Universal features of cluster numbers in percolation

Abstract

The number of clusters per site $n(p)$ 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 $n(p)$ 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 new results, including precise values for $n(p_c)$ for several systems, a clear demonstration of the singularity in $n''(p)$, 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.