Uniqueness of the Gibbs measure for the $4$-state anti-ferromagnetic Potts model on the regular tree

Abstract

We show that the $4$-state anti-ferromagnetic Potts model with interaction parameter $w\in (0,1)$ on the infinite $(d+1)$-regular tree has a unique Gibbs measure if $w\ge 1-\frac{4}{d+1}$ for all $d\ge 4$. This is tight since it is known that there are multiple Gibbs measures when $0\le w<1-\frac{4}{d+1}$ and $d\ge 4$. We moreover give a new proof of the uniqueness of the Gibbs measure for the $3$-state Potts model on the $(d+1)$-regular tree for $w\ge 1-\frac{3}{d+1}$ when $d\ge 3$ and for $w\in (0,1)$ when $d=2$.

Subsection 1.2 and Section have been merged and slightly rewritten. Proof of Lemma 1.3 has been moved to an appendix, fixed a small mistake in the proof of this lemma. Some small other changes have been made. No significant changes. Accepted in CPC

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• Uniqueness of the Gibbs measure for the anti-ferromagnetic Potts model on the infinite $\Delta$-regular tree for large $\Delta$