Uniqueness of the Gibbs Measure for the Anti-ferromagnetic Potts Model on the Infinite $$\Delta $$-Regular Tree for Large $$\Delta $$

Abstract

In this paper we prove that for any integer $$q\ge 5$$ q ≥ 5 , the anti-ferromagnetic q-state Potts model on the infinite $$\Delta $$ Δ -regular tree has a unique Gibbs measure for all edge interaction parameters $$w\in [1-q/\Delta ,1)$$ w ∈ [ 1 - q / Δ , 1 ) , provided $$\Delta $$ Δ is large enough. This confirms a longstanding folklore conjecture.