Journal of the European Mathematical Society


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Volume 21, Issue 5, 2019, pp. 1509–1570
DOI: 10.4171/JEMS/866

Published online: 2019-02-01

Long-range order in the 3-state antiferromagnetic Potts model in high dimensions

Ohad N. Feldheim[1] and Yinon Spinka[2]

(1) The Hebrew University of Jerusalem, Israel
(2) The University of British Columbia, Vancouver, Canada

We prove the existence of long-range order for the 3-state Potts antiferromagnet at low temperature on $\mathbb Z^d$ for sufficiently large $d$. 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 Koteck√Ĺ conjecture.

Keywords: Potts model, long-range order, phase transition, rigidity

Feldheim Ohad, Spinka Yinon: Long-range order in the 3-state antiferromagnetic Potts model in high dimensions. J. Eur. Math. Soc. 21 (2019), 1509-1570. doi: 10.4171/JEMS/866