Vol. 4, No. 3-4, 2016

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Some results on the asymptotic behavior of finite connection probabilities in percolation

Massimo Campanino and Michele Gianfelice

Vol. 4 (2016), No. 3-4, 311–325
Abstract

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.

Keywords
random-cluster model, Ornstein–Zernike behavior for connectivities, renormalization, Ruelle operator, local limit theorem, invariance principle
Mathematical Subject Classification 2010
Primary: 60K35, 82B43, 60K15, 60F17
Milestones
Received: 1 April 2016
Revised: 1 May 2016
Accepted: 31 May 2016
Published: 17 December 2016

Communicated by Raffaele Esposito
Authors
Massimo Campanino
Dipartimento di Matematica
Università degli Studi di Bologna
I-40127 Bologna
Italy
Michele Gianfelice
Dipartimento di Matematica e Informatica
Università della Calabria
Campus di Arcavacata
I-87036 Arcavacata di Rende
Italy