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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 Z Φ be 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 of Z Φ for translation-equivariant perfect matching rules? We prove that, for every d ≥ 2, there exists a translation-equivariant perfect matching rule Φ such that EZ Φ 2/3-ε < ∞ for every ε > 0.
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