Electronic Communications in Probability

Symmetric 1-dependent colorings of the integers

Alexander Holroyd and Thomas Liggett

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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 $\sqrt{q}/2$.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 31, 8 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G10: Stationary processes
Secondary: 05C15: Coloring of graphs and hypergraphs 60C05: Combinatorial probability

Random colorings one-dependent processes

This work is licensed under a Creative Commons Attribution 3.0 License.


Holroyd, Alexander; Liggett, Thomas. Symmetric 1-dependent colorings of the integers. Electron. Commun. Probab. 20 (2015), paper no. 31, 8 pp. doi:10.1214/ECP.v20-4070. https://projecteuclid.org/euclid.ecp/1465320958

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