Translation-Equivariant Matchings of Coin Flips on ℤd

Abstract

Consider independent fair coin flips at each site of the lattice ℤd. A translation-equivariant matching rule is a perfect matching of heads to tails that commutes with translations of ℤdand is given by a deterministic function of the coin flips. LetZΦbe the distance from the origin to its partner, under the translation-equivariant matching rule Φ. Holroyd and Peres (2005) asked, what is the optimal tail behaviour ofZΦfor translation-equivariant perfect matching rules? We prove that, for everyd≥ 2, there exists a translation-equivariant perfect matching rule Φ such that EZΦ2/3-ε< ∞ for every ε > 0.