 00245 K. S. Alexander
 Powerlaw corrections to exponential decay of connectivities and correlations
in lattice models
(184K, AMSLATeX 1.2)
May 26, 00

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Abstract. Consider a translationinvariant 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 powerlaw correction to the
exponential decaythere exist $A$ and $C$ such that
$e^{m(\theta)x} \geq P(0 \leftrightarrow x) \geq
Ax^{C}e^{m(\theta)x}$
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 powerlaw lower bound holds for the
FortuinKasteleyn 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.
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