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

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 $q\ge 2d+1$, 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.

Minor revisions, 14 pages, 2 figures