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arXiv · 1903.11685

Mixing properties of colorings of the $\mathbb{Z}^d$ lattice

Noga Alon·Raimundo Briceño·Nishant Chandgotia·Alexander Magazinov·Yinon Spinka
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Abstract

We study and classify proper $q$-colorings of the $\mathbb Z^d$ lattice, identifying three regimes where different combinatorial behavior holds: (1) When $q\le d+1$, there exist frozen colorings, that is, proper $q$-colorings of $\mathbb Z^d$ which cannot be modified on any finite subset. (2) We prove a strong list-coloring property which implies that, when $q\ge d+2$, any proper $q$-coloring of the boundary of a box of side length $n \ge d+2$ can be extended to a proper $q$-coloring of the entire box. (3) When $q\geq 2d+1$, the latter holds for any $n \ge 1$. Consequently, we classify the space of proper $q$-colorings of the $\mathbb Z^d$ lattice by their mixing properties.

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Noga Alon, Raimundo Briceño, Nishant Chandgotia, Alexander Magazinov, Yinon Spinka. 2019-03-27. Mixing properties of colorings of the $\mathbb{Z}^d$ lattice. https://arxiv.org/abs/1903.11685

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