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Abstract
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We review results of two previous papers on the asymptotic behavior of finite
connection probabilities in three or more dimensions for Bernoulli percolation and
the Fortuin–Kasteleyn random-cluster model. In the introduction, we prove a
multidimensional renewal theorem that is needed for these results and previous
results on Ornstein–Zernike behavior; the proof is significantly simpler than that
originally derived by Doney (1966) and those of other subsequent works on this
subject.
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Keywords
random-cluster model, Ornstein–Zernike behavior for
connectivities, renormalization, Ruelle operator, local
limit theorem, invariance principle
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Mathematical Subject Classification 2010
Primary: 60K35, 82B43, 60K15, 60F17
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Milestones
Received: 1 April 2016
Revised: 1 May 2016
Accepted: 31 May 2016
Published: 17 December 2016
Communicated by Raffaele Esposito
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