Symmetrization for finitely dependent colouring

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.