The Annals of Probability

Finitary coloring

Alexander E. Holroyd, Oded Schramm, and David B. Wilson

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Suppose that the vertices of $\mathbb{Z}^{d}$ are assigned random colors via a finitary factor of independent identically distributed (i.i.d.) vertex-labels. That is, the color of vertex $v$ is determined by a rule that examines the labels within a finite (but random and perhaps unbounded) distance $R$ of $v$, and the same rule applies at all vertices. We investigate the tail behavior of $R$ if the coloring is required to be proper (i.e., if adjacent vertices must receive different colors). When $d\geq2$, the optimal tail is given by a power law for $3$ colors, and a tower (iterated exponential) function for $4$ or more colors (and also for $3$ or more colors when $d=1$). If proper coloring is replaced with any shift of finite type in dimension $1$, then, apart from trivial cases, tower function behavior also applies.

Article information

Ann. Probab., Volume 45, Number 5 (2017), 2867-2898.

Received: June 2015
Revised: May 2016
First available in Project Euclid: 23 September 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G10: Stationary processes 05C15: Coloring of graphs and hypergraphs 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]

Coloring finitary factor tower function shift of finite type


Holroyd, Alexander E.; Schramm, Oded; Wilson, David B. Finitary coloring. Ann. Probab. 45 (2017), no. 5, 2867--2898. doi:10.1214/16-AOP1127.

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