## The Annals of Probability

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

### Finitary coloring

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

#### Abstract

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

**Source**

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

**Dates**

Received: June 2015

Revised: May 2016

First available in Project Euclid: 23 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1506132029

**Digital Object Identifier**

doi:10.1214/16-AOP1127

**Mathematical Reviews number (MathSciNet)**

MR3706734

**Zentralblatt MATH identifier**

1385.60048

**Subjects**

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

**Keywords**

Coloring finitary factor tower function shift of finite type

#### Citation

Holroyd, Alexander E.; Schramm, Oded; Wilson, David B. Finitary coloring. Ann. Probab. 45 (2017), no. 5, 2867--2898. doi:10.1214/16-AOP1127. https://projecteuclid.org/euclid.aop/1506132029