On the mean-field equations for ferromagnetic spin systems

TL;DR

This paper derives mean-field equations for ferromagnetic spin systems, providing explicit error bounds and connecting them to random matrix theory, with applications to various models.

Contribution

It introduces a novel approach linking mean-field equations with free convolution in random matrix theory, including explicit error bounds for finite volumes.

Findings

Derived mean-field equations with explicit error bounds

Linked mean-field equations to free convolution formalism

Applied results to Kac interactions, diluted models, and external fields

Abstract

We derive mean-field equations for a general class of ferromagnetic spin systems with an explicit error bound in finite volumes. The proof is based on a link between the mean-field equation and the free convolution formalism of random matrix theory, which we exploit in terms of a dynamical method. We present three sample applications of our results to Ka\'{c} interactions, randomly diluted models, and models with an asymptotically vanishing external field.