Finitely dependent cycle coloring

TL;DR

This paper introduces stationary finitely dependent colorings of cycles, using a necklace insertion process and Eden growth model, providing simpler proofs for their marginal distributions.

Contribution

It constructs new finitely dependent cycle colorings analogous to integer colorings, with novel descriptions and simplified proofs of their properties.

Findings

Constructed stationary finitely dependent cycle colorings

Described colorings via necklace insertion and Eden growth models

Provided simpler proofs for 1- and 2-color marginals

Abstract

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.