Phase Transition in Long-Range $q-$state Models via Contours. Clock and Potts Models with Fields

TL;DR

This paper establishes phase transitions in long-range $q$-state models, including Clock and Potts models, using a novel contour method that accounts for the models' group structure and applies to various interaction decay rates.

Contribution

It introduces a new contour definition for long-range spin systems and proves phase transitions for a broad class of models with power-law interactions and external fields.

Findings

Proves phase transition for long-range ferromagnetic models with $eta > 0$.

Demonstrates phase transition in Potts models with decaying and random fields.

Extends contour methods to multidimensional and long-range settings.

Abstract

Using the group structure of the state space of $q-$state models, a new definition of contour for long-range spin-systems in $\Z^d$ ($d\geq 2$), and a multidimensional version of Fr\"{o}hlich-Spencer contours, we prove phase transition for a class of ferromagnetic long-range systems which includes the Clock and Potts models. Our arguments work for the entire region of exponents of regular power-law interactions, namely $\alpha > d$, and for any $q \geq 2$. As an application, we prove phase transition for Potts models with decaying fields when the field decays fast enough and in the presence of a random external field.