Long-range order in the 3-state antiferromagnetic Potts model in high dimensions

TL;DR

This paper proves the existence of long-range order and multiple Gibbs measures in the high-dimensional 3-state antiferromagnetic Potts model at low temperatures, confirming a long-standing conjecture.

Contribution

It establishes the high-dimensional case of the Kotecký conjecture by demonstrating spontaneous magnetization and multiple Gibbs measures.

Findings

Existence of six extremal ergodic Gibbs measures.

Spontaneous magnetization in high dimensions.

Confirmation of the Kotecký conjecture in high dimensions.

Abstract

We prove the existence of long-range order for the 3-state Potts antiferromagnet at low temperature on $\mathbb{Z}^d$ for sufficiently large $d$. In particular, we show the existence of six extremal and ergodic infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one bipartition class have a much higher probability to be in one state than in either of the other two states. This settles the high-dimensional case of the Koteck\'y conjecture.