Mixing properties of colourings of the ℤdlattice

Noga Alon, Raimundo Briceño, Nishant Chandgotia, Alexander Magazinov, Yinon Spinka

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)360-373
Number of pages14
JournalCombinatorics Probability and Computing
Issue number3
StatePublished - 19 May 2021


FundersFunder number
National Science FoundationDMS-1855464
Simons Foundation
Bonfils-Stanton Foundation2018267
Horizon 2020 Framework Programme678520
European Research Council1289/17, 1570/17, 1702/17, 676970
Comisión Nacional de Investigación Científica y Tecnológica
Fondo Nacional de Desarrollo Científico y Tecnológico3190191
Israel Science Foundation281/17


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