Finitary Coloring

Suppose that the vertices of 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≥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.