We construct stationary finitely dependent colorings of the cycle which are analogous to the colorings of the integers recently constructed by Holroyd and Liggett. These colorings can be described by a simple necklace insertion procedure, and also in terms of an Eden growth model on a tree. Using these descriptions we obtain simpler and more direct proofs of the characterizations of the 1- and 2-color marginals.
"Finitely dependent cycle coloring." Electron. Commun. Probab. 23 1 - 12, 2018. https://doi.org/10.1214/18-ECP118