Scaling studies of percolation phenomena in systems of dimensionality two to seven: Cluster numbers

Cluster statistics are obtained by computer simulation for percolation processes on $d$-dimensional lattices with $d=2\mathrm{}$ through 7. For all $d$, $n_s$, the number of $s$-site clusters per site, is found to satisfy reasonably well the scaling hypothesis first proposed by Stauffer. The scaling functions are analyzed for dimensional dependence, and it is found that as $d$ increases they approach very rapidly the exactly known result for the Bethe lattice corresponding to $d=\infty$. Corrections to scaling are also studied, and at the upper critical dimension $d_c=6\mathrm{}$, a deviation from scaling consistent with a logarithmic correction is obtained. Some universal quantities such as the ratio of the amplitudes $C_{+}$ and $C_{-}$ of the "susceptibility" (second moment of $n_s$) below and above the percolation threshold $p_c$ are also found to approach the Bethe lattice limit very quickly. In addition, our data suggest that $\frac{C_{+}}{C_{-}}$ already assumes the limiting value of unity for $d=6\mathrm{}$. This is consistent with the exact relation for the asymmetric decay of $n_s$ proved by Kunz and Souillard if we adopt the hypothesis that the asymmetry of $n_s$ about $p_c$ only enters the corrections to the leading scaling term for $d>~6$.