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1 results

  • Translation-Equivariant Matchings of Coin Flips on ℤd

  • Part of
  • Terry Soo
  • Journal:
    Advances in Applied Probability / Volume 42 / Issue 1 / March 2010
    Published online by Cambridge University Press:
    01 July 2016, pp. 69-82
    Print publication:
    March 2010
    • Article
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  • 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 ZΦ 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 ZΦ for translation-equivariant perfect matching rules? We prove that, for every d ≥ 2, there exists a translation-equivariant perfect matching rule Φ such that EZΦ2/3-ε < ∞ for every ε > 0.