  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 substantially improves previous bounds. The
construction has two major parts: first we define a sequence of coarser and
coarser partitions of $\Z^d$ in an equivariant way, such that with high
probability the cell of a fixed point is like a cube, and the labels in it are
i.i.d. Then we define a matching for a fixed finite cell, which stabilizes as
we repeatedly apply it for the cells of the consecutive partitions. Our methods
also work in the case when one wants to match points of two Poisson processes,
and they may be applied to allocation questions.
