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Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Nov;70(5 Pt 2):056116. Epub 2004 Nov 18.

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

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  • 1Fakultät für Physik, Universität Bielefeld, D-33501 Bielefeld, Germany.


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 ln L at d=6 . The predictions for d>6 depend on the boundary conditions, and the results there may vary between L(d-6) and L0. While simulations in six dimensions are consistent with this prediction [after including corrections of order ln(ln L)], in five dimensions the average number of spanning clusters still increases as ln L 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.

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