Phase transition of the long range Ising model in lower dimensions, for $d < \alpha \leq d + 1$, with a Peierls argument

TL;DR

This paper proves the existence of phase transitions in the long range Ising model in lower dimensions using a Peierls argument, extending previous results and employing recent techniques from related models.

Contribution

It introduces a novel Peierls argument for the long range Ising model in lower dimensions, building on recent developments in the random-field Ising model.

Findings

Phase transition occurs for the long range Ising model in specified dimensions.

Extension of previous results to lower dimensions using new contour methods.

Application of a recent argument from the random-field Ising model to the long range case.

Abstract

We extend previous results due to Ding and Zhuang in order to prove that a phase transition occurs for the long range Ising model in lower dimensions. By making use of a recent argument due to Affonso, Bissacot and Maia from 2022 which establishes that a phase transition occurs for the long range, random-field Ising model, from a suggestion of the authors we demonstrate that a phase transition also occurs for the long range Ising model, from a set of appropriately defined contours for the long range system, and a Peierls' argument.