# Invariant matchings of exponential tail on coin flips in $\Z^d$

@article{Timr2009InvariantMO, title={Invariant matchings of exponential tail on coin flips in \$\Z^d\$}, author={{\'A}d{\'a}m Tim{\'a}r}, journal={arXiv: Probability}, year={2009} }

Consider Bernoulli(1/2) percolation on $\Z^d$, and define a perfect matching between open and closed vertices in a way that is a deterministic equivariant function of the configuration. We want to find such matching rules that make the probability that the pair of the origin is at distance greater than $r$ decay as fast as possible. For two dimensions, we give a matching of decay $cr^{1/2}$, which is optimal. For dimension at least 3 we give a matching rule that has an exponential tail. This… Expand

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