00-245 K. S. Alexander
Power-law corrections to exponential decay of connectivities and correlations in lattice models (184K, AMS-LATeX 1.2) May 26, 00
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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.

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