Translation-Equivariant Matchings of Coin-Flips on Z^d

Consider independent fair coin-flips at each site of the lattice Z^d. A translation-equivariant matching rule is a perfect matching of heads to tails that commutes with translations of Z^d and is given by a deterministic function of the coin-flips. Let X_R be the distance from the origin to its partner, under the translation-equivariant matching rule R. Holroyd and Peres asked what is optimal tail behaviour of X_R, for translation-equivariant perfect matching rules. We prove that for every d>1, there exists a translation-equivariant perfect matching rule R such that X_R has a finite p-th moment for every 0 < p < 2/3.