Power-law corrections to exponential decay of connectivities and correlations in lattice models

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 <----> 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 a = 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 \) greater than or equal to P(0 <----> x) greater than or equal to 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.