Mixing properties of colourings of the ℤ d lattice

We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When , 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 , any proper q-colouring of the boundary of a box of side length can be extended to a proper q-colouring of the entire box. (3) When , the latter holds for any . Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.