Proper -colorings of 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.

A central limit theorem for square ice

“…We begin with introducing some notations, following closely with that of [7,19]. Given an even domain D ⊆ Z 2 , we denote by P 0 D (resp.…”

“…We recall the following estimate for the number of loops in an annuli, stated in [19] (see also [7]). Proposition 2.6.…”

“…Lemma 3.2 follows from a monotone coupling of the uniform square ice height functions introduced in [19], and an application of the ballot theorem for random walks, stated below. Proposition 3.3.…”

“…Let us also mention that the uniform square-ice model has an explicit bijection to the uniform proper 3-colorings on the dual graph. For proper 3-colorings in Z 2 , an algebraic rate of mixing and Bernoullicity of the measure, which is suggested by the logarithmic fluctuation of the square-ice, was proved in [19].…”

“…The proof of Theorem 1.1 and 1.3 is based on an application of the martingale central limit theorem. The key inputs include a Russo-Seymour-Welsh estimate for the level line of the height function (established in [7]), the FKG inequality for the absolute value height function and the monotone coupling (used in [7] and [19]), and random walk estimates. We will summarize these tools in the next section, and the proof of Theorem 1.1 and 1.3 will be given in Section 3 and 4.…”