Number of spanning clusters at the high-dimensional percolation thresholds

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.