Universal features of cluster numbers in percolation

The number of clusters per site $n\left(p\right)$ in percolation at the critical point $p=p_c$ is not itself a universal quantity; it depends upon the lattice and percolation type (site or bond). However, many of its properties, including finite-size corrections, scaling behavior with $p$, and amplitude ratios, show various degrees of universal behavior. Some of these are universal in the sense that the behavior depends upon the shape of the system, but not lattice type. Here, we elucidate the various levels of universality for elements of $n\left(p\right)$ both theoretically and by carrying out extensive studies on several two- and three-dimensional systems, by high-order series analysis, Monte Carlo simulation, and exact enumeration. We find many results, including precise values for $n\left(p_c\right)$ for several systems, a clear demonstration of the singularity in $n^{\prime \prime }\left(p\right)$, and metric scale factors. We make use of the matching polynomial of Sykes and Essam to find exact relations between properties for lattices and matching lattices. We propose a criterion for an absolute metric factor $b$ based upon the singular behavior of the scaling function, rather than a relative definition of the metric that has previously been used.

Figure 1

Second derivative of the cluster density $n_L\left(p\right)$ for square lattices of size $L\times L$ for $L=8,16,...,1024$. Error bars are much smaller than the line width. The vertical dashed line marks the percolation threshold $p_c$.

Figure 2

An example of a plot of the MC and exact-enumeration data, used to find coefficients given in Table 1: $n_L^{\prime \prime }\left(p_c\right)=2C$ for site percolation on square lattices vs $L^{-1/2}$. The line is a fit of (9c), which yields values for $C_0$ and $C_1$. Error bars of the MC data are much smaller than the size of the symbols.

Figure 3

$M_L\left(p\right)=L^2[n^{\mathrm{SQ}}\left(p\right)-n^{\mathrm{NNSQ}}(1-p)-\varphi \left(p\right)]$ vs $p$ from exact enumeration results for $L=3,4,5,6,7$ (solid lines) and $L=8,12,16,24,32,48$ from MC (dashed lines), plotted as a function of $p$, and (inset) as a function of the scaling variable $(p-p_c)L^{1/\nu }$, yielding $f\left(z\right)-f(-z)$. In the limit $L\to \infty , M_L\left(p\right)$ becomes a step function.