TY - JOUR

T1 - Number of spanning clusters at the high-dimensional percolation thresholds

AU - Fortunate, Santo

AU - Aharony, Amnon

AU - Coniglio, Antonio

AU - Stauffer, Dietrich

N1 - Funding Information:
We acknowledge support from the German-Israeli Foundation and from the Humboldt Foundation, P. Grassberger and J. Adler for helpful discussion, and the computer centers of the Technion (Israel, M. Goldberg) and Julich (Germany) for time on their large computers. S.F. acknowledges the support of the DFG Forschergruppe under Grant No. FOR 339/2-1. A.C. acknowledges support from MIUR-PRIN and FIRB 2002, CRdC-AMRA, and EU Network MRTN-CT-2003-504712.

PY - 2004/11

Y1 - 2004/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=45849155423&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.70.056116

DO - 10.1103/PhysRevE.70.056116

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AN - SCOPUS:45849155423

SN - 1539-3755

VL - 70

SP - 056116-1-056116-7

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

IS - 5

M1 - 056116

ER -