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September 2017 Finitary coloring
Alexander E. Holroyd, Oded Schramm, David B. Wilson
Ann. Probab. 45(5): 2867-2898 (September 2017). DOI: 10.1214/16-AOP1127


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.


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Alexander E. Holroyd. Oded Schramm. David B. Wilson. "Finitary coloring." Ann. Probab. 45 (5) 2867 - 2898, September 2017.


Received: 1 June 2015; Revised: 1 May 2016; Published: September 2017
First available in Project Euclid: 23 September 2017

zbMATH: 1385.60048
MathSciNet: MR3706734
Digital Object Identifier: 10.1214/16-AOP1127

Primary: 05C15 , 37A50 , 60G10

Keywords: coloring , Finitary factor , shift of finite type , tower function

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 5 • September 2017
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