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

TL;DR

This paper proves that the unique measure of maximal entropy for proper 3-colorings of the 2D integer lattice is Bernoulli, meaning it can be generated from i.i.d. variables through a translation-equivariant function.

Contribution

It establishes the Bernoulli property for the zero-slope Gibbs measure of proper 3-colorings on 2, a significant advancement in understanding their probabilistic structure.

Findings

The measure is Bernoulli, expressible as a function of i.i.d. variables.

Provides estimates on the mixing properties of the measure.

Confirms the measure's maximal entropy and Bernoulli nature.

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.