K. S. Alexander
Power-law corrections to exponential decay of connectivities and correlations
in lattice models
(184K, AMS-LATeX 1.2)
ABSTRACT. 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 $e^{-m(\theta)|x|}$ 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
$e^{-m(\theta)|x|} \geq P(0 \leftrightarrow x) \geq
A|x|^{-C}e^{-m(\theta)|x|}$
for all nonzero $x$. In four or more dimensions, a similar bound
holds with $|x|^{-C}$ replaced by $e^{-C(\log |x|)^{2}}$. 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.