Number of spanning clusters at the high-dimensional percolation thresholds

Santo Fortunate*, Amnon Aharony, Antonio Coniglio, Dietrich Stauffer

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


A scaling theory is used to derive the dependence of the average number ⟨k⟩ of spanning clusters at threshold on the lattice size L. This number should become independent of L for dimensions d<6 and vary as lnL at d=6. The predictions for d>6 depend on the boundary conditions, and the results there may vary between Ld−6 and L0. While simulations in six dimensions are consistent with this prediction [after including corrections of order ln(lnL)], in five dimensions the average number of spanning clusters still increases as lnL even up to L=201. However, the histogram P(k) of the spanning cluster multiplicity does scale as a function of kX(L), with X(L)=1+const∕L, indicating that for sufficiently large L the average ⟨k⟩ will approach a finite value: a fit of the five-dimensional multiplicity data with a constant plus a simple linear correction to scaling reproduces the data very well. Numerical simulations for d>6 and for d=4 are also presented.

Original languageEnglish
Article number056116
Pages (from-to)056116-1-056116-7
Number of pages7
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Issue number5
StatePublished - Nov 2004


FundersFunder number
Alexander von Humboldt-Stiftung
European CommissionMRTN-CT-2003-504712
Deutsche ForschungsgemeinschaftFOR 339/2-1
German-Israeli Foundation for Scientific Research and Development
Technion-Israel Institute of Technology


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