Finitary Coloring

TL;DR

This paper studies the probabilistic rules for coloring infinite grid vertices with finite colors, revealing different tail behaviors depending on the number of colors and dimension, including power law and tower functions.

Contribution

It characterizes the tail behavior of the radius in finitary colorings of  lattices, establishing optimal bounds for proper colorings with various numbers of colors.

Findings

Power law tail for 3 colors in  

Tower function tail for 4 or more colors in  

Similar tail behaviors extend to shift of finite type in 1D

Abstract

Suppose that the vertices of ${\mathbb Z}^d$ are assigned random colors via a finitary factor of independent identically distributed (iid) 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 (that is, if adjacent vertices must receive different colors). When $d\geq 2$, 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.