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We prove the existence of long-range order for the 3-state Potts antiferromagnet at low temperature on for sufficiently large . In particular, we show the existence of six extremal and ergodic infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one bipartition class have a much higher probability to be in one state than in either of the other two states. This settles the high-dimensional case of the Kotecký conjecture.
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Ohad N. Feldheim, Yinon Spinka, Long-range order in the 3-state antiferromagnetic Potts model in high dimensions. J. Eur. Math. Soc. 21 (2019), no. 5, pp. 1509–1570