Resumen de Finitary coloring
Alexander E. Holroyd, Oded Schramm, David B. Wilson
Suppose that the vertices of ZdZd are assigned random colors via a finitary factor of independent identically distributed (i.i.d.) vertex-labels. That is, the color of vertex vv is determined by a rule that examines the labels within a finite (but random and perhaps unbounded) distance RR of vv, and the same rule applies at all vertices. We investigate the tail behavior of RR if the coloring is required to be proper (i.e., if adjacent vertices must receive different colors). When d≥2d≥2, the optimal tail is given by a power law for 33 colors, and a tower (iterated exponential) function for 44 or more colors (and also for 33 or more colors when d=1d=1). If proper coloring is replaced with any shift of finite type in dimension 11, then, apart from trivial cases, tower function behavior also applies.
