Geometry of contours and Peierls estimates in d=1 Ising models

TL;DR

This paper develops a geometric framework using triangles to analyze one-dimensional long-range Ising models, providing a direct proof of phase transitions at low temperatures through Peierls bounds.

Contribution

It introduces a novel geometric description of spin configurations and establishes Peierls bounds for long-range interactions in 1D Ising models.

Findings

Proves phase transitions at low temperatures for the model

Establishes Peierls bounds using geometric contours

Provides a new geometric perspective on long-range interactions

Abstract

Following Fr\"ohlich and Spencer, we study one dimensional Ising spin systems with ferromagnetic, long range interactions which decay as $|x-y|^{-2+\alpha}$, $0\leq \alpha\leq 1/2$. We introduce a geometric description of the spin configurations in terms of triangles which play the role of contours and for which we establish Peierls bounds. This in particular yields a direct proof of the well known result by Dyson about phase transitions at low temperatures.