We review results of two previous papers on the asymptotic behavior of finiteconnection probabilities in three or more dimensions for Bernoulli percolationand the Fortuin–Kasteleyn random-cluster model. In the introduction, we provea multidimensional renewal theorem that is needed for these results and previousresults on Ornstein–Zernike behavior; the proof is significantly simpler than thatoriginally derived by Doney (1966) and those of other subsequent works on thissubject.

SOME RESULTS ON THE ASYMPTOTIC BEHAVIOR OF FINITE CONNECTION PROBABILITIES IN PERCOLATION

GIANFELICE, Michele
2016

Abstract

We review results of two previous papers on the asymptotic behavior of finiteconnection probabilities in three or more dimensions for Bernoulli percolationand the Fortuin–Kasteleyn random-cluster model. In the introduction, we provea multidimensional renewal theorem that is needed for these results and previousresults on Ornstein–Zernike behavior; the proof is significantly simpler than thatoriginally derived by Doney (1966) and those of other subsequent works on thissubject.
Random Cluster model; Ornstein-Zernike behaviour for connectivities; Local limit theorem; Renormalization; Ruelle operator; Invariance principle
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11770/144060
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