Open Access
2015 Symmetric 1-dependent colorings of the integers
Alexander Holroyd, Thomas Liggett
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Electron. Commun. Probab. 20: 1-8 (2015). DOI: 10.1214/ECP.v20-4070


In a recent paper, we constructed a stationary $1-$dependent $4-$coloring of the integers that is invariant under permutations of the colors. This was the first stationary $k-$dependent $q-$coloring for any $k$ and $q$. When the analogous construction is carried out for $q>4$ colors, the resulting process is not $k-$dependent for any $k$. We construct here a process that is symmetric in the colors and $1-$dependent for every $q\geq 4$. The construction uses a recursion involving Chebyshev polynomials evaluated at $\sqrt{q}/2$.


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Alexander Holroyd. Thomas Liggett. "Symmetric 1-dependent colorings of the integers." Electron. Commun. Probab. 20 1 - 8, 2015.


Accepted: 29 March 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 06473030
MathSciNet: MR3327870
Digital Object Identifier: 10.1214/ECP.v20-4070

Primary: 60G10
Secondary: 05C15 , 60C05

Keywords: one-dependent processes , random colorings

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