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Translation-Equivariant Matchings of Coin Flips on ℤd

Published online by Cambridge University Press:  01 July 2016

Terry Soo*
University of British Columbia
Postal address: Department of Mathematics, University of British Columbia, Vancouver BC, V6T 1Z2, Canada. Email address:
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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.

Stochastic Geometry and Statistical Applications
Copyright © Applied Probability Trust 2010 


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