## Abstract

We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When q ≤ 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 ≥ d+2, any proper q-colouring of the boundary of a box of side length n ≥ d + 2 can be extended to a proper q-colouring of the entire box. (3) When q ≥ 2d+1, the latter holds for any n ≥ 1. Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.

Original language | English |
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Pages (from-to) | 360-373 |

Number of pages | 14 |

Journal | Combinatorics Probability and Computing |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - 19 May 2021 |

### Funding

Funders | Funder number |
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National Science Foundation | DMS-1855464 |

Simons Foundation | |

Bonfils-Stanton Foundation | 2018267 |

Horizon 2020 Framework Programme | 678520 |

European Research Council | 1289/17, 1570/17, 1702/17, 676970 |

Comisión Nacional de Investigación Científica y Tecnológica | |

Fondo Nacional de Desarrollo Científico y Tecnológico | 3190191 |

Israel Science Foundation | 281/17 |

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