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 Zd and is given by a deterministic function of the coin-flips. Let XR 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 XR, for translation-equivariant perfect matching rules. We prove that for every d>1, there exists a translation-equivariant perfect matching rule R such that XR has a finite p-th moment for every 0 < p < 2/3.
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