Entropy-driven phase transition in low-temperature antiferromagnetic Potts models

TL;DR

This paper proves the existence of long-range order at low temperatures in the three-state Potts antiferromagnet on certain plane quadrangulations, demonstrating spontaneous magnetization and providing rigorous bounds supported by computer calculations.

Contribution

It establishes the presence of multiple Gibbs measures and spontaneous magnetization in low-temperature antiferromagnetic Potts models on specific lattices, including a rigorous bound for the diced lattice.

Findings

Existence of at least three infinite-volume Gibbs measures.

Spontaneous magnetization in sublattices at low temperatures.

Rigorous lower bounds on magnetization probabilities for the diced lattice.

Abstract

We prove the existence of long-range order at sufficiently low temperatures, including zero temperature, for the three-state Potts antiferromagnet on a class of quasi-transitive plane quadrangulations, including the diced lattice. More precisely, we show the existence of (at least) three infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one sublattice have a higher probability to be in one state than in either of the other two states. For the special case of the diced lattice, we give a good rigorous lower bound on this probability, based on computer-assisted calculations that are not available for the other lattices.