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 infinite $(d+1)$ -regular tree has a unique Gibbs measure if $w\geq 1-\dfrac{4}{d+1_{_{\;}}}$ for all $d\geq 4$ . This is tight since it known there are multiple measures when $0\leq w\lt 1-\dfrac{4}{d+1}$ and moreover give new proof of uniqueness $3$ 1-\dfrac{3}{d+1}$ 3$ $d=2$