94-16 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) Jan 21, 94

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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 $p\nearrow 1$.

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