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

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.