Marchetti D.H.U.
UPPER BOUND ON THE TRUNCATED CONNECTIVITY 
IN ONE-DIMENIONAL BETA /|x-y|^2 PERCOLATION MODELS 
AT BETA LARGER THAN 1
(59K, LaTeX)

ABSTRACT.  We consider one-dimensional Fortuin-Kasteleyn percolation models generated
by the bond occupation probabilities 
$$
p_{(xy)}={\cases{p & if $|x-y|=1$ \cr
1-e^{-\beta / |x-y|^2} & otherwise \cr}} 
$$
and a real parameter $\kappa $. We prove that for any $\beta >1$ and $\kappa
\geq 1$ the percolation density $M$ is strictly positive provided  $p$ is
sufficiently close to 1. We also prove, under the same assumptions, that the
following upper bound for the truncated connectivity
$$
\tau ^{\prime }(x,y)\leq C|x-y|^{-\overline{\theta }} 
$$
holds with $\overline{\theta }=\min (2(\beta \eta -1),2)$ where $\eta 
=\eta (p)\nearrow 1$ as $p\nearrow 1$.