Matchings of exponential tail on coin flips in Z^d

We construct a translation-invariant matching between vertices of different labels of Bernoulli(1/2) percolation on Z^d, in such a way that the probability that a vertex is at distance > r from its pair decays as an exponential function of the d-2'nd power of r. Our method implies earlier results about matching n uniformly distributed red points with n uniformly distributed blue points in the unit square with minimal average distance. Other consequences to invariant matchings in R^d and optimal matchings in the unit cube are also presented.