Proper 3-colorings of ${\mathbb{Z}}^2$ are Bernoulli

Abstract

We consider the unique measure of maximal entropy for proper 3-colorings of ${\mathbb{Z}}^2$, or equivalently, the so-called zero-slope Gibbs measure. Our main result is that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translation-equivariant function of independent and identically distributed random variables placed on ${\mathbb{Z}}^2$. Along the way, we obtain various estimates on the mixing properties of this measure.

22 pages, 2 figures