Study of a long range perturbation of a one-dimensional Kac model

TL;DR

This paper investigates a one-dimensional Ising model with long-range $1/r^2$ interactions, using coarse graining to analyze its properties and compare with mean field theory, extending techniques from higher-dimensional models.

Contribution

It introduces a coarse graining approach for a 1D long-range Kac model and explores its relation to mean field theory, which is novel for this setting.

Findings

Coarse graining effectively describes the system's basic properties.

The model exhibits behavior consistent with mean field predictions.

Long-range interactions significantly influence the phase structure.

Abstract

We consider a one dimensional ferromagnetic Ising spin system with interactions that correspond to a $1/r^2$ long range perturbation of the usual Kac model. We apply a coarse graining procedure, widely used for higher-dimensional finite range Kac potentials, to describe the basic properties of the system and the relation with the mean field theory.