We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When $q\le d+1$ , there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when $q\ge d+2$ , any proper q-colouring of the boundary of a box of side length $n \ge d+2$ can be extended to a proper q-colouring 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-colourings of the ℤd lattice by their mixing properties.

中文翻译：

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ℤd 晶格的着色混合性质

我们研究和分类适当q- ℤ 的着色dlattice，确定了三种不同组合行为的状态。(1) 当$q\le d+1$, 存在冻结着色, 即适当q-ℤ的着色d不能在任何有限子集上修改。(2) 我们证明了一个强大的列表着色性质，这意味着，当$q\ge d+2$, 任何适当的q-边长盒子边界的着色$n \ge d+2$可以扩展到适当的q- 整个盒子的着色。(3) 当$q\geq 2d+1$, 后者适用于任何$n \ge 1$. 因此，我们对适当的空间进行分类q- ℤ 的着色d晶格的混合特性。