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Translation-Equivariant Matchings of Coin Flips on ℤd

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Terry Soo
Terry Soo*
Affiliation:
University of British Columbia
*
∗ Postal address: Department of Mathematics, University of British Columbia, Vancouver BC, V6T 1Z2, Canada. Email address: tsoo@math.ubc.ca
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Abstract

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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.

Keywords

MatchingBernoulli random field

MSC classification

Primary: 60G55: Point processes 60G60: Random fields
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 1 , March 2010 , pp. 69 - 82
Copyright
Copyright © Applied Probability Trust 2010 

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