, which enables an extensive study of the phase diagram in some examples. It is worth mentioning that the model considered in this article covers, in particular, both the case of i.\,i.\,d.\ random external fields (also known under the name of random field Curie-Weiss models) and the case of dependent random external fields generated by e.\,g.\ Markov chains or dynamical systems.