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

11/11/2020

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by   David de Boer, et al.

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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 w≥ 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 w∈ (0,1) when d=2.