Proper 3-colorings of Z(2) are Bernoulli

We consider the unique measure of maximal entropy for proper 3-colorings of 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 Z(2). Along the way, we obtain various estimates on the mixing properties of this measure.