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

We show that the 4 -state anti-ferromagnetic Potts model with interaction parameter w∈(0,1) on the infinite (d+1) -regular tree has a unique Gibbs measure if   ≥1− ^4/_d+1
for all d ≥ 4 . This is tight since it is known that there are multiple Gibbs measures when 0 ≤ w <1− ^4/_d+1 and d ≥ 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 ≥1− ^3/_d+1 when d ≥ 3 and for  ∈ (0,1) when d =2