05-221 D. H. U. Marchetti, V. Sidoravicius, M. E. Vares
Oriented percolation in one-dimensional beta / |x-y|^2, beta > 1 random-cluster model (508K, postscript) Jun 20, 05
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Abstract. We consider the one-dimensional long-range Fortuin--Kasteleyn random-cluster model, generated by the edge occupation probabilities p_{<x,y>} = p if |x-y| = 1, 1 - exp{-beta |x-y|^2} otherwise, and weighting factor kappa \geq 1. We prove the occurrence of oriented percolation when beta>1 and kappa \geq 1, provided p is chosen sufficiently close to 1. We also show that the oriented truncated connectivity tau ^{prime}(x,y) satisfies tau ^{prime }(x,y) \leq C |x-y|^{-theta } with theta = min(2(beta eta -1),2) where eta = eta(p) \nearrow 1 as p \nearrow 1.

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