The Peierls argument for higher dimensional Ising models

Abstract

The Peierls argument is a mathematically rigorous and intuitive method to show the presence of a non-vanishing spontaneous magnetization in some lattice models. This argument is typically explained for the $D=2$ Ising model in a way which cannot be easily generalized to higher dimension. The aim of this paper is to present an elementary discussion of the Peierls argument for the general $D$-dimensional Ising model.