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)

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.