Consider a translation-invariant bond percolation model on the
integer lattice which has exponential decay of connectivities, that is, the
probability of a connection $0 \leftrightarrow x$ by a path of open bonds
decreases like $\exp\{-m(\theta)|x|\}$ for some positive constant $m(\theta)$
which may depend on the direction $\theta = x/|x|$. In two and three
dimensions, it is shown that if the model has an appropriate mixing property
and satisfies a special case of the FKG property, then there is at most a
power-law correction to the exponential decay—there exist $A$ and $C$
such that $\exp\{-m(\theta)|x|\} \ge P(0 \leftrightarrow x) \ge A|x|^{-C}
\exp\{-m(\theta)|x|\}$ for all nonzero $x$ . In four or more dimensions, a
similar bound holds with $|x|^{-C}$ replaced by $\exp\{-C(\log |x|)^2\}$. In
particular the power-law lower bound holds for the Fortuin-Kasteleyn random
cluster model in two dimensions whenever the connectivity decays exponentially,
since the mixing property is known to hold in that case. Consequently a similar
bound holds for correlations in the Potts model at supercritical
temperatures.