Symmetrization for finitely dependent colouring
Probability
2023-05-24 v1 Combinatorics
Abstract
We prove the existence of a finitely dependent proper colouring of the integer lattice Z^d that is fully isometry-invariant in law, for all dimensions d. Previously this was known only for d=1, while only translation-invariant examples were known for higher d. Moreover we show that four colours suffice, and that the colouring can be expressed as an isometry-equivariant finitary factor of an i.i.d. process, with exponential tail decay on the coding radius. Our construction starts from known translation-invariant colourings and applies a symmetrization technique of possible broader utility.
Cite
@article{arxiv.2305.13980,
title = {Symmetrization for finitely dependent colouring},
author = {Alexander E. Holroyd},
journal= {arXiv preprint arXiv:2305.13980},
year = {2023}
}