Finitary coloring

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.