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Proper $3$-colorings of $\mathbb {Z}^{2}$ are Bernoulli
Published online by Cambridge University Press: 27 April 2022
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.
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- © The Author(s), 2022. Published by Cambridge University Press